THE THEORY OF SINGULAR PERTURBATIONS
E.M. DE JAGER, emeritus Department of Mathematics, Computer Science, Physics and Astronomy University ofAmsterdam The Netherlands JIANG
FURU
Shanghai Institute of Applied Mathematics andMechanics Shanghai University People's Republic of China
1996 ELSEVIER AMSTERDAM
9L A U S A N N E
9N E W Y O R K ~ O X F O R D ~ S H A N N O N
~TOKYO
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ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 EO. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0-444-82170-8
9 1996 ELSEVIER SCIENCE B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B. V., Copyright& Permissions Department, P.O. Box 521, lO00 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and~or damage to persons orproperty as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.
This book is printed on acid-free paper. PRINTED IN THE NETHERLANDS
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To o u r wives
Carien and Yongzhen
Tai
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PREFACE Much scientific endeavour is aimed at the relation between causes and their effects. This becomes the more intriguing whenever the cause is small and the effect large. The study of this relation in the field of the theory of perturbations in mathematical or physical systems has already a respectable history, which can be retraced to the time of Lindstedt, Poincar6 and Prandtl about a century ago. Despite this long history the subject is still in state of a vigorous development and it is known as the theory of singular perturbations, where the meaning of a "small" perturbation causing a "large" impact is to be made explicitly clear. This book is about singular perturbation problems, depending on a small parameter such that the solutions show a nonuniform behaviour as the parameter tends to zero. Because of a very large variety of succesful applications of perturbation methods in the physical and engineering sciences and the recognition of the subject in pure and applied mathematics there exists a vast amount of literature on singular perturbations among which several treatises and textbooks. However, it is not well possible to present in a single volume a comprehensive survey including the latest developments. Instead of this we give here an introductory selfcontained text that acquaints the reader not only with topics well treated in other books but also with topics which to our knowledge have not been recorded up till now in already existing textbooks; the latter subjects have been chosen according to our experience and interest. We consider in the first half of the book singular perturbation problems of so-called cumulative type, where the influence of the small perturbation is observable only after a long time interval. The methods of averaging and multiple scales are treated in detail together with several applications from the theory of nonlinear oscillations. The second half is devoted to singular perturbations of boundary layer type, where the nonuniformity is observable only in a small neighbourhood of the boundary or parts of the boundary; as well ordinary as partial differential equations of elliptic and hyperbolic type, either linear or quasilinear have been considered. For a much more detailed description of the contents of this textbook the reader is referred to the table of contents. In the treatment of all these subjects the emphasis lies primarily on rigorous mathematical proofs for the justification of the perturbation methods to obtain valid approximations of the solutions of perturbation problems. The theory of singular perturbations is from the mathematical point of view a very interesting subject because it is possible to apply with success the results of the more abstract theory of differential equations, in particular one needs a priori estimates of solutions which may be obtained by Gronwall's lemma, maximum principles, energy
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integrals, fixed point theorems or Gs inequality. We hope that we meet in this way not only students and researchers who are mainly interested in the mathematical aspects of the theory, but also physicists and engineers who want to understand the mathematical justification of their clever and well devised formal approximations of solutions of practical perturbation problems. This book is the result of lectures given by both authors at their home universities and by the first author E.M. de Jager at the University of Shanghai, already in 1986. In fact it was in consequence of the latter occasion and with the support of the President of this University, dr Chien Weizang, that a contract has been signed between the University of Amsterdam and the Shanghai University of Technology concerning the cooperation of the authors. We are both indebted to the Royal Netherlands Academy of Arts and Sciences (KNAW) and the National Natural Science Foundation of China (NSFC) for the financial support needed for the expenses of travelling and lodging. Also both universities are acknowledged for providing their hospitality. Further we are very much indebted to the department of mathematics of the University of Amsterdam for putting at our disposal the help of a secretary in the person of mrs Y. Voorn; she prepared the whole typescript with great dedication and without complaints; therefore she deserves here a special word of thanks. The second author Jiang Furu is very grateful to professor H. van der Tweel and his wife for their very generous hospitality during his two visits to Amsterdam, in particular during his second stay of about half a year in 1989-1990, in which also his wife enjoyed this token of hospitality. Finally, it is our duty to acknowledge the pleasant cooperation with the publication manager D.J.N. van der Hoop and the technical editor E. Oosterwijk of the NorthHolland division of Elsevier Science, which has resulted in the publication of this book in the North-Holland Series in Applied Mathematics and Mechanics. We are aware that this text could not have been written without the use of many sources from the literature and so we are not in the least very much indebted to many colleagues as well in our own countries as abroad.
E.M. de Jager, Amsterdam Jiang Furu, Shanghai
CONTENTS Preface ........................................................................... vii C h a p t e r 1 General I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Asymptotic Expansions ............................................... 9 1 Order Symbols ...................................................... 9 2 G a u g e F u n c t i o n s a n d A s y m p t o t i c Sequences . . . . . . . . . . . . . . . . . . . . . . . . 12 3 A s y m p t o t i c Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 C o n v e r g e n c e versus A s y m p t o t i c C o n v e r g e n c e . . . . . . . . . . . . . . . . . . . . . . . 16 5 E l e m e n t a r y O p e r a t i o n s on A s y m p t o t i c E x p a n s i o n s . . . . . . . . . . . . . . . . . 18 6 O t h e r T y p e s of E s t i m a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 Generalized Asymptotic Expansions ................................ 21 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 3 Regular Perturbations ............................................... 27 1 Regular Perturbations ............................................. 27 2 A N o n l i n e a r I n i t i a l Value P r o b l e m C o n t a i n i n g a S m a l l P a r a m e t e r . . . 28 3 Applications ....................................................... 37 3.1 Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 T h e M o t i o n of a P l a n e t a r o u n d t h e S u n . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 C h a p t e r 4 The M e t h o d of the Strained Coordinate ............................. 43 1 Introduction ....................................................... 43 2 A p p l i c a t i o n s of t h e M e t h o d of t h e S t r a i n e d C o o r d i n a t e . . . . . . . . . . . . . 44 2.1 T h e N o n l i n e a r Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 T h e M e t h o d of t h e S t r a i n e d P a r a m e t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Lighthill's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5 Temple's Method .................................................. 55 6 L i m i t a t i o n s of t h e L i n d s t e d t - P o i n c a r ~ M e t h o d . . . . . . . . . . . . . . . . . . . . . . 57 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 5 The Method of Averaging ........................................... 61 1 Introduction ....................................................... 61 2 The Krilov-Bogoliubov-Mitropolski Theorem ....................... 63 2.1 I n t r o d u c t i o n to F i r s t O r d e r A v e r a g i n g . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2 G e n e r a l i z a t i o n of T h e o r e m 2; K . B . M . T h e o r e m - S e c o n d V a r i a n t .. 66 2.3 T h e K r i l o v - B o g o l i u b o v - M i t r o p o l s k i T h e o r e m for N o n p e r i o d i c Fields; K.B.M. Theorem-Third Variant ................................ 70 3 W e a k l y N o n l i n e a r Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1 T h e G e n e r a l Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 T h e Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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3.3 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4 T h e L i n e a r O s c i l l a t o r w i t h S m a l l D a m p i n g . . . . . . . . . . . . . . . . . . . . . 78 3.5 T h e Free van der Pol E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4 W e a k l y Forced N o n l i n e a r Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 T h e Case w i t h o u t D a m p i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 T h e Case w i t h D a m p i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5 A Linear Oscillator with Increasing Damping . . . . . . . . . . . . . . . . . . . . . . . 87 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 C h a p t e r 6 The M e t h o d o f Multiple Scales ...................................... 91 1 Introduction ....................................................... 91 2 W e a k l y N o n l i n e a r Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.1 T h e Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.2 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3 The Linear Oscillator w i t h Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 T h e E q u a t i o n of M a t h i e u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 F l o q u e t ' s T h e o r y for L i n e a r E q u a t i o n s w i t h P e r i o d i c Coefficients 102 4.3 A p p l i c a t i o n to Hill's E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4 A p p l i c a t i o n to M a t h i e u ' s E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 T h e T r a n s i t i o n C u r v e s for t h e M a t h i e u E q u a t i o n . . . . . . . . . . . . . . 106 4.6 T h e A p p r o x i m a t i o n of t h e S o l u t i o n O u t s i d e t h e T r a n s i t i o n C u r v e s 110 5 T h e G e n e r a l Case a n d t h e E r r o r E s t i m a t e . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 E s t i m a t e of t h e R e m a i n d e r T e r m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6 A v e r a g i n g a n d M u l t i p l e Scales for P e r t u r b e d W a v e E q u a t i o n s . . . . . 123 6.1 T h e A p p r o x i m a t i o n by C h i k w e n d u a n d K e v o r k i a n . . . . . . . . . . . . . 123 6.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2.1 W a v e E q u a t i o n w i t h L i n e a r D a m p i n g . . . . . . . . . . . . . . . . . . . 126 6.2.2 W a v e E q u a t i o n w i t h C u b i c D a m p i n g . . . . . . . . . . . . . . . . . . . . 127 6.3 J u s t i f i c a t i o n of t h e C h i k w e n d u - K e v o r k i a n P r o c e d u r e . . . . . . . . . . . 130 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 C h a p t e r 7 Singular Perturbations o f Linear Ordinary Differential Equations . . . . 137 1 T h e initial Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.3 T h e A P r i o r i E s t i m a t e of t h e Solution of a S i n g u l a r l y P e r t u r b e d O r d i n a r y Differential E q u a t i o n w i t h G i v en I n i t i a l D a t a . . . . . . . . 140 1.4 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . 142 2 T h e B o u n d a r y Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2 T h e M a x i m u m P r i n c i p l e for O r d i n a r y Differential O p e r a t o r s . . . 145 2.3 A n A P r i o r i E s t i m a t e of t h e S o l u t i o n of t h e B o u n d a r y Value Problem ....................................................... 146 2.4 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.5 T h e A P r i o r i E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s 151 3 B o u n d a r y Value P r o b l e m s w i t h T u r n i n g P o i n t s . . . . . . . . . . . . . . . . . . . . 158 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
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3.2 T h e T u r n i n g P o i n t P r o b l e m w i t h f ' ( x ) < 0 . . . . . . . . . . . . . . . . . . . 158 3.3 T h e A s y m p t o t i c A p p r o x i m a t i o n a r o u n d t h e T u r n i n g P o i n t a n d t h e Case / 3 r m=0,1,2 ..................................... 161 3.4 T h e A s y m p t o t i c A p p r o x i m a t i o n in t h e Case of R e s o n a n c e . . . . . 164 3.5 T h e T u r n i n g P o i n t P r o b l e m w i t h .f'(x) > 0 . . . . . . . . . . . . . . . . . . . 168 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 C h a p t e r 8 Singular Perturbations of Second O r d e r Elliptic Type. Linear Theory 175 1 Introduction ...................................................... 175 2 T h e M a x i m u m P r i n c i p l e for Elliptic O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . 177 3 The Formal Approximation ....................................... 179 4 E s t i m a t i o n of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . . . 185 5 Domains with Characteristic Boundaries .......................... 191 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.2 T h e S i n g u l a r P e r t u r b a t i o n P r o b l e m in a R e c t a n g l e . . . . . . . . . . . . 194 6 Elliptic B o u n d a r y Value P r o b l e m s w i t h T u r n i n g P o i n t s . . . . . . . . . . . . 200 6.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2 E x a m p l e s of T u r n i n g P o i n t P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.2.1 C u r v e s of T u r n i n g P o i n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.2.2 I s o l a t e d T u r n i n g Points; N o d e s . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.3 A Saddle T u r n i n g P o i n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 C h a p t e r 9 Singular Perturbations of Second Order Hyperbolic Type.
Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4
Introduction ...................................................... Characteristics and Subcharacteristics ............................. The Formal Approximation ....................................... A P r i o r i E s t i m a t e s of Solutions of I n i t i a l Value P r o b l e m s for P a r t i a l Differential E q u a t i o n s w i t h a S i n g u l a r P e r t u r b a t i o n of Hyperbolic Type .................................................. 5 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
209 209 210 213
215 223 227
C h a p t e r 10 Singular Perturbations in Nonlinear Initial Value Problems of
Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction ...................................................... 2 A Fixed Point Theorem ........................................... 3 T h e Q u a s i l i n e a r I n i t i a l Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 4 A G e n e r a l N o n l i n e a r I n i t i a l Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 5 Q u a s i l i n e a r I n i t i a l Value P r o b l e m s w i t h a S i n g u l a r P e r t u r b a t i o n of Second O r d e r H y p e r b o l i c T y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . .
229 229 230 232 232 232 235 239 239 240 244 250 250 250 253
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Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a p t e r 11 Singular Perturbations in Nonlinear Boundary Value Problems of
Second Order
.....................................................
1 Introduction ...................................................... 2 B o u n d a r y Value P r o b l e m s for Q u a s i l i n e a r O r d i n a r y Differential Equations ........................................................ 2.1 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 3 T r a n s i t i o n Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Autonomous Conservative Equations .............................. 5 A M o r e G e n e r a l Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 B o u n d a r y Value P r o b l e m s for Q u a s i l i n e a r P a r t i a l Differential E q u a t i o n s of Elliptic T y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 T h e N o n l i n e a r G e n e r a l i z a t i o n of t h e M a x i m u m P r i n c i p l e . . . . . . . 6.3 Elliptic E q u a t i o n s w i t h o u t F i r s t D e r i v a t i v es . . . . . . . . . . . . . . . . . . 6.4 Elliptic E q u a t i o n s w i t h F i r s t D e r i v a ti v es . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a p t e r 12 Perturbations of Higher O r d e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction ...................................................... 2 P e r t u r b a t i o n s of H i g h e r O r d e r in O r d i n a r y Differential E q u a t i o n s .. 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 T h e F o r m a l A p p r o x i m t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Elliptic P e r t u r b a t i o n s of Elliptic E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Elliptic P a r t i a l Differential E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 S o b o l e v Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Elliptic O p e r a t o r s , Bilinear F o r m s a n d G s Inequality 3.2.3 G e n e r a l i z e d Dirichlet P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 E x i s t e n c e a n d G e n e r a l i z e d Solutions . . . . . . . . . . . . . . . . . . . . . 4 Elliptic S i n g u l a r P e r t u r b a t i o n s of H i g h e r O r d e r . . . . . . . . . . . . . . . . . . . 4.1 T h e B o u n d a r y Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 E x i s t e n c e a n d A P r i o r i E s t i m a t e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 T h e a p p r o x i m a t i o n of t h e S o l u t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 T h e E s t i m a t e of t h e R e m a i n d e r a n d F i n a l R e s u l t s . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index ..................................................................
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Chapter 1 GENERAL
INTRO D U CTIO N
The theory of perturbations, in particular of singular perturbations, has a memorable history. As so many branches of mathematics it has its roots in remarkable phenomena in physics. These phenomena are characterized by transitions in the observables which are due to a small parameter in the m a t h e m a t i c a l model. Let us write this model for the moment symbolically as an equation
Ps[ue] = 0 ,
(1.1)
where ue is the relevant physical quantity and e the small parameter. Physicists developed an approach for calculating ue in the form of an expansion into powers of e and this expansion is continuous for e > O, but it may well be discontinuous for e = 0. This is related to the circumstance t h a t the so-called reduced problem P0[u0] = 0
(1.2)
is in general of another type as the problem (1.1) and so it is not a priori sure whether the solution of (1.2) is a reasonable approximation of ur even for e very small. Problems of this kind are related to the well-known question:
"Is the limit of the solution equal to the solution of the limit?' or phrased in the case of differential operators: "Is the limit of the integral equal to the integral of the limit?' The subject of this textbook is the study of perturbation problems where the solution is not uniform in e whenever e approaches zero t h a t is lim u~ r u0.
e--~0
(1.3)
We distinguish two classes of p e r t u r b a t i o n problems, viz. singular perturbations of cumulative type and singular perturbations of boundary layer type.
Singular Perturbations of C u m u l a t i v e T y p e This class concerns oscillating systems where the influence of the small parameter becomes observable only after a long time, for instance after an interval of order O(~).
Chapter 1. General Introduction Let us take a nonlinear spring as an example; the displacement of its mass is given by the equation
d2ue +u~+Eu 3=0, dt 2
0
0<6<<1
(1.4)
with the initial conditions u~(0) - 1 and d---~ (0) -- 0. An asymptotic approximation dt of the solution may be obtained by the method of stretching the coordinate t. This method was already introduced at the end of the nineteenth century by Lindstedt and Poincar~ in connection with their studies of perturbation problems in celestial mechanics [98], [99], [116]. Substituting t - (1 + CW 1 -1t- ~2W 2 -~-'" ")T (1.5) and
u~(t) = SO(T) + EUl (T) + . . .
(1.6)
into (1.4) one obtains after taking together equal powers of ~ a recursive set of linear initial value problems. The constants {wi} are chosen such that the expansion (1.6) does not contain so-called "secular terms", which kill the asymptotic property of the expansion (1.6) for large values of t, e.g. ul (T) should not contain a factor T, otherwise
~ ( ~ ) # o(~).
The result becomes in first approximation
u~(t) = cos{(1 +
3E)t} 4- 0(~),
(1.7)
uniformly valid in a time interval of length O(1). This expression reveals an important property of the simple system of our nonlinear spring. First of all lim ue(t) =/: so(t) for t -- O ( 1 ) ,
~-~0
(1.3)
and so there exists a nonuniformity in the behaviour of the solution whenever e -+ 0. Further we remark that there are involved two time scales in the motion of the point mass: a fast one t and a slow one et; the occurrence of several time scales in one phenomenon is often encountered in nature, also in biological systems. This brought already in the early times Lindstedt and Poincar~ [98], [99], [116] to the so-called multiple scale technique which has been elaborated, refined and applied later on by several others, [85], [120]. The multiple scale technique consists in its most simple form in the expansion
~ ( t ) = so(t, ~) + ~u~(t, ~) + . . . , with T = r
(1.8)
Substitution into the differential equation that models the system, e.g.
d2ue due, dt---T + u~ = cf(t, u~, --~-)
(1.9)
yields again after taking together equal powers of r a recursive set of linear equations for ui(t, T). The introduction of the extra "independent" variable ~- makes it possible
Singular Perturbation of Boundary Layer Type to determine ui(t, T) in such a way that the expansion (1.8) becomes asymptotically meaningful, also for large values of t, i.e. no secular terms appear. Another method closely related to the multiple scale technique is based on the averaging pinciple of Krilov, Bogoliubov and Mitropolski [14]. Suppose the vector valued function u(t) satisfies the initial value problem due
dt = ~f(t, ue),
u(0) = u0,
t > 0,
(1.10)
where f satisfies some regularity conditions and f is periodic in t with period T, independent of e. Then u~(t) is approximated during a time interval of O(~) by the solution of the averaged equation dv
dt = cfo(v),
with
v(0) - u0,
1/
t > 0
(1.11)
T
fo(v) -- ~
f(t, v)dt,
(1.12)
o where the integration is performed as if v were a constant. This principle was already used by Lagrange who averaged certain quantities varying slowly in time. Also Gausz applied the principle in his study of the mutual influence of the planets during their motion; he distributed the mass of each planet over its orbit in proportion to the time and replaced the attracting force of each planet by that of a ring. The principle is also applied in modern developments in statistical physics.
Singular Perturbation of Boundary Layer Type There are several phenomena in physics which are characterized by a rapid transition of the observable quantity such as for instance occur in shock waves in gas motions, in boundary layer flow along the surface of a body and in edge effects in the deformation of elastic plates. The mathematical models describing these phenomena contain a small parameter e and the influence of this parameter reveals itself in a sudden change of the dependent variable u~, taking place within a small layer. The most famous prototype is from Prandtl's boundary layer theory [118], [125]. The two-dimensional flow around a finite plate is described by the Navier-Stokes equation. For the streamfunction r it reads -
Oy Ox
Ox Oy
6V 2 V2r
- 0,
(1.13)
where 6 denotes the inverse of the Reynolds number and so it is proportional to the viscosity u. The boundary conditions are r
0) - 0, - c ~ < x < +c~, r
y ) - y for x --4 - c ~
(1.14)
Chapter 1. General Introduction and 0r
Oy
(x, 0) - 0 for 0 < x < 1.
(1.15)
The latter boundary condition is very important in our discussion because it means the absence of a slip stream along the plate which is an effect due to the presence of viscosity. In case the viscosity is small one is tempted to consider in a first approximation the differential equation
P0[r
(0r
0
0r 0 ) v~r = 0,
= \ Oy Ox
Ox Oy
(1.16)
but because the order of P0 is lower than that of Pe it is impossible to satisfy all three boundary conditions (1.14)-(1.15). Since the condition (1.15) is a consequence of the viscosity it is natural to drop this condition and there results r y) -- y. This simple potential flow was to be expected, but it can never be a good approximation of the solution near the plate surface because the boundary condition (1.15) does not hold and one has again lime-+0 ~oy =/: ~ " The same reasoning applies also to other geometrical configurations and it is remarkable that the solution r y) of the so-called reduced equation gives a satisfactory description of the flow despite the fact that the viscosity has not been taken into account; this is d'Alemberts paradox since it is known that nonviscous fluids do not exert a force on bodies in the flow. It was Prandtl who solved this paradox by assuming that the effect of the viscosity is only observable in a narrow layer adjacent to the body. So it is of importance to investigate what happens near the surface and this is done with the aid of a microscope in the form of coordinate stretching and in this case one uses the transformation
y = v~,
r
= vqV.
(1.17)
After substitution into the equation, a subsequent calculation involving the lost boundary condition (1.15) and a matching with the potential flow r for r/ large, yields a result at least formally valid as well near the body as at a large distant from the body; for details of the calculations, see M. van Dyke [26, Ch.7]. Another physical example where we meet the same phenomenon is from the theory of elasticity. The deflection of a thin plate 12 clamped along its edge 012 and subjected to lateral pressure satisfies the boundary value problem
eA2ue - Au~ = h(x,y), with
(x,y) e it,
Oue ue[o ~ = 0 and -~n--n[o~ - 0;
(1.18)
(1.19)
e is a measure of the plate thickness and n denotes the direction normal to the edge. For e = 0 one obtains the membrane problem
AUo = - h ( x , y), with only UOlo~ -- O.
(x, y) e fl
(1.20)
Singular Perturbation of Boundary Layer Type The solution of this reduced problem describes the deflection of the plate rather well, but it can never be a good approximation in the neighbourhood of the edge of the plate because the second boundary condition of (1.19) is in general not satisfiedby u0. One needs a correction and this correction is obtained by stretching the local coordinates in the direction normal to the boundary. Investigating the transformed problem it appears that the correction is only significant in the neighbourhood of the edge and a composite expression consisting of u0 and the correction term satisfies the boundary value problem (1.18)-(1.19) in good approximation. The solution of the problem of the boundary layer flow and that of the elastic plate is characterized by the fact that the perturbation with e small has an observable effect only in the neighbourhood of the boundary and therefore one uses the term "singular perturbations of boundary layer type". However, it can also happen that the perturbation is observable in a thin layer not in the neighbourhood of some boundary or edge and in this case we have a "singular perturbation of free layer type" The two examples above lead to a mathematical generalization which is formulated as follows. Consider a boundary value problem of the form
eL2[ue(x)]+Ll[ue(x)]=f(x),
xE~tC]R~,
0<e<
(1.21)
where L2 and L1 are differential operators of the orders m and k respectively with m > k > 0, while u~(x) satisfies the boundary conditions
B~[ue(x)] = qo~(x),
x E af~,
g = 0,1,2,....
(1.22)
The solution u0 of the reduced equation with e - 0 cannot satisfy in general all boundary conditions and it is clear that the solution ue will show a nonuniform behaviour for E -+ 0. Just as in the phsyical examples above this nonuniform behaviour manifests itself only in a small layer in the neighbourhood of the boundary or of some part of the boundary, but also a free layer may occur. It is possible to write ue as a composite expression consisting of two asymptotic expansions, one valid outside the layer, usually called the "outer expansion" and the other valid inside the layer, usually called the "inner expansion". The inner expansion is obtained by expanding the operator eL2 + L1 after the application of a stretching of the local coordinates, i.e. by using the microscope. Much attention has been paid to the matching of the two expansions which should lead to the overall solution u~. Van Dyke [26, Ch.5] introduced certain asymptotic matching principles according to which both asymptotic expansions can be matched and Kaplun and Lagerstrom [90] introduced the hypothesis that there is a common region where both asymptotic expansions are valid and that an intermediate matching can be applied. Both methods and their interrelation have been studied and clarified by W. Eckhaus [30], [33]. The matching procedure may be difficult and it may lead to very unattractive calculations, see e.g. [53, Ch.6], [62, section 8], but on the other hand there are also many cases where the matching can be attained in a rather easy intuitive way. Readers interested in the history of singular perturbations of boundary layer type are referred to the books by M. van Dyke [26], J.D. Cole [19], J. Kevorkian and J.D. Cole
Chapter 1. General Introduction [85], P.A. Lagerstrom [90], R.E. O'Malley [112], the paper by K.O. Friedrichs [45] and the SIAM-Review, Vol 36, 1994. While applied mathematical research was aimed at asymptotic approximations of solutions and procedures were invented to construct these approximations in a more or less formal way there was also the mathematical question regarding the validity of these procedures. Early investigations in this direction were carried out by a . o . W . Wasow [139] in 1944, N. Levinson [95] in 1950, O.A. Oleinik [114] in 1954, I.M. Visik and L.A. Lyusternik [137], [138] in 1957. Meanwhile the subject has received a broad international interest stimulated from many research activities and there exists nowadays an overwhelming vast quantity of literature. We do not aim at a complete bibliographical survey and therefore we present in the list of references a number of publications composed according to the subjective taste and knowledge of the authors, see [100, 70, 29, 30, 111, 112, 55, 64, 9, 41,101]. Readers will certainly miss some names and important papers which should also be mentioned. However many additional references will be found in the publications quoted and in those still to be quoted in the chapters to follow where we explain a large numbers of topics of the theory of singular perturbations. In this textbook we give our attention primarily to the construction of formal approximations to solutions of initial and boundary value problems and to the validity of these formal approximations. The latter is justified by a careful investigation of remainder terms, being the difference between the solution and its formal approximation. This involves quite a number of mathematical techniques, such as the use of Gronwall's lemma, the contraction principle in Banach spaces, a priori estimates of solutions of boundary and initial value problems using the maximum principle and energy estimates. Although a number of physical applications are included, mostly in the context of oscillation problems, the emphasis is on mathematical analysis. Excellent texts on the applications of singular perturbation theory in many examples from mathematical physics and engineering are the books by J.D. Cole [19], J. Kevorkian and J.D. Cole [85], R.E. O'Malley [112], D.R. Smith [127] and A. Nayfeh [108]. The book can be roughly divided into three parts. The first part is of an introductory nature; we give the essentials of the theory of asymptotic expansions and the theory of regular perturbations; these are perturbation problems where the nonuniformity does not show up. In the second part we deal with singular perturbations of cumulative type and the methods of Poincar6-Lindstedt, averaging and multiple scales are treated in detail together with applications to several oscillation problems. Finally, the third part is devoted to singular perturbations of boundary layer type. In particular we consider differential equations of the type
r
+ Ll[u~(x)] = f(x),
or
r
y)] + Ll[Ue(X, y)] = f(x, y),
(1.23)
where ue is a function of the independent variable x C I C R or of (x,y) E f~ C JR2 with I and 12 bounded. L1 and L2 are differential operator with the order of L2 higher than that of L1 and 6 is the small positive parameter. Moreover ue should satisfy initial or boundary conditions. The operator L1 is perturbed by
Singular Perturbation of Boundary Layer Type the operator eL2 and this perturbation is singular because the reduced equation with - 0 is of lower order than that of the full equation. In general we do not have lime_~oue--uo for all x in I or for all (x,y) in ~t. Asking for solutions and in particular for asymptotic expansions of ue in terms of order functions of ~ one has to specify the differential operators L1 and L2. It is evident that according to the order and the type of L1 and L2 a large variety of perturbation problems turns up and a certain ordering is required. It is not quite possible to treat exhaustively all kinds of possible perturbation problems that may occur in theory or in practical applications. Therefore we limit our considerations to the following problems i) L1 and L2 are linear ordinary differential operators with L2 of order 2 and we consider initial and boundary~value problems. ii) L1 and L2 are linear partial differential operators with L2 of order 2; in case L2 is elliptic we study boundary value problems and in case L2 is hyperbolic we investigate initial value problems. iii) L1 is a nonlinear ordinary or partial differential operator of the first order and L2 is a linear operator of order 2. iv) L1 and L2 are ordinary differential operators of order k and m respectively with m > k :> 0; L1 and L2 are partial elliptic differential operators of order 2k and 2m respectively with m > k > 0. These problems form only a small part of possible singular perturbation problems. One might study for instance the cases with L1 elliptic, hyperbolic or parabolic and L2, independently of L1, elliptic, hyperbolic or parabolic. So we have for 6 = 0 the transition from one type of a differential equation to another type. Moreover one might envisage linear and nonlinear problems. Besides all this there are still many more problems by considering several boundary conditions along the boundaries of bounded or unbounded domains. For a broad bibliographical survey of several perturbation problems the reader is referred to the book by J.L. Lions [100]. As to more details on the text of this book the reader may consult the table of contents.
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Chapter 2 ASYMPTOTIC 1.
Order
EXPANSIONS
Symbols
In the preceding introduction we have acquainted the reader with some problems concerning the approximation of solutions of differential equations and therefore we devote our attention first to approximations of functions in general. In case one has a sequence of numbers, say { f ( n ) } ~ = 1 , the number which may be approached by this sequence for n --4 oo is well known and is denoted by lim f ( n ) , in case this limit n--+ cx)
exists. If
lim f ( n ) = f , then we have the following situation: given any ~ > 0 there n - - - ~ OO
exists a number N(e), dependent on e, such t h a t If(n) - fl < e,
Vn > N(e).
(2.1.1)
This definition tells us something about the behaviour of f ( n ) for n --4 c~, namely that f ( n ) approaches the number f as close as we want by increasing the number n, but it does not tell _how f ( n ) approaches the number f, and the definition does not give much information about the difference between f (n) and f when n becomes very large; it follows only that this difference approaches zero. In order to describe the behaviour of sequences of numbers in a more precise way, one compares different sequences of numbers in the following way DEFINITION 1
i) The order of { f ( n ) } ~ = 1 is for n -+ c~ not higher than t h a t of {g(n)}~=l, whenever there exist numbers A and N such t h a t
If(n)l
< A[g(n)l,
Vn > N.
(2.1.2)
This is expressed by the Bachmann-Landau notation [92]
f ( n ) = O(g(n)) for n sufficiently large
(2.1.3)
ii) The order of { f ( n ) } ~ = 1 is for n --4 cx~ lower than t h a t of {g(n)}~=l, whenever there exists for any positive number r/, a number N(r/), dependent on r/, such that If(n)] < ~]g(n)l , kin > g(r/). (2.1.4)
10
Chapter 2. Asymptotic Expansions This is expressed by the B a c h m a n n - L a n d a u notation
f(n) = o(g(n)) for n sufficiently large.
(2.1.5)
iii) The order of { f ( n ) } ~ = 1 is for n -+ co equal to t h a t of {g(n)}~= 1, whenever f(n) : O(g(n)) and f(n) :fi o(g(n)) for n sufficiently large. This is expressed by the notation f(n) = 08(g(n)) for n sufficiently large. (2.1.6)
Remarks 1. In case g(n) is not zero for n sufficiently large, and in the particular circumstance t h a t the limit of I f--(-~ ] exists, the defining relations (2.1.2) and (2.1.6) may be 9(-) replaced by l i r n [ f(n) g - ~ I = ~ # 0.
(2.1.7)
The reader should be warned t h a t the relation (2.1.7) is much more restrictive t h a n (2.1.2), and so it cannot be used as a definition of the "0" symbol 2. Similar definitions hold whenever the discrete variable n is replaced by a continuous variable w with w E R and w -+ co. The same applies also when w is replaced by the variable ~ with e E R and E $ 0 (define e = 1/w). Examples 1. s i n ( n ) = 0 ( 1 ) for h E N and n - - + c o , sinw:0(1) for w E R and w--+co. 2. s i n ( 1 / n ) = o ( 1 ) for h E N and n - - + c o , s i n E = o ( 1 ) for ~611( and E--+0. 3. l o g ( l + e ) : 0 8 ( e ) and l o g ( l + e ) = o ( e ~'), # < 1 for e E R and e - - + 0 4. e ~ - l = 0 8 ( e ) for e - + 0 and e ~ - l = 0 8 ( e ~) for w - - + + c o 5. e -1/~ : O(e"), Vn, e E R and e $ 0. e -1/~ is called "asymptotically equal to zero for e $ 0." The definitions are easily extended to sequences of functions f(x, n) depending on a variable x and the discrete p a r a m e t e r n or to sets of functions f(x, w) or f(x, e) depending on x and the continuous p a r a m e t e r w and ~ respectively. In the following we restrict our t r e a t m e n t only to the case e E R and e --+ 0, and the functions f ( x , e ) and g(x,e) to be compared with each other are defined for x-(S) inadomain DCR". DEFINITION 2 i) The order of f ( x , e) is not higher t h a n t h a t of g(x, ~) at the point x = x0 E D for $ 0 whenever there exist positive functions k(x) and e(x) such t h a t
If(x0,e)l
< k(xo)[g(xo, e)l for 0 < e < e(x0).
Notation:
f(xo, e) = 0(g(x0, e)), ~ $ 0. If (2.1.8) holds at each point x 6 D, we write f(x,~) -- O(g(x,~)),
x E D,
~ $ 0.
(2.1.8)
1. Order Symbols
11
ii) The order of f(x,e) is lower t h a n t h a t of g(x,e) at the point x0 6 D for e $ 0 whenever for any positive number 7/ there exists a positive function 6n(x0) , depending on ~ and x0 such t h a t If(xo, e)[ < nlg(xo, e)l for 0 < e <
(2.1.9)
6n(xo ).
Notation: f(xo, 6) = o(g(xo, e)),
e $ 0, If (2.1.9) holds at each point x0 E D we write
f(x,e) = o(g(x,e)), iii) The order of f(x, x0 6 D whenever
x e n,
e $ O.
e) is equal to t h a t of g(x, e) at the point
f(xo,6) = O(g(xo,6)) and f(xo,6) # o(g(xo,6)),
(2.1.10)
6 ,~ O.
Notation: f ( x 0 , 6 ) = Os(g(xo,6)). If (2.1.10) holds at each point x0 6 D we write
f(x,e) = O,(g(x,e)),
x e D,
e $0.
In case the function k(x) has a finite upper b o u n d in D with K = sup k(x) and the functions e(x) and en(x ) have a positive lower b o u n d with xED
e(x) respectively e_n -- inf 6n(x ) it is possible to make a uniform comparison x6D ' between the functions f(x, e) and g(x, e) as e $ 0, which is independent of the point
e -
inf
x6D
x 6 D. So we come finally to the last definition of comparison: DEFINITION 3
i) The order of f(x,6) is not higher t h a n t h a t of g(x,e) uniformly in a domain D C R '~ for e $ 0, whenever there exist positive numbers e and K such t h a t
If(x,6)l < K ig(x,6),
Vx 6 D, and 0 < 6 < e.
(2.1.11)
Notation: f(x, e) = O(g(x, e)), uniformly in D, as 6 $ 0. ii) The order of f(x, e) is lower t h a n t h a t of g(x, e) uniformly in a d o m a i n D C IRn for e $ 0 whenever there exists for any positive n u m b e r small, a positive number 6 n such t h a t If(x, 6)1 < ~lg(x, 6)1,
Vx ~ D and 0 < 6 5 6 n.
~,
however (2.1.12)
Notation: f(x,e) = o(g(x,e)), uniformly in D as e $ 0. iii) The order of f ( x , 6) is equal to t h a t of g(x, e) uniformly in a domain D C ]R~ for e -+ 0 whenever f ( x , e) =
O(g(x,e)) uniformly in D as e $ O,
but
f (x, e) --/:o(g(x, e)) in D. Notation: f(x,e) = Os(g(x,e)) uniformly in D as e $ 0.
12
Chapter 2. Asymptotic Expansions
Examples 1 1. ~ - ~ - 0 ( 1 ) as e $ 0 for each value of x in (0,1] , but ~-~ 1 is not 0(1) uniformly in (0, 1], because k(x) - 1/x has not a finite upper b o u n d in (0, 1]. 2. log(sin ex) - 0(log 2ex ) uniformly in (0, 1] as ~ $ 0. We have 2~x 0 < - W - < s i n c x for 0 < x < l in 0 < ~ < 1, and therefore [log(sinr < [log (2~---~-~)[, Vx 9 (0,1] and 0 < e < 1. It follows t h a t k(x) maybe taken identically equal to 1, K - 1 and e_ - 1 and hence log(sinex) 0 ( log 2~, ) uniformly in (0, 1] e $ 0. One may also prove log(sinex) = 08( log 2e__xx) uniformly in (0, 1],
e .~ 0.
71"
This is left as an exercise to the reader. 3. e x p ( - x / r = o(r VN" at every point of (0, c~), but this result is not true uniformly, because the condition (2.1.12) cannot be fulfilled in all points of the open interval (0, c~). W h e n we take for example x = 61+~', # > 0 we have e -x/e : e -e" > e -1 for 0 < e < 1.
2. G a u g e Functions and A s y m p t o t i c
Sequences
The order definitions give us a tool to compare the values of two functions which depend on the variable x and the p a r a m e t e r e. By taking a special privileged set of functions we get a set of comparison functions with the values of which we m a y compare the values of a large class of functions f(x,r This special set is chosen as simple as possible (depending on the class of functions f(x,r and with this set we have obtained, so to say, a yard stick or measuring rod to be used to measure the values of our functions f(x, r as ~ $ 0. The elements of the special set are called gauge functions. DEFINITION 4
A gauge function 5(~) is a function of the p a r a m e t e r e with the properties of being positive, monotoneously decreasing or increasing for ~ $ 0 and continuously differentiable in a right neighbourhood (0, 60) of e - 0. Sets of frequently used gauge functions are the positive and the negative powers of ~ : g,`(c) -- c,`, n - 0, + l , .... Other useful sets are e.g.: e~'gn(6), ]loge[Ogr,(e) or exp(-1/e)g,~(e) with a,/3 9 R and a > O. After this introduction of gauge functions we introduce some ordering in sets of gauge functions. So we are led to the following definition of ordering: DEFINITION
5
The sequence {5,, (~)}~=0 of gauge functions is called an asymptotic sequence whenever 5,~+1(~) = o(5n(r
Vn, as ~ $ 0.
(2.2.1)
Examples {e,`}~--0, {r with p > 0, {[loge[~e"}.~1760 with j3 e R, {8,`e-1/e}n~176 exp(-l/r = o(r for all values of n and hence e x p ( - X / e ) is smaller t h a n all gauge functions of the set {~n}n~176 0. Therefore we call e x p ( - 1 / r asymptotically zero and this is denoted by e x p ( - 1 / e ) ~ 0.
3. Asymptotic Series
13
3. A s y m p t o t i c S e r i e s We start immediately with some definitions. DEFINITION 6
Let f(x,e) be a function defined in D • (0, e0], and {6n(e)}n~__0 an asymptotic sequence as 6 $ 0. If there exists a function gg(x) and functions {a,~(x) },~=1 g finite in x 0 E D , such that N
f(xo, s) -- Zan(Xo)5~(~) + 9N(xO) " O(hN(~)), as e $ 0,
(2.3.1)
n--O
then we call the right-hand side "the asymptotic expansion of f(x, ~) at xo E D up to the order 5N(~ ) aS e .~ 0". In case (2.3.1) is valid for all x0 E D, then we drop the suffix 0 and we call the right-hand side the asymptotic expansion of f(x,r in D up to the order 6N(r as e$0. DEFINITION 7
If (2.3.1) is valid for all x0 E D and for any positive integer N, then we write oo
f(x,e) ,.~ Z a , ( x ) 6 , ( e ) ,
x E D,
e $ O,
(2.3.2)
n----O
and the infinite series in the right-hand side is called "the asymptotic series of f (x, e) in x E D with respect to the asymptotic sequence 5~(e) as e $ 0". The series is called
asymptotically convergent. DEFINITION 8
In case (2.3.1)ia valid for all x E D and 9N(X) is uniformly bounded in D by some number KN the relation (2.3.1) maybe written as N
f(x,e) = Za,~(x)6,(r
+ o((~N(6)) ,
VX E
D as ~ $ 0,
(2.3.3)
n--O
and the asymptotic expansion of f(x,r up to the order 6N(r is uniformly valid in D. In case (2.3.1) is valid for all x E D and all natural numbers N and the set of functions {gu(x)} is uniformly bounded in D by some number K the asymptotic "equality" (2.3.2) is uniformly valid and the infinite series (2.3.2) is called "the uniformly asymptotic series of f (x, e) with respect to the asymptotic sequence 6, (e) as e $ 0". oo
In the special case {6n (e)}oon=0 = { en }oon=0 we call the asymptotic series y] an (x)r n the n--0
Poincar6 expansion of the function f(x, e) at the point x E D. It is easily shown that the absolute uniform convergence of the series (x)
f(x,r
= Zan(x)6n(r n--O
with 0 < e < Co,
14
Chapter 2. Asymptotic Expansions
uniform with respect to x E D, implies t h a t this series is also a uniformly asymptotic series for f(z,e). An example of an asymptotic series not uniformly valid in its domain of definition is given by the function
f (x,6) = "~/x + 6 = v/'x 9
+-,
x>O.
X
This function has for all x with x > 0 the asymptotic expansion 6
. 6. .2.
+ (-1)'-l(2n-
~-,~
f(x, 6),~ V'~(1 + 2--x - 8x 2
3)!! 6 n
(;) +...),
uniformly valid in any left bounded interval x > ~ > 0; however, the expansion is not uniformly valid in x > 0 because the remainder after n terms is not 0(6~), uniformly in x > 0. ( ( 2 n - 3 ) ! ! - 1 - 3 . 5 . . . ( 2 n 3)). From the definition of the o-symbol and from definition 7 it follows t h a t the coefficients an(x) of the asymptotic series (2.3.2) are uniquely determined by the function f ( x , 6); one proves simply t h a t these coefficients are defined by the recurrent relations
ao(x)
=
lim
f (x, 6)
~,o ~o(~)
and
(2.3.4) n--1
f (x, 6) - ~ ai(x)e~i(c) an (x) = lim ~,0
i=0 ~n(e)
,
n=l,2,....
However the inverse is not true as is clear from the following counterexample. Suppose oo
f(x,6)
has the asymptotic expansion,
~ an(x)6 n, with respect to the asymptotic n=0
sequence {6n},~~176, then the function f ( x , 6) + exp ( - 1 / 6 ) has the same asymptotic expansion with respect to {6n}n~176. This consideration leads to the following definition. DEFINTION 9 Two functions which have the same asymptotic expansion with respect to the same asymptotic sequence are called asymptotically equal with respect to this sequence. A function is called asymptotically equal to zero with respect to a certain asymptotic sequence whenever all coefficients an(x) are zero. This definition m a y be applied as follows: i) Two functions are asymptotically equal in a point x E D. ii) Two functions are asymptotically equal in all points x E D. iii) Two functions are uniformly asymptotically equal in D. T h e latter applies only in the case t h a t b o t h functions possess uniform asymptotic expansions.
3. Asymptotic Series
15
We finally remark, that the asymptotic expansion changes, when the asymptotic sequence changes . We illustrate this with the following example, given by the function ( 1 - l +exe )-1 ' defined for Ixl < 1.
f(x,r r
Expanding with respect to 1-~ we obtain oo
-C ) nxn
S(x,e) = E ( 1
+ e
r~--O
and N
oo
Z( n=N+l
If(~' ~) - ~ - ~ ( 1 + ~ n=O
l+e
)N+llxIN+lf(x'c) <-- (1
<-(l+e
e g+l e )g+l -- (1 -it- ~) N ~ 2(1 +e _
e )N+I(1 e )-1 + e 1+ e
for
O<e_
N },
VN, uniformly in
N
It follows that f (x, e) -
y~. x ~ ( y ~ ) ' ~ + 0{ ( ~ ) rt=0
Ixl <_ 1 and hence f(x, e) possesses in Ixl < 1 with respect to the asymptotic sequence n O0 { (Yg T) }~=0 the uniform asymptotic expansion oo
x"(1 +e
f(x,e),..,
1",
so.
r~=0
We may expand f ( x , e )
also in another way:
l+e f ( x , e ) = 1 + e(1 - x) = (1 + r
oo - x) n n--0
oo
oo
= E(-1)ne'~(1
- x) '~ + E ( - 1 ) n e ' ~ + l ( 1
n=O
- x) '~
n=O
= 1 + E(-1)'~e"(1
- x) n + E ( - 1 ) n - l e ~ ( 1
n--'- 1
- x) ~-1
n=l
oo
= 1 + E(-1)'~-le"(1
- x ) n - l { 1 - ( 1 - x)}
n--1 oo
1)n-ze '~
= 1 + Ex(xn---1
and so we get with respect to the asymptotic sequence expansion f ( x , e) ~., 1 + ~ _ x ( x - 1)n--lr n, n=l
S .~ O,
en
the uniform asymptotic I~1 _< 1.
16
Chapter 2. Asymptotic Expansions
4. C o n v e r g e n c e Versus A s y m p t o t i c C o n v e r g e n c e We have now at our disposition two notions of convergence and we will show that these notions correspond to quite different concepts. We do this by giving an example concerning the error function which is defined by: oo
erf A = 1
2 / _t ~ v/~ e dt. A
Substituting
t 2 - T we obtain the alternative expression
1/ e-rT-1/2dT"" c~
erf A = 1 - -
A2
We denote oo
F,(A) =/e-~T-(2"+l>/2d.r,
(n -- 0,1, 2, . . .) ,
A2
and integration by parts yields the following recursion formula _A 2
e F,(A) - h2,+ 1
2n+l 2
F,+I(A),
(n = 0, 1 , 2 , . . . ) .
From this result we get oo
-A 2
e - r T - 1 / 2 d T = F0(A) - e
h
1 2 FI(A) . . . .
A2
---- e
-A2[ 1 A
1 1"3 2A 3 + 2-5~A 5 -}-
-}-(_I)N_ 1 1 . 3 . 5 . . .
(2N-
+ (_1) N 1 . 3 . 5 . . . ( 2 N ' 2N
3) 1)
]
FN(A).
Hence _A a
erf A = 1
N
e ) , - 1 1 " 3.2~(_2~ v ~ ~'-~(-1
3)
1 2,~-1 (X) + RN(A)
n--1
with
1 1)g+ 11. 3... 2~ ( 2 N - 1) F~(A). R~(A) = --~(Because
/
c~
FN(A) -
A2
1 /
r
e -rT-(2N+l)/2dT" < A(2N+I )
A2
e
-A 2
e-'dT = h2,+ 1
4. Convergence Versus Asymptotic Convergence
17
we have 1 1 . 3 . . . ( 2 N - 1) e - n 2 2N A2N+I '
IRN(A)I <
and therefore
e-A2( ~--~(-1) N n-1
erf A -- 1 - ~
1.3.
2~(2_~_in-
1 2n-1 1 } 3) ( ~ ) + O(A2g_ 1 )
n--1
as A -4 co and this result is valid for each value of N. It follows now that erf A ~ 1
~1. ~ _~ (_1),~_ 1 1 . 3 .2: (_21n- - 3) e -As _ A2n_ 1 as A -4 co.
Despite the fact that this series
diverges for any value
(2.4.1)
of A the result yields nevertheless
a correct useful
asymptotic series with respect to the asymptotic sequence { A2n+l ~-A2 }. This example illustrates that the concepts of convergence and asymptotic convergence of a series are quite different. Using the former concept of convergence one considers the remainder RN(A) as function of N keeping A constant, and using the concept of asymptotic convergence one consider the remainder RN(A) as function of A keeping N constant. The same applies, of course, to functions f(x, A) depending also on the variable x or to functions f(x,r Convergent series and asymptotically convergent series represent totally different points of view; a well convergent series representation of a function may be bad from the asymptotic point of view and a correct asymptotic series may be even divergent. An interesting question is the following: given a divergent series which is asymptotically convergent, find an estimate of the number of terms which gives an optimal approximation of f(r or f(x, 6) where e is a given fixed small number. This is from the practical point of view an interesting question which has been solved by a . o . I , van den Berg, using methods of nonstandard analysis; see Ref. [5]. In the literature, in particular, in the literature of special functions, many examples can be found of useful asymptotic approximations of functions. We give here the well-known example of the Bessel-function Jo(x). This function has the convergent expansion. oo
= n--0
22~(n!) 2
which is uniformly convergent within any bounded segment of IR. For have also the well-known asymptotic expansion, [148, p.368], 2
Jo(x) ",-' ~
c o s ( x - lr n=o + sin(x
(_1) n l 2. 3 2 . . . ( 4 n 26n (2n)!
Ix[ large we
1) 2 1 x2~
7r,~v,( ] ~ 12.32...(4n+1) 2 1 ] 'A---''-l'r~ 26"+3(2n + 1)' x2"+----~ J ~r~,-'-0
D
18
Chapter 2. Asymptotic Expansions
While the convergent expansion is rather useless for getting values of Jo(x) for large values of x, the asymptotic series is very useful. In order to obtain an approximate value of J0(3) up to three numerals one needs eight terms of the convergent expansion and only one term of the asymptotic expansion.
5. Elementary Operations on Asymptotic Expansions In practise it may be necessary to perform some elementary operations on asymptotic series such as addition, multiplication, integration and differentation. In this section we investigate whether these operations do not destroy certain asymptotic properties and we give the conditions on which they may be applied. Let us assume f ( x , e ) ,,~ Ea,(X)hn(r
and g(x, r ,',., Ebn(x)5,(6)
n=0
(2.5.1)
n=0
at x C D as c $ 0. Without restricting generality we may assume 60(e) = 1. We have the following trivial theorem THEOREM 1
If f (x,e) and g(x,e) have the asymptotic expansions (2.5.1) then oo
f(x,e) + g(x,e) has the asymptotic expansionsion E {an(x) + bn(x)}5,(e). n--O
The multiplication of two asymptotic expansions (2.5.1) with respect to the asymptotic sequence {fin(e)} may lead to a problem, because the result will, in general, give an expansion with respect to another asymptotic sequence. The reason is the appearence of terms with a factor 5n(r which may not belong to the set {6n(r Therefore, we introduce the following definition. DEFINITION 10 The asymptotic sequence {6,,(~)} is called multiplicable whenever 5. (6)6m (r = 5n+.~ (e).
(2.5.2)
We have now the following two simple theorems. THEOREM 2
If the asymptotic sequence {hn(~)} is multiplicable, then oo
f(x,e)g(x,r
~ Ecn(x)6~(r
at x E D as r $ 0
(2.5.3)
n--0 n
with Cn(x) = ~ am (x)bn_m(x). m--O
THEOREM 3
If the asymptotic sequence {6n(r
is multiplicable and ao(x) ~ O, then
1 o~ f(x, 6) "" Ed"(x)6"(r r~"-O
(2.5.4)
19
5. Elementary Operations on Asymptotic Expansions
where dn(x) is defined by the relations n
ao(x)do(x) = 1 and E a m ( x ) d n _ m ( X ) = O,
n-
1,2, . . . .
m"-O
Consecutively we deal with the nonalgebraic operations of integration and differentation. Because integration is very easy we give first two theorems concerning this operation; the proofs are left to the reader. THEOREM
4
If f (x, ~) is integrable in D along an arbitrary piecewise continuous path C and if it has in D the uniformly asymptotic expansion (2.5.1) with coemcients an(x) also integrable along C, then oo
C
5
n=0
In case C connects the points xo and x we have
/
/
f(~,~)d~ ~ ~ { n--O
xO
a~(~)d~}~(~),
~ S O,
(2.5.4*)
Xo
where do" is the infinitesimal path element along C. THEOREM
5
0
n=0
0
The differentiation of the asymptotic series (2.5.1) with respect to formulated with the aid of Theorems 4 and 5 respectively. THEOREM
If f(x,e) oo
Oa
Y~ ~(
X)
x
or
c
can be
6
has the asymptotic expansion (2.5.1),with an differentiable in D
and if
Of
an(e) is uniformly asymptotic convergent in D, then -8-~,(x, e)
n--O
exists and it has the asymptotic expansion Of
~
oo Oan (x)Sn(c )
9
(2.5.6)
n--O
The differentiation of an asymptotic series with respect to e is a little bit tricky as is clear from the following example, taken from [16]. The function f ( s ) = e -1/~ sine 1/~ has with respect to the sequence {en}~=0 the asymptotic expansion e -1/~ sin(e 1/~) ,,~ 0 (all coefficients are zero).
20
Chapter 2. Asymptotic Expansions
Differentiation with respect to e yields for the right-hand side zero, but for the left-hand side 1 cos(el/~ ) ~-2
1 -1/~ sin(el/~) e-~e
which is even unbounded for e ~ O. Therefore, the differentiation with respect to e should be formulated carefully and we have the following theorem: THEOREM
7
If f ( x , e ) has the asymptotic expansion (2.5.1), of( and d 6 n ( e ) (n -- 0,1, " " .) exist for x E D and for 0 < e < eo and if of has the asymptotic Og expansion Of oo d6,` n--0
then a , ( x ) = an(x), n - O , 1 , 2 , . . . . The proof follows easily from the uniqueness of asymptotic expansions and from T H E O R E M 5 above. For the particular case 6,, (s) - ~,` it follows that under the assumptions (x)
f(x,e),,~ ~-~a,`(x)c n and Of
oo ,`--0
we have
na,`(x) = b,-l(X),
n = 1, 2, . . .
(see also [16])
6. O t h e r T y p e s of E s t i m a t e s In the foregoing section we considered asymptotic approximations of the type N
f(x,r
= ~-~an(x)6n(r
RN(X,r
(2.6.1)
n--0
with RN(X.r when e.g.:
= O(6N(r
uniformly in some region D C R,`. This situation occurs
N
f (x, ~) -- ~ an (x)6n (~) sup xED
n--0
6N(~)
< e u,
#>0,
0<e<e0.
(2.6.2)
The relation (2.6.1) with R g ( x , e ) = o(6g(e)) uniformly in D gives information on the magnitude of the error when f ( x , e) is approximated. This error is measured by the supremum of the absolute value of the difference between f ( x , e ) and the first
21
7. Generalized asymptotic Expansions
N terms of the asymptotic expansion. However, it is also possible to measure this error in another way. A general procedure which will appear to be very useful later on, is provided by using a little bit of functional analysis. Let us assume t h a t as well f(x,~) as {a,~(x)} belong to a certain Banach space B(D) of functions defined on D; it follows t h a t the error RN(X,r also belongs to B(D) and hence it can be measured by the n o r m [IRN(',e)IIB(D) defined on B(D). In case B(D) is the space C(D) of functions defined and continuous on a b o u n d e d closed d o m a i n D we have
IIR~(~, ~)llc(m - m~x IR~(~, ~)1
(2.~.3)
and this definition leads to the estimate (2.6.2) dealt with above. W h e n B(D) is the space CI(D) of functions defined and continuously differentiable in the b o u n d e d closed domain D one can measure the error Ry(x, c) by
IIR~(~,~)II~(,) - ~ x
IR~(~,~)+
I - b ~ (~,~)1
(2.6.4)
i--1
or even alternatively
{
IIRN(~,~)IIC~(D) = m~x IRN(~,~) + i=1
}
.
(2.6.5)
If ]lRg(x, e)llcl(D ) -- o(bg(~)) one has not only a pointwise estimate of DRy(x, e)l , but also a pointwise estimate of the derivatives of Ry(x, ~): It will appear later t h a t the n o r m (2.6.5) is very useful. A n o t h e r way of measuring the m a g n i t u d e of the error is given by the Banach space L2(D) of functions square integrable on some d o m a i n D. In this case one gets
IIRN(x,e)IIL2(D) =
{/ IRg(x,e)2ldx
(2.6.6)
D
One does not obtain a pointwise estimate, but an average estimate of the error. Even if IIRy(x,e)llL2(D) is very small, it m a y h a p p e n t h a t it is very large in a neighbourhood of some point x0 E D. Despite this disadvantage the n o r m (2.6.6) and related integral norms have been used frequently in the theory of singular perturbations; see e.g. [138], [100].
7. Generalized asymptotic Expansions It will appear later on t h a t is not always possible to give an asymptotic expansion of the form
S(x,c) ~ Ea,~(x)bn(~).
(2.7.1)
n=0
A more general situation is given by oo
f(x,~) ~ Ean(x,~)bn(e). n=0
(2.7.2)
22
Chapter 2. Asymptotic Expansions
An example to be used frequently is (x)
x>0,
0<6<60
n---0
with gn (-~) = P,-,(~) exp(-x/6) and P . (~) a polynomial of degree equal to or smaller than n. In this case every term of the asymptotic series is 08(6 n) uniformly on x > O; this follows immediately from
x n e_Z/s
max [ ~=>0
I = m~x I,'"~-~1--08(1). ~_>0
A series of the type (2.7.2) is not of Poincar6 type and so we call it a generalized
asymptotic expansion. Exercises 1. Show for 6 small and positive and for all n _ 0
x"e -~- = 0 ( 6 " ) uniformly in any x interval in [0, c~). 2. Determine the asymptotic expansion of the function G
f(x; 6) = (x + 6)(1-- x)'
0 < x < 1,
with respect to {6"}n~176 0 < 6 << 1. Is this expansion uniformly valid in (0, 1)? 3. Let f(x) be a function which is differentiable up to any order in some interval O<x
f (x) ~ ~-" l f(") (O)xn. z...~n!" n--O Note that the right-hand side is only a formal Taylor series, which does not necessarily converge. Hint: use the remainder of the finite Taylor expansion x
n~(~) =
1 / ( x - t)gf(g+l)(t)dt. 0
4. Suppose f(x)
satisfies the condition of exercise 3 above, then oo
f e-"Xf(x)dx ~ ~-'~f(n)(O)w - ~ - 1 as w -+ +oo. 0
n--O
23
Exercises Hint: the remainder in the right-hand side may be written as
SN(W) -
e_.,~ f(x)d x _ ~ 0
f(")(O)n! e-~*x"dx
n=0
--
e-~RN(x)dx 0
0
N/(-) (0) n!
n=O
e-~x~dx. a
5. Watson 's lemma Suppose oo
f(t)-- Eant(~-l)
,
0_~t_~a+5,
a>0,
5>0,
n=l
and
f ( t ) < K e bt f o r t _ > a ,
K>0,
b>0,
then o0
oo
F(s) - / e - ~ t f ( t ) d t ~ 0
~-~a"r(n-)s-e'r n=l
uniformly valid for Is[ large and [arg s[ _< 2 - A, ~ > O. Hint" the remainder of the right-hand side of F(s) cx~
reads
N
RN(S) = / e - ~ t { f ( t ) 0
Ea~t(~-l)}dt n=l
and there exists a constant C such that
N If(t)- Eant(~-l)
J ~_ Ct (N+~-l)ebt,
Vt>0.
n=l
6. Give the asymptotic expansion of oo
/ e-wy dy, F(w) = 1 + y o
w --+ +cr
Derive from this result oo
g(~) .= f0
l +~-~ xt
dt ~ E ( - 1 ) ' ~ n ! x n n~0
0
24
Chapter 2. Asymptotic Expansions for x small and positive. The right-hand side is a formal Taylor expansion of g(x), which diverges for x r 0 and is as such useless, but it has a definite meaning as a useful asymptotic series, see also Exercise 3.
7. Show that the asymptotic expansion of the function oo
f(x) = f costt dt for x -4 +cx~ ~g
is given by
f(x) ~ (
x -~ x 3
x5 ~...
sinx+
~-~-x4 +x6
cosx.
Hint" use repeated integration by parts. 8. The error function is defined as
2/
x
erf x -- - ~
e -t2 dt.
o Power series expansion of the integrand yields 2 ~
err X
( - 1 ) " x 2"+1
n--O
2n + 1
n!
This convergent expansion is not suitable for the numerical calculation of erf x for large values of x. On the other hand one has erfcx:--l-erfx=
~
2/
(x)
oo
1
e-t:dt-
~e-
e-t(t+
x
x2) - ~1d t .
0
Prove e v--x-2- ,
o~
, 1 ,
erfcx ~ ~ / ~ j ( - - 1 ) n ( ~ ) n v
x
--2n--I
for x --+ +cx).
n--O
9. Let y(t) be the solution of the initial value problem d2---~Y+2e~t + y = O ,
dt 2
t>O,
y(O)=O;
-dy ~-(0)=1,
0<~<<1
Give a first order approximation with error O(E), uniformly valid for all values of t>0.
Exercises
25
10. Let y(t) be the solution of the initial value problem
d2y dy e-d--t-ff+2-d-~+y=O ,
t>O,
y(O)=l;
dy -~-(0)=0,
O<e<
a. Show
y(t) - e x p [ - 2 t] + O(e), uniformly valid for 0 _< t _< t*, with t* an arbitrary finite positive number, independent of e. b. Show
y(t) = e x p [ - ~I t ] + e
{1 1 1 2 1 } )t] g e x p [ - 2 t ] - 4 e x p [ ( - - +e
+O(e2).
uniformly valid for 0 _ t _ t*. c. Show
1
y(t) = e x p [ - - ~t] + c
1
1}
gexp[-- ~t] -- ~ e x p [ + ~ t ]
+0@2),
uniformly valid for 0 < 5 _< t _< t*, with 5 independent of e. To what extent satisfies the expression e x p [ - ~I t ] + e
1 1 1 2 1 } g e x p [ - ~ t ] - ~ e x p [ ( ~- + )t]
the differential equation and the initial conditions? 11. Let y(x) be the solution of the boundary value problem
d2y
e~-x-sx2+ 2
+y=0,
O<x
y(O)=O,
y(1)=1,
O<e<
Show
y(x) = e x p [ -
1
(x - 1)] + O(e),
uniformly for 5 _< x _< 1-5, with 5 arbitrarily small and positive, but independent of
e.
This Page Intentionally Left Blank
Chapter 3 REGULAR
1.
PERTURBATIONS
Regular P e r t u r b a t i o n s
In follow up of the introductory Chapter 1, we consider a boundary or initial value problem L~[u~(x)] - O,
x = (Xl,X2,...,x~)
Be[u~(x)] = O,
x e S C OFt,
e gt,
(3.1.1) (3.1.2)
where (3.1.1) represents some differential equation and (3.1.2) some boundary or initial condition for the unknown function ue. Both equations contain in general a small parameter e. Ft is the domain of the independent variables, OFt its boundary containing some part S along which the boundary or the initial conditions are specified. The problem (3.1.1)-(3.1.2) is a perturbation problem and the pertaining reduced problem is obtained from (3.1.1)-(3.1.2) by putting 6 equal to zero; this leads to the boundary (initial) value problem L o [ u o ( x ) ] = O,
x e Ft,
B0[u0(~)] = 0,
9 e S C 03.
(3.1.3) (3.1.4)
The question to be investigated is whether the solution of (3.1.3)-(3.1.4) is a good approximation of the solution of (3.1.1)-(3.1.2) when 6 is very small, and further whether it is possible to improve this approximation and to give an estimate of the error. In case 6 appears in a "simple" way in the equations (3.1.1)-(3.1.2) the most naive procedure to construct an approximation for u~(x) is to assume that it is possible to expand u e ( x ) in an asymptotic expression N
u~(~) - Z ~ " , , ( ~ ) +
R~(~; ~),
(3.1.5)
rt--0
and to substitute (3.1.5) into (3.1.1)-(3.1.2). Equating equal powers of 6 may lead to a system of boundary value problems for the fuuctions u n ( x ) , n - 1 , . . . , N , which might be less difficult to solve than the original problem (3.1.1)-(3.1.2). If it is possible
28
Chapter 3. Regular Perturbations
to give also an estimate for the error RN(X; ~), e.g. as 0(~ N+I) or as o(r N) uniformly valid in f~, then we call the perturbation (3.1.1)-(3.1.2) with respect to (3.1.3)-(3.1.4) a regular perturbation and the method for obtaining (3.1.5) the regular perturbation method. In the next sections we give important examples and point out some serious limitations of this naive method. 2.
A Nonlinear Initial Value Problem Containing a Small Paramter
We consider for the function x = x0 the initial value problem
y(x) to be defined in some neighbourhood of dy d---x= f(x, y;---c),
(3.2.1)
with
v(~0) = c ( ~ ) .
(3.2.2)
We assume that f, defined on ]I{2 • [0, 1] and C defined on [0, 1], have the following asymptotic expansions for ~ $ O: oo
f(x,y; ~) ,,~ ~ e n f , , ( x , y ) ,
(3.2.3)
tt=O
uniformly valid in any bounded domain of R2, and oo
C(:) ~ ~-~enC,,.
(3.2.4)
n--O
Moreover, we suppose that the functions f~ are infinitely differentiable in R2. The purpose of this section is to construct an asymptotic expansion of y(x) with respect to the small parameter ~. First we give a purely formal expansion by means of the regular perturbation method and so we put
v(~) ~ v0(~) + ~u,(~) + - . . + ~ " v , ( ~ ) + . . .
9
Substitution into the right-hand side of (3.2.1) yields oc
oo
s(~, y; :) ~ ~ " s . ( ~ , Z~y~(~)) n=O
~Z:"
k=O
:-(~, ~0) + -~v
n--O
k--1
1 02fn
oo
+ ~--~y~ (~, vo) ( ~ ~ w (~)) ~ k--1
+ . . . } _~
(3.2.5)
2. A Nonlinear Initial Value Problem Containing a Small Paramter
: :0(~, v0) + ~ ~Ofo (~ , yO)yl(~) + fl (~, y0)
29
]
o f 0 (~, y0)y~(~) + 1 02fo (~, y0)y~(~)
+ ~
Of1
-~- -~-y (X, YO)Yl(X) -t- f2(x,
}
YO) -Jr-
-O~y (~, yo)y,(~) + f~(~, yo, y l , . . . , y~-l)
}
.........................................
fp(x, Yo, Yl,..., yp-1) x, Yo,... Yp-1 and which can
where
is some function which depends only on be constructed explicitly. Hence
f (~, y; ~) ~ fo(~, yo) + ~c~ n
{OfO(x W ' yo)Yn(X) ..~_]n(X,yO,... Yn_1) }
(3.2.6)
n=l
Substitution of (3.2.6) into the right-hand side of (3.2.1) yields after equating coefficients of equal powers of e the following system of initial value problems
dyo = f0(~, y0); yo(~0) = c0, dx dy,~ = ~Ofo ( X , yo)yn+fn(X, yO,...Yn-1); dx Oy
(3.2.7)
yn(Xo) =C,~,
n--1,2,....
(3.2.8)
Our perturbation problem is now reduced to the initial value problem (3.2.7), i.e. (3.2.1)(3.2.2) with e = 0, and a set of linear initial value problems which can be solved easily one after the other when the solution of (3.2.7) has been constructed. The function fo(x,y) is given to be infinitely differentiable in R2 and so we obtain by the well-known theorem on existence and uniqueness of solutions of ordinary differential equations (Ince [74], Ch.III) the existence of a unique C~-solution of (3.2.7) valid,in some neighbourhood, say I x - x01 _ a, of the point x - xo. If fo(x,y) is defined in the region 7:) - (Ix-xol) <_o~, ly-Col
fO o x
yn(x) -- Cn
exp[
-~y (~, yo(~))d~]
Xo
~xo
valid for I x - x 0 1 _ a .
-0-~y(~',yo(~'))e~']e~,
~(~,Yo(~),...Yn-l(~))exp[ (
(3.2.9)
Chapter 3. Regular Perturbations
30
formal expansion
In this way we have obtained a
of the unknown function y(x):
oo
~
(3.2.5) n--0
however, we have still to Therefore we put
prove
that this formal expansion is asymptotically correct. N
y(z) -- Eenyn(x) + RN(X;
e),
(3.2.10)
n--O
and we should show that RN(X; c) = o ( E N + I ) , uniformly in I x - x01 < a. Therefore our next task is to establish the existence of Rg(x; ~) for e sufficiently small in Ix - xol < a, which yields also the existence of y(x) in I x - x0] < a for sufficiently small, and consecutively we have to estimate the value of RN(X; e) in this interval. This is done with the aid of Liouville's method of successive approximations. Substitution of (3.2.10)into (3.2.1)-(3.2.2) and using (3.2.7)-(3.2.8) we get for RN(x; e) the differential equation
dx = f(x,
N
N
n=0 N
n=0
E c n y n + RN; c ) -
f(x, E e n y n ; e ) +
dyn N
n=0
~N(X; e),
(3.2.11)
n=0
with
N
N
n=0 N
n=0
dyn
= f(x, Ecny,~; ~)- fo(x, yo) n----O
- - E ~ n {OfO(x 'yO)yn-t-]n(x'yO'
. . Y.n -.i ) }
(3.2.12)
-- O ( e N + I ) = e N + l / z ( e )
(3.2.13)
n=l
The initial condition for
RN(X;
e) reads N
RN(XO; e) = C(e) - EenCn n--0
with u(e) some bounded function of e. First, we estimate qOy(x; e). In the same way as we have obtained the asymptotic expansion (3.2.6) for
f(x, y; e)
N
we get for
f(x, Y~ enY,~; e)
an expansion which has
n--0
the same first (N + 1) terms and therefore
qON(X; e) = o(eN+l),
uniformly in I x - x01 _< a
(3.2.14)
2. A Nonlinear Initial Value Problem Containing a Small Paramter
31
The differential equation (3.2.11) with the initial condition (3.2.13) is recast into the equivalent integral equation
RN(X; E) = EN+I/2(~) Jr-
(flN(~; ~) + f(~, E E n y n ( ~ ) + RN(~; C); E) n-0
xo
N
- s(~, ~~y~(~); ~)}e~,
(3.2.15)
n--O
valid for I x - x01 < a. Successive approximation of
RN(X; r
gives
R ( ~ ) ( x ; c) -- g'Nl-1/](c)
R(m-t-1)(x; ~) __ ~N-I-1//(g.)
+
~N(~; ~) + f(~, } ~ " v . ( ~ ) Xo
+
(~; ~);
n=0 N
- f(~,Ee'~y,~(~);--c)}d~, m = 0 , 1 , 2 , . . .
(3.2.16)
rt--O
Due to the smoothness of the functions integer M
{fj(x,y)}
we have, due to (3.2.3), for each
M
f(~, y:; ~) - f(~, y~; ~)1 _< ~-~'~lI~(~, y:) - f~(~, y,)l +
o(~M+~)
n--O
~_ LMlY2 -- Yll
+
O(g'M-I-1),
uniformly in any bounded domain of R2,
(3.2.17)
where LM is a suitable Lipschitz-constant independent of c. Hence it follows from (3.2.16) that R(Nm) - o ( ~ g + l ) , uniformly in Ix--xol <_ a implies R(Nm+l) -o(~N+l), uniformly in IX--Xol <_ a; induction on m yields that R(Nm) - - O ( ~ g + l ) uniformly in I x - x01 _< a for each value of m. Further we have N
IR~>(x; ~) - R(~>(x; 6)l <_
I~ON(~;rl)+ f(~, Ec'~y,~(~) + ~N+I~'(C);~)
Xo
n--O N -
Ida, n--O
and so by (3.2.17) with M -- N + 1.
32
Chapter 3. Regular Perturbations
In~)(x; e) - R(~)(x; e)l -< BeN+l, uniformly for Ix -- x0[ _< a, with B constant independent of e, which we take larger than 1. Further we obtain for I x - x0[ < a
n=0
a suitable
]
xo
" rt--0
Using (3.2.17) again we get similarly
IR(~)(~; ~)- R(~)(~; ~)l-< nN+~ sup IR(~)(~;6)- R(~)(~; ~)II~- ~ol Ix-xol
+ I~- ~01o(~~+~) < (L~+,B + O(~))I~- ~01~~§ <_M B I x - x0le g+l, with M - LN+I + 1. Repeating this progress again we get IR(N3)(x; e ) - R(~)(x; e)l _< M2B -~ 1 I~ - ~01 ~ + ~ and in general R(m+l)(x; e) N
--
R y ) ( x ; e)[ < M m B l [ x. --
--
xo['r'eg+l
~
m
~-
0,1,2,
(3.2.18)
uniformly valid for [x - Xo[ < a, and for e sufficiently small say 0 < e < co, e0 is independent of m. The sequence [R ( r e + l ) - R(Nm)[ is majorized by the uniformly 1 imBeN+1 convergent sequence {Mm~.lx-xo } and so we get the result that the sequence
{m
}
j=l
converges uniformly in Ix - x0[ < a and we obtain 1Lrno,' IR(N"')(x; e)l < ceN+le Ma, where C is a suitable constant, indepedent of c. It follows that the integral equation (3.2.15), and therefore also the initial value problem (3.2.11)-(3.2.13) has a solution, namely
nn(x; e ) -
lim n(m)(x; e ) - o ( e g + l ) , m--+
cx)
(3.2.19)
33
2. A Nonlinear Initial Value Problem Containing a Small Paramter
uniformly in [x - x0[ < a. Using once again the "Lipschitz-property" (3.2.17), it can be shown that, apart from an expression that is asymptotically zero, RN is the only solution satisfying (3.2.11)(3.2.13). The final result is that the initial value problem (3.2.1)-(3.2.2) has a unique ssolution y(x) for [x - x0[ < a with the property that we have for all values of N N
N
y(~) = ~ ~ y ~ ( ~ ) + R~(~; ~) ~ ~ ~ y ~ ( ~ ) + o(~ ~§ n=0
(3.2.20)
n=0
or oo
y(x) ~ ~-'~r
uniformly in I x - xo[ _< a.
n--0
This result is stated in the following theorem THEOREM 1 If the initial value problem
dyo dx = fo(x, yo),
yo(~o) -Co
with fo c C ~ (R2) has its unique solution defined in the bounded segment [x - xo[ ~_ a, then the initial value problem (3.2.1)-(3.2.4) dy oo d-~ : f ( x , y ; ~ ) ~ ~ f ~ ( x ,
y)~
n--0 oo
y(~0) : c(~)~ ~ c ~
~,
n--0
with {fn} C C ~ ( R 2 ) has also a unique solution for [ x - x o [ possesses in I x - xo[ ~_ a the uniform asymptotic expansion
~_ a.
This solution
N
y(x) ~ ~-~y,~(x)r n + 0(r
N = 0, 1, 2 , . . . ,
(3.2.20)
n'-'O
where the coefficients yn (x) (n = 1, 2 , . . . )
are determined by the recursiye set of linear
initial =lue V~oblem~ (3.2. Z) ~,d (3.ZS). Remarks 1. If we make instead of the assumption (3.2.3) oo
f(~, y; ~) ~ ~ - f ~ ( ~ ,
y),
n~0
uniformly in any bounded domain of JR2, the hypothesis oo
cO'f (x, y; e) ,',., y ~ e ~ g,.,(x, y) Oy n'--O
(3.2.21)
34
Chapter 3. Regular Perturbations uniformly in any bounded domain of R2 with {gn} C C~ then the assumption (3.2.3) holds again (see Theorem 4 of Ch.2). However, in this case the estimate (3.2.20) may be differentiated term by term with respect to x. This follows immediately from the differential equation (3.2.11) and the use of the mean value theorem, which yields t h a t also ddn-~(x; ~) = 0(6N+1), uniformly in I x - x0[ < a. Hence we have besides the expansion (3.2.20) also N
~-~Y _
~dY"" +-gz~R~ (~; ~)~ ~ d y . . ~ n=0
+O(~§
(3.2.22)
n=0
uniformly in Ix - x0[ < a. 2. In case the functions fn(x, y) do not belong to C~176 but are only differentiable a finite number of times the asymptotic approximation (3.2.20) should be terminated after a finite number of terms. This leads to the following corollary. COROLLARY 2
If the initial value problem dyo = f0(~, y0), dx
y0(~0) = c0
with fo E C1(R2) has its unique solution defined in the bounded segment I x - xo[ <_ a then the initial value problem dy = f0(~, y) + ~f~ (~, y), dx with f l E
C1(]~2)
u(~0) - c0 + o(~),
has also a unique solution in I x - xo[ (_ a and
[y(x) - yo(x)l = 0(~), u n i f o r m l y in I x - xol <_ a. The proof is left as an exercise for the reader. 3. The theorem may be generalized to the case where y(x) and f ( x , y; 6) are vector valued. The formal construction of the asymptotic expansion and the proof of its asymptotic correctness run along the same lines as above. 4. The theorem is not valid for [ x - x0[ unbounded. This follows from the fact that the iteration procedure is no longer useful. The estimates [R ('~+1) - R(m)[ , see (3.2.18), are in unbounded intervals no longer uniformly o(eg+l). The corollary above to be derived from the general Theorem 1 may also be proved directly. We give this proof for the case that y(x) and f ( x , y; ~) are n-vectors. First we derive a well-known lemma, namely Gronwall's/emma.
2. A Nonlinear Initial Value Problem Containing a Small Paramter
35
LEMMA 3 Let ~ and r be nonnegative continuous functions defined for xo <__x <_ xo + a and satisfy the relation x
qo(x) _< 51 [ r
+ 52,
(3.2.23)
Xo
with 51 and 52 positive constants. Then the function qo(x) satisfies the estimate
~(x) < 52 exp[51 f r162162
xo _~ x _< xo + a.
(3.2.24)
XO
Proof
It follows from (3.2.23) that
~(~)
_ 1,
x
51 f r162 ~o and therefore
/
xo < x _ x o + a ,
+ 52
51~)(~)(P(~)
~o 51 f r
]
d ~ < 51
+ 52
r
xo
Xo
x
or
log[51 / r
/ + 52] - log 52 < 51
Xo
r Xo
Exponentiating both sides we get 51
]
~b(~)qo(~)d~+ 52 < 52 exp[51
x0
]
~b(~)d~]
x0
and using finally (3.2.23) we obtain the result (3.2.24).
V]
We consider now the following perturbation problem dy = f(~, y) + ~g(~, y; ~), dx
y(~o) = yo
(3.2.25)
with y, f and g vector valued and f continuously differentiable in Ix0, c~) • ][r and g continuous in [x0, (x)) • ]R~ • [0, 60]; further as well f as g are Lipschitz-continuous in y with Lipschitz-constants independent of x and r THEOREM 4 Let y(x) be the solution of the initial value problem (3.2.25) with f and g satisfying the conditions stated above.
36
Chapter 3. Regular Perturbations
Moreover, let z(x) problem
with xo <_ x <_ xo + a be the solution of the reduced initial value
dz -- f ( x , z), z(xo) -- Yo, dx then there exists a positive constant xl with xo < xl < xo + a such that l y ( x ) - z(x)l = O(e), u n i f o r m l y in xo < x < xl
(3.2.26)
(3.2.27)
n
(lyl 2 -
~y~,
etc.).
i--1
Proof Let us choose an arbitrary compact and convex set 79 in [x0, cr x ]I~ with (x0, Y0) in its interior. According to the existence theorem for solutions of ordinary differential equations there exists a segment [x0, xl(e)] such that y(x) satisfies (3.2.25) for x0 < x _ xl (e) with y(x) E l) and such that z(x) satisfies (3.2.26) for x0 < x < x0 + a with z(x) E 1). Because of the continuity of g(x, y, e) in 1) x [0, e0] there can be chosen a number Xl independent of e such that y(x) satisfies (3.2.25) for x0 < x < xl, see [74], Chapter III, 3.1 and 3.3. We put y(x) = z ( x ) + w(x), and we take xl such that x0 < x < Xl < x0 + a. The function w satisfies the initial value problem
dw dx = { f ( x , z(x) + w(x)) - f ( x , z(x))} + ~g(x, z(x) + w(x); ~)
of
= Oz (x, z(x) + O(x). w ( x ) ) w ( x ) + eg(x, z(x) + w(x); e),
(3.2.28)
where o/ is the n x n-matrix with entries o_/x 0 < Oj(x) < 1 and the initial condition Oz Ozj ' is W(Xo) = O. Because we have taken 1) as convex, the argument (x, z(x) + O(x). w(x)) also belongs to 1) since y(x) and z(x) belong to 1). Due to f E C1(1)) we have that
Of
z + ~ w)ll
" ~ jl
Ofi (x, z-+-0.w)[
i----1 "is
bounded by a positive constant C 1 ,
and due to g E C(1) x [0, ~0]) we have also
Ig(~, z + ~; ~)1 < c~ with C2 another positive constant, independent of ~. Integrating the differential equation (3.2.28) we get with the aid of Cauchy's inequality
Of (~, z(~) -+- 0(~). w(~))w(~) A- ~g(~, z(~) A- w(~); ~)}d~[ Xo .ill?.
< c , / I w ( ~ ) l d ~ + ~C2(x~ - xo), XO
xo <_ x <_ x~.
37
3. Applications Applying finally Gronwall's lemma there results the a priori estimate
~ EC2(x 1 - - X 0 ) e x p [ C l ( x - x0)],
]w(~)]
x0 _< x _< X l ,
which yields ly(x) - z(x)[ = 0(6), uniformly for x0 < x <
3. 3.1
X 1.
Applications Dufllng Equation We consider the initial value problem for the nonlinear spring d2x
dt 2
-~- X -Jr-C X 3
=
O,
x ( O ) - 1,
0 < t < CXD, 0 < ~ << 1, dx
-~- (0) = O,
(3.3.1) (3.3.2)
and we construct an approximation of x ( t ) for small values of ~ by using the regular perturbation method. This initial value problem is easily brought into the form (3.2.1)(3.2.2) with y ( x ) and f ( x , y; c) two-vectors, so that we may apply Theorem 1, Remark 3 or Theorem 4. We simply put x - zl and -~dx = z2, and the equations (3.3.1) and (3.3.2) become: dzl dt = z2 dz2 = _Zl dt
w gZ31
with Zl(0) = 1, z 2 ( 0 ) - 0 . From the foregoing section it follows that the approximation can only be uniformly valid in bounded t-intervals, say 0 _< t < b, with b independent of ~. So we make the "Ansatz""
9(t) ~ ~o(t) + ~ ( t )
+...
+ ~'~n(t) + . . .
(3.3.3)
Substituting this into (3.3.1)-(3.3.2), and collecting equal powers of ~ we obtain a linear system of recurrent initial value problems for x i ( t ) , i - O, 1, 2, . . . . d2xo + Xo = 0 dt 2
x o ( O ) - 1,
dt (0) = 0
d2xl dt 2
}
(3.3.4)
} ~- x l = - x 3
(3.3.5)
~(o) =o,
dxl
--z-(o) = o
38
Chapter 3. Regular Perturbations
Solving these problems consecutively we obtain in any bounded t-segment
xo(t) = cost, x l (t) =
3_t sin t +
-8
and hence
x(t) = cost + e [ - 3 t s i n t + 8
1
~ 1
(cos 3t - cos t)
(cos 3t -
'
cost)] +
0(e 2)
(3.3.6)
uniformly in any bounded segment 0 _< t _< b. The t e r m - ~3t sin t shows clearly t h a t the expansion (3.3.6) can have only a meaning, whenever t is bounded. W h e n t = 0(1) the second t e r m is no longer 0(e) and when t = 0 ( ~ ) the second t e r m blows up. Terms of this type are called secular terms. Because it is not possible to give a regular asymptotic expansion uniformly valid in the whole time interval we call the initial value problem (3.3.1)-(3.3.2) a singular
perturbation problem. The interval considered in this example is the hairline t > 0. In C h a p t e r 5 we extend the theory in such a way t h a t we obtain an approximation for x(t) with error 0(e), uniformly valid in an interval [0, be-i], with b an arbitrary number, independent of e.
3.2
The Motion of a Planet around the Sun
In this subsection we study the motion of a planet under the influence of the m u t u a l force of gravitational attraction. It follows from mechanics t h a t the planet moves in a fixed plane and we introduce in this plane polar coordinates (r, 0) with the center of the sun as origin
P
J O Figure 3.1 We neglect the influence of the other planets on the motion and we use Newton's law, viz:
Mm F = G ~ r2
'
where F is the force of attraction, G the constant of gravity, and M and m the masses of the sun, respectively the planet. An elementary calculation yields for the
39
3.2. The Motion of a Planet around the Sun radial and the transverse components of the acceleration the formulae:
ar
d2r dt 2
(dO) 2 r --~
1 d(dO) a # - - r - ~ r2
and so it follows from Newton's law of motion:
d2r dt 2
(dO) 2 Mm r -~ - -G r 2 , r2
r dt
-~
(3.3.7)
-- O.
(3.3.8)
Integrating the second equation we obtain the second law of Kepler, i.e.
r 2dO -~- = h = constant,
(3.3.9)
which means t h a t the angular m o m e n t u m of the planet is constant. We put 1
~(~) = ~(~),
where r - r(0) denotes the trajectory of the planet. We get the following relations
du _ du ~ d O _ dO dt / dt and
1 dr / h _ r 2 dt / -r 2
d2u d du ~ d O dO 2 - dt ( - ~ ) -~ =
1dr h dt ,
l d2r/ h, h-dr ~ -~
or with the aid of (3.3.7)"
d2u dO 2
Mm h2
1 r
or
d2u Mm +u=G~ dO 2 h2 9 Finally, we introduce a characteristic distance ~ and the dimensionless variable - ~u; ~ satisfies the differential equation
d2~t dO 2
- -
with a = G Mm~ h 2
-~-fi=
a
(3.3.10)
9
The solution is readily obtained and we get the result = a + ( b - a)cos(O - O0),
(3.3.11)
40
Chapter 3. Regular Perturbations
where the constants b and ~0 are determined by the initial conditions, which specify the position and the velocity of the planet when e.g. v9 = 0. It follows t h a t r(0)
is given by
r(vq) = a + (b - a)cos(~ - 0o)
(3.3.12)
and the trajectory of the planet is for b < a a fixed ellips with the centre of the sun at one of its loci. However, observation has revealed t h a t the major axis of the ellips rotates slowly around the sun and so the result (3.3.12) is not completely correct. It was a verification of Einstein's theory of gravity that this rotation could be explained by using the theory of general relativity and the so-called Schwarzschild metric (see e.g. [7], Ch.XIV). In this theory the equation (3.3.10) should be replaced by d2'5 d,02 t- '5 - a + ~'52
(3.3.13)
with e---- 3 Gc2e M m , c the velocity of light, and so 6 is a small positive number. Therefore we are led again to a p e r t u r b a t i o n problem which can be handled in the same way as the nonlinear spring. The equation (3.3.13) is supplemented with the initial conditions d'5 '5(0) -- c~, ~-~ (0) -- ,6. (3.3.14) Applying the theory of this chapter we make the assumption
~(~) = u0(O) + ~ , ( ~ ) + . . . + ~"u.(O) + . . .
(3.3.15)
Substitution into (3.3.13) and collecting the terms with equal powers of e we get a linear system of recurrent initial value problems for the functions ui(tg), i - 0, 1, 2 , . . .
d2uo d0---5- + u0 = a,
u0(0) - c~,
duo - - ~ ( 0 ) --/3
(3.3.16)
d2ul dO----5- + ul = u20, ul(0) = 0,
dul --d-~(0) = 0
(3.3.17)
Solving these problems consecutively we obtain: (3.3.18)
Uo(O) : a + (o~ - a)cos vq + ~ sin z9 d
d
ul (~) = sin 0 / [ u o (~o)]2 cos ~od~o- cos ~ / [ u o ( ~ ) ] 2 sin ~d~o o
o
= a(c~ - a)O sin 0 - afh9 cos 0 + . . . .
(3.3.19)
Exercises
41
According to Theorem 3.3.1 we have the result
~(~) = ~0(~) + ~
(3.3.20)
(~) + 0(~ ~)
uniformly in any bounded domain, say o < ~ < t~b, with Zgb independent of ~. We see again the appearance of "secular" terms ~)sin zg, t9 cos~9 in the contribution cul (v~), and thereforethe expansion (3.3.20) is only meaningful in bounded ~9-intervals. In the next chapter we explain as a consequence of (3.3.20) the perihelium rotation and in Chapter 5 we extend the range of uniform validity. Exercises
1. Let y(x; ~) be the solution of the initial value problem
dy =l+y2+r dx
x>0,
y(0)=r
0<~<<1,
valid in a right neighbourhood of x = 0. Give an approximation of y(x; ~) within an error O(~ 2) and its interval of uniform validity. 2. Let y+ (t; c) be the solution of the initial value problem
d2y+ + dt 2
dy+ --~+y+
dy+ =0,
t>0,
y+(0)=0,
--~(0)=1,
with 0 < ~ < < 1 . Give by using the regular expansion method an approximation of y+ (t; c) within an error O(~ 2) and its t-interval of uniform validity. Discuss the reason why the result is not satisfactory. 3. Consider the Duffing initial value problem with a small forcing term
d2u dr----~ + u
--
- f u
3
-~-eA cos t,
t > O,
u(O) -- 1,
du
-d-~-(O)= O,
0 < ~ ~ 1.
For which values of the amplitude A is the resonance only observable in the third term of the asymptotic approximation?
This Page Intentionally Left Blank
Chapter 4 THE METHOD OF THE STRAINED COORDINATE 1.
Introduction
In the preceding chapter we showed that the naive regular perturbation method has the severe restriction that it is only applicable for finite time intervals and so we cannot construct by this method asymptotic approximations uniformly valid in "large" time intervals, that is to say for intervals of e.g. O(~) or even for all values of t > 0. This becomes apparent in the occurrence of secular terms; see the examples of the nonlinear spring and the motion of a planet. Nevertheless, by using a modification of the regular perturbation method involving a stretching of the time coordinate as t-
( 1 + ewl + e2w2 + ''')T
(4.1.1)
it is possible to determine the constants {wi} such that the solution of the perturbed differential equation does no longer contain secular terms; in this way one obtains an asymptotic expansion valid for time intervals of O(~). With the aid of this modification it is also possible to obtain information on certain important quantities such as for instance the frequency of the nonlinear spring and the perihelium precession of the planet. The method of straining the coordinate originates from Lindstedt [98, 99] and Poincar~ [116], who introduced it in studies of perturbation problems in celestial mechanics. Therefore it is also called the Lindstedt-Poincard or shortly the L.P.-method. We illustrate the method by treating again the nonlinear spring and the motion of a planet around the sun. The question of the asymptotic correctness of the Lindstedt-Poincard procedure will be dealt with in the next chapter. We conclude this chapter with a procedure introduced by Lighthill [96, 97], who generalized the stretching transformation (4.1.1) by setting t -- T + e f l ( T ) + e2f2(T ) + . . . , (4.1.2) where the stretching functions {fi} are chosen such that an asymptotic expansion of the solution of the perturbation problem becomes possible. This transformation appears to be useful for a class of perturbation problems for which the reduced differential equation with e - 0 contains a singularity, e.g. (t -1t- "CU)du + q ( t ) ~ = ~(t), t > o.
44 2.
2.1.
Chapter 4. The Method of the Strained Coordinate o f the Strained C o o r d i n a t e
Applications of the Method
T h e N o n l i n e a r Spring We consider the initial value problem d2x
dt 2 t - x + ~ x 3 = 0 ,
t>O
(4.2.1)
x(0) = a and -~- (0) = 0.
(4.2.2)
w i t h the initial conditions dx
We introduce the following transformation of the independent variable: (4.2.3)
t = (1 + ewl + e2w2 + " ' ) %
where the w~, i - 1, 2, 3 , . . . , are as yet undetermined constants to be chosen in such a way that the secular terms in the asymptotic expression for x(t) disappear, when x is represented as a function of r. Substituting (4.2.3) into (4.2.1) we get the initial value problem for x as function of T : d2x dT----~ -~- (1 + ewl + g2w2 -~---')2(x -[- Ex 3) -- 0,
T > 0,
(4.2.4)
dx
x(0) = c~,
~--~T(0) = 0.
(4.2.5)
This problem is again of the form treated in the preceding section, whenever 1 + ewl + e2w2 + " "
(4.2.6)
is asymptotically convergent. Under the assumption that this is the case, we may apply the regular perturbation method, valid for finite T intervals. So we expand x(~-) as
~(~) = ~0(~) + ~ ( ~ )
+ ~(~)
(4.2.7)
+-
Inserting this expression into (4.2.4)-(4.2.5) and taking together equal powers of e, we obtain the following recursive set of linear initial value problems: d2xo a~---3- + ~o = o,
dxo -~j(o)
~o(O) = ~,
d 2x 1
dT_-----Y -f- xl = -(Xo3 + 2wlxo),
(4.2.8)
= o, dx 1
xl(0) = 0,
--~-T(0) = 0,
(4.2.9)
d2x2 dT---Y + x2 = --3x2x, -- 2w1(x1 -[- XO3) -- (2~2 -[-W12)Xo, dx2 -jj~ (o) = o
~ ( o ) = o, .
.
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(4.2.10)
45
2.1. The Nonlinear Spring The solution of (4.2.8) reads x0(r) = a c o s T
(4.2.11)
and we obtain for Xl(T) the initial value problem:
d2xl d ~ + ~1 = - { ~ ( ~ o s ~) ~ + 2 ~ 1 ~os~} C~3 -
- c ~ , , (_3 4 62
+ 2wl) cos T -- --4 cos 3T.
The first term of the right-hand side gives rise to the appearance of a secular term in x l ( T ) unless we take 3 O~2 . (4.2.12) CO1 DE 8
Taking this value for W1 we obtain: C~3
Xl(T) = ~--~(cos 3T
-
COST).
(4.2.13)
Substitution of (4.2.11), (4.2.12) and (4.2.13)into (4.2.10) yields for X2(T) the initial value problem:
d2X2dT 2+
(~57
x2 = a ~ l ~ g
~ 4 _
2w2)COST
+
N.S.T.,
(4.2.14)
where N.S.T. denotes the terms which will not lead to a secular contribution in x2(T); so N.S.T. does not contain a term such as cos T or sin T. Avoiding again the appearance of secular behaviour of X2(T) we should take 57.64" 256 In principle this process may be repeated indefinitely; up to the second order we have 092 ---
OL3
X(T) = a cos T + C~-~ (COS3T -- COST) + 0(~2),
(4.2.15)
uniformly valid in any finite T interval 0 _< T _< TO with TO independent of ~, where the strained coordinate T is given by T = (1 -
57 o4c 2 -~-'-' ) - i t .
c~2~ + 256
(4.2.16)
However, it should be noted that the result (4.2.15) is only mathematically justified if the series (4.2.6) converges asymptotically. Assuming for the moment that this is the case we obtain for the circular frequency of our nonlinear spring w = 1 + _3 862~-
21 O4e 2 -~-0(e3). (4.2.17) 256 We remark that this frequency depends on the initial amplitude a and of course also on The expression (4.2.15) still makes sense for large values of T and in the next chapter we shall prove ~'.
x(t) = a cos{(1 + g3626)t} + 0(~), uniformly valid for 0 < t < to with to a positive number, independent of ~.
(4.2.18)
46 2.2.
Chapter 4. The Method of the Strained Coordinate The Perihelium
Precession
The differential equation for the trajectory in polar coordinates planet reads according to Example 2 of the preceding chapter:
r - r(0)
of the
d2u dO---~ + u = a + eu 2,
0>0
(4.2.19)
with u = ~ and ~ is a characteristic length; a is a positive constant and e the small parameter. The initial values are chosen as
du
(4.2.20)
u(O) = cr and ~-~(0) = ft. We introduce again the strained coordinate
(4.2.21)
0 = (1 + ewl + e2w2 + ' " ) 9 ~ , and the initial value problem becomes d2u + (1 + COdI + ~2~d 2 ~ - ' " .)2 (U -- a -- Cu 2) ~- 0 d~p2
(4.2.22)
with
du
u(0) = a and - 7 ( 0 ) = fl(1 + ewl + e2w2 + ' " "). aT
(4.2.23)
Under the assumption t h a t the coefficients wi will be chosen such t h a t (1 -Jr CO)1 -~ E2W2 - ~ ' - . ) is asymptotically convergent we are again in the situation t h a t the regular p e r t u r b a t i o n m e t h o d may be applied. So we set
u(~) = ~0(~) + ~Ul(~) + ~ ( ~ )
+...
(4.2.24)
Substitution into (4.2.22) and (4.2.23) yields again a recursive system of linear equations: d2u______oo_ u0 = a; dT 2
u0(0) = a,
d2ul d~ 2 + u~ = u 0~ - 2 ~ 1 ( u 0 - ~ ) ;
duo - ~ - (0) = fl, ~(0)
= 0,
(4.2.25)
dltl
-d~ (0) =Wlfl.
(4.2.26)
The solution of (4.2.25) is simply obtained UO(T) = a{1 + e cos(qo -- (P0)},
(4.2.27)
where e and ~o0 are determined by a and /3. The function r0(T) = ~o(~) 1 is an ellips, if e > 0, with the origin as one of its loci.
47
3. The Method of the Strained Parameter
Substitution of u0(cp) in the right-hand side of the differential equation (4.2.26) yields
d2Ul d~o------5-
+
u:
--
a2(1 + l e 2 ) + 2ae(a
-
w:)cos(~o
-
~Oo)
+ l a 2 e 2 cos 2 ( ~ 0 - qo0).
2 This equation gives again rise to a secular term unless we take wl - a.
(4.2.28)
Therefore we get
u = a + ae cos(~o - ~o) + 0(:), uniformly in any bounded ~ interval 0 ~ ~ < 9~0 with q:, = (1 + a: + w2:2 + . . . ) - 1 0 . This process may, in principle, be repeated indefinitely and the asymptotic approximation is valid if the series 1 + a : + 022:2 -~- "'" is asymptotically correct. Assuming the latter condition we have finally u(O) -- a + ae cos{(1
-
ae)O
-
0o} + 0(:),
(4.2.29)
uniformly valid for 0 < 0 _ t~b with 0 b a n arbitrary positive number independent of In the next chapter we shall prove the asymptotic correctness of (4.2.29), which is even uniformly valid for 0 < 0 < o__~bwith ~qb again some positive number, independent of :. The peri- and aphelium of the planet is obtained for the values 0 -- Op with du
--0
(u is maximal, respectively minimal)
dO
i.e. 0p satisfies approximately the relation sin{(1 - a : ) O p - 00} = 0, and it follows that (1 - a:)vqp -- t90 + 7rn, n = 0, 1, 2. Two consecutive values of the perihelium angle : ~ ~ 21r + 21ra:, and so the perihelium precession is approximately given by differ y 2, the formula G2M2m 2 2~ra: - 6 1 r ~ . (4.2.30) c2h 2
3.
T h e M e t h o d of t h e S t r a i n e d Parameter
It may happen that in a differential equation, apart from the small parameter e, another parameter, say p, appears which depends on :, but this dependence is quantitatively unknown. The requirement of a certain behaviour of the solution of the differential equation, e.g. no secular terms should be present, may lead to an asymptotic
48
Chapter 4. The Method of the Strained Coordinate
expansion of the p a r a m e t e r p into powers of e. The m e t h o d for obtaining an asymptotic approximation for as well the solution as for the unknown p a r a m e t e r is sometimes called the m e t h o d of the strained parameter. We give as an illustration the following example. Stokes [147, p.486] considered in 1849 weakly nonlinear wave propagation on the surface of a liquid. His model is one dimensional and the height u ( x , t) of the fluid measured from equilibrium satisfies an equation of Klein-Gordon type utt - o~2uxx -~- ,),2~t = / ~ u 3,
(4.3.1)
where a, /~ and 7 are certain constants. We are interested in obtaining the so-called d i s p e r s i o n r e l a t i o n for p e r i o d i c t r a v e l l i n g waves; the dispersion relation relates the wave number and the frequency. The linear case with fl = 0 is almost trivial. Putting u = u(~), ~ = x - ct, (4.3.2) with c > c~ > 0 the wave velocity, we get for u(~) the ordinary differential equation: (c 2 - a 2 ) u '' + 72u = 0, and it follows t h a t
u(~) is a linear combination of sin{ ~/c 27--0~2 ( x -
ct)}
and
c o s { ~/C . . . 72 -- Ot2 ( x - ct)}, and so the wave number is given by k=
X/C
7
-- C~ 2
and hence w 2 = k2c 2 = a2k 2 + 72,
(4.3.3)
which relates the wave number k and the frequency w. It is to be noted t h a t this relation is independent of the amplitude of the wave. We consider now the full nonlinear equation, which reads (C 2 -- O~2)U t' ~- ')'2U = ~ U 3
(4.3.4)
with u = u(~) = u ( x - ct). We assume t h a t the amplitude a of the wave is small and so we put u = aft
(4.3.5)
(c 2 - a2)~" % 72fi - ~a2fi 3,
(4.3.6)
and there results and this would present again a p e r t u r b a t i o n problem of the kind considered before with a 2 instead of e, whenever the wave velocity were to be independent of the amplitude. But as is well known from nonlinear wave p h e n o m e n a the speed of the wave depends on the amplitude. Because a is small we expand c into powers of a C -- CO + acl A- a2c2 + ' ' ' ,
(4.3.7)
49
3. The Method of the Strained Parameter where the coefficients c~ should be chosen properly with co > c~. We bring (4.3.6) into the form ~" U
-V2~2 + ~a2u3 --
c2 _
(4.3.8)
ol 2
and we expand c2_1~2 into powers of a. If the series (4.3.7) converges asymptotically we are again in the situation of C h a p t e r 3 and we may apply the regular p e r t u r b a t i o n method. We assume now t h a t (4.3.7) is indeed asymptotically correct and we put (4.3.9)
~(~) = ~1 (~) + a~2(~) + a2~3(~) + " " or what amounts to the same
u(~) = aul(~)+ a2u2(~) + a3u3(~) + " "
9
(4.3.10)
Instead of substituting (4.3.9) and (4.3.7) into (4.3.8) and taking together equal powers of a, we may as well substitute (4.3.10) and (4.3.7) directly into equation (4.3.6). Taking together equal powers of a, there results the following recursive set of linear equations for the functions ui(~), i = 1, 2 , . . . .
(~] _ ~2)~, + ~ 2 ~ = 0
(4.3.11)
(~ - ~)~
(4.3.12)
+ ~
(~0~ - - ~ ) u g + ~
= -2~o~ (-2~o~
--
--
C l~' )U
"1
--
2CoClU
"2
-'k
Z~.
(4.3.13)
We start with a particular solution of (4.3.11), viz. (4.3.14)
~1 = cos(k~) with k2 _ -
,)12
c2 _ a2 or c~ = c~2 + ~'2k-2,
(4.3.15)
which corresponds with the dispersion relation (4.3.3) for the linear case with fl = 0. We substitute Ul(~) into (4.3.12) which gives
,
2c0cl k2
u~. + k2u2 = Co ~ _ ~.
cos(k~).
Searching for waves a2u2 of small amplitude we should eliminate secular terms and so we must take Cl - 0 and the equation for u2 becomes the same as t h a t for ul 9 The equation for u3 becomes now
,,
U3
~
k2u3
--
c~
-
1 oz 2 (2coc2 k2 +
p) cos(k~) + 4(c2 ~- c~2) cos(3k~) "
Eliminating the secular t e r m again we should take 3
3p
4
8cok2
2coc2 k2 -~- -:~ = 0, or c2 --
(4.3.16)
50
Chapter 4. The Method of the Strained Coordinate
and the differential equation for u3 reduces to u3" + k~u3 = 4 ( ~ -f~
~)
cos(3k~).
(4.3.17)
Under the condition that (4.3.7) is asymptotically correct, i.e. c - c o -4- a2c2 + 0(a 3) - Jo~ 2 A- ~,2]~2 _
3~a2 8k 2 J ~ 2 ~_ ~,2k-2 -4- O(a 3)
we have obtained the approximation u ( x - ct) = a cos k ( x - ct) + O(a2).
(4.3.19)
with the dispersion relation = kc = V/c~2k2 + 72 -
3f~a2 + O(a3). 8 j ~ 2 k 2 A- ,72
(4.3.20)
The dependence of w on the amplitude appears to be only quadratic. It is not difficult to integrate (4.3.6) directly; multiply with fi', integrate and one obtains u as an elliptic function of ~, (cnoYdal wave). Taking the amplitude small one arrives at a result similar as (4.3.20); see also [147], p.486-489. 4.
Lighthill's method
An interesting class of perturbation problems concerns the case where the reduced equation with ~ = 0 is singular while the perturbed equation is not singular. A typical perturbation problem of this type is (x + r
du + q(x)u = r(x), -~x
0 < x _< 1,
(4.4.1)
with u ( 1 ) = b and q and r smooth functions, q ( O ) # O. Using the Poincar6-Lindstedt method the uniformity of an asymptotic expansion does not hold in a neigbourhood of x-O. Putting x = (1 + eWl + ~20.'2 + ' " ")~
~(~) =
~0(~) + ~u~ (~) +
~ ( ~ ) +...
leads to the recursive set of singular linear differential equations ~--~
+ q(~)u, = f,(~,
uo, u l , . . .,
Un--1)~
with solutions not suitable in the expansion of u(~) in the neighbourhood of ~ = 0. To illustrate this situation consider the very simple example (z + r
-du -~x+U--O,
O<x
u(1)---1,
0
(4.4.2)
4. Lighthill's method
51
This equation may be written as a H a m i l t o n - s y s t e m
dx
.-7 = (x + 6u), qa
du
,-7 = - u , ga
u(1) = 1
(4.4.3)
with the solution
or
- x + Vrx 2 + 26 + e 2
u =
.
(4.4.4)
6
This solution is regular in x for all values of x with x _ 0 and for all positive values of 6, but it has only for x > 5 > O, with 5 independent of 6, the asymptotic relation u - sx + 0 ( 6 ) . Let us now make the straightforward expansion
= ~o(~) + ~ ( ~ )
(4.4.5)
+....
Substitution into (4.4.2) yields after equating coefficients of equal powers of e o and 61"
duo
x--~x + u 0 - - - 0 ,
dul
u0(1)=l,
duo
x - ~x- + Ul = - u o - ~ -x ,
Ul(1) = O,
and therefore u0=-
1
andul=
1
( 1 - x -2)
The t e r m u0 is singular and Ul is even more singular at x -- 0 and so the regular expansion procedure is useless in the neighbourhood of x - 0. The stretching of the coordinate x according to x = (1 + ~W 1 ~- 82W 2 - ~ ' " ")~
(4.4.5)
is, of course, of no help since x is only multiplied by a factor of O(1). Lighthill [96, 97] devised a generalization of the coordinate stretching (4.4.5) such t h a t the singularity no longer occurs in the expansion. Instead of the simple linear coordinate transformation, he introduced the transformation (4.4.6)
- ~ + ~f~(~) + e2f2(~) + . . . where now the straining functions fi(~) of class C ~
~(~) = ~0(r + ~ ( r
+ ~2(r
are chosen such t h a t as well
+..-
(4.4.7)
as (4.4.6) become b o t h uniform expansions in the domain considered, in our example 0 < x < 1. This is guaranteed by the requirements t h a t
um(~c) and fm(~c) u~_~(~) f~_~(~)
(4.4.8)
52
Chapter 4. The Method of the Strained Coordinate
are uniformly bounded in the ~-interval corresponding with the x-interval [0, 1]. Despite the fact that the method is cumbersome and involves many tedious calculations, it has been applied in a large variety of perturbation problems concerning wave propagation; we refer the reader to an extensive list in the book by Nayfeh, [108, p.78, 79]. Therefore we illustrate the procedure using the simple example above, which was also given by Tsien [135]. Substitution of (4.4.6) into (4.4.2) gives
{(: + :::(:) + :s,(:) +.-.) + : u }du: + df l ..c2df 2 {1 + e-d-~-(~) + -d-~-(~)+ . . . }u = o,
(4.4.9)
and substitution of (4.4.7) into (4.4.9) yields as before a recursive system o f / / n e a r differential equations: d d~ (~u0) = 0 d
(4.4.10) d
~(~1 = -~(S~o)-
duo
~0 d~
(4.4.11) To get the boundary conditions it is necessary to invert the expansion (4.4.6); we have to find the expansion of ~ = ~*, the value of which corresponds with the value x - 1. In other words, we should construct a root of the equation 1 = ~* + 6fl (t~*) + e2f2(~ *) + - - "
(4.4.12)
9
Its approximate solution is obtained with the aid of the regular perturbation method, i.e. we assume ~* = 1 + ble + b2e 2 + " " (4.4.13) and insert this into (4.4.12): (bl~ + b2~2 + - - ' ) q- {~.fl(~*) q- ~2f2(~*) + ' " "} = 0.
(4.4.14)
Expanding fi(~*) into a Taylor expansion around ~ = 1 we have f~(~*) -- fi(1) + f~(1)(bl~ + b2c 2 + . . . ) + 1f~'(1)(b16 + b262 + " ' ) 2 + . . . and (4.4.14) becomes (bl~ + b2~2 + ' " ") + ~{fl(1) + f~(1)(bl~ + b2~2 + ' " ") + ' " "}
+ ~{f~(1) + A(1)(b: + b: ~ +
) +.--}
+ ................................... --0.
,
53
4. Lighthill's method Taking finally equal powers of 6 together we get bl = - f l ( 1 ) b2 -- - b l f ~ ( 1 ) - f2(1) - fl(1)f~(1) - f2(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and so there results formally ~* = 1 - efl(1) + e2{fl(1)f~(1) - f2(1)} + " "
(4.4.15)
9
The boundary condition ux=l = u(~*) = 1 becomes
uo(~*) +
eUl (~*) + e2u2(~c*) + . . . .
or
e2(fl(1)f;(1) -
uo(1) + u~(1){ - efl(1) + 1 + ~u~(1){ - ell(l)+
1
f2(1)) + ' "
}
e2 (fl(1)fl(1) - f2(1)) + " "
}2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ C[Ul(1) + u~(1)( - e l l ( l ) + c2(fl(1)f;(1) - f2(1)) + ' "
}
1 "(1)( - efl(1)+ e2(fl(1)f~(1)- f2(1)) + . . . }2 + ~ul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+
", . . . . . . . . . . . . . . .
(1) + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ .....................................................
]
]
1.
Taking again equal powers of c together, we get the recursive set of boundary conditions pertaining to (4.4.10), (4.4.11 ) .... u0(1) = 1,
Ul(1) = u~)(1)f(1), . . . .
(4.4.16)
The solution of (4.4.10), (4.4.16) is 1
u0(~) = ~,
(4.4.17)
and the boundary value problem for Ul(~) becomes according to (4.4.11) and (4.4.16)
d
1
1
1 dfl
d--~(~ux) = ~-5 + ~-~fl(~)- ~ - ~ ( ~ )
(4.4.18)
u~(1) = - f 1 ( 1 ) .
(4.4.19)
with From the equations (4.4.10)-(4.4.11) it follows after multiplication of the first with Ul and of the second with u0 and subtraction that
d(ul)
l(dfl
fl
1)
54
Chapter 4. The Method of the Strained Coordinate
The stretching function fl
is still free, but the requirement (4.4.8) induces
dfl
1
1
(4.4.20)
d~ We take the solution of this equation as simple as is possible and we choose 1
1
fl(~) = ~ ( ~ -
~).
(4.4.21)
The b o u n d a r y value problem (4.4.18)-(4.4.19) for Ul has now only the trivial solution u l - 0 and we have, finally, constructed in a formal way: 1 u(~) = ~ -~- e2u2(~) -~- .-.
(4.4.22)
with x:~+
e(~-~)+e2f2(~c)+....
(4.4.23)
Solving ~c from the latter equation, neglecting the third t e r m in the right-hand side, we get x + ~/x 2 + 2E + ~2 = , (4.4.24) 2+~ where we have taken the positive root, because x = 1 corresponds with ~* = 1 + O ( e 2 ) . The m a p (4.4.24) is a 1 - 1 m a p from x +-> ~c and it is graphically presented in Figure 4.1 and the segment O_<x_< 1 is m a p p e d onto the segment
~ ~_ <~ <~* _ ,I V
/
Fig. 4.1 The graph of x = x(~C) 9~C0-- ~/ 2ue v Inserting (4.4.24)into (4.4.22) there results 1 u(~) ~ ~ = ~ + J~
2 + ~ + 2~ + ~
- x + ~/x 2 + 2e + ~2 =
5. Temple's method
55
which equals the exact solution (4.4.4) of (4.4.2). Hence the transformation 1
1
- ~ + ~ ( ~ - ~),
u(~)=
1
yields the right solution of our perturbation problem (4.4.2), uniformly valid for 0 __ x _< 1, and Lighthill's technique yields already the solution after an expansion up to O(~); apparently f i ( ~ ) = 0 and u i ( ~ ) - O for i > 2. Because Lighthill's method has been widely applied to a large variety of perturbation problems we have chosen to give a detailed treatment of a simple problem. Our derivations have been formal and we do not give a proof of the validity of the method; we remark only that Wasow [143] has investigated conditions for the convergence of Lighthill's expansion for the solution of the problem (4.4.1). 5.
Temple's method
A method of approximation closely related to that of Lighthill has been given by Temple [132]. Consider the perturbation problem f (x, u;-c) du - g(x,u;~),
X < Xl,
U(Xl)-C, 0 < c < < l .
(4.5.1)
Suppose that it is possible to replace the differential equation by the equivalent system dx = X ( x , u; ~), d~
du -w~ = U(x, u; c), a~
(4.5.2)
where X and U have uniform asymptotic expansions in powers of e with C ~ coefficients in x and u. Then it is possible to apply the regular approximation procedure as discussed in Chapter 3, and in this way one obtains in a systematic way the functions {ui(~)} and {fi(~)} appearing in Lighthill's expansion. The possibility to split the differential equation in the form (4.5.2) was already present in the example of the preceding section, see (4.4.2) and (4.4.3). Here we give another example, but this one does not lead to a simple integrable Hamiltonian system. We consider
(x + cu) du dxx + ( 2 + x ) u = 0 '
(4.5.3)
u ( 1 ) - e -1.
From this equation it follows that u(x) > 0 for 0 _< x _< 1, and so the differential equation does not contain a singularity for 0 _< x _< 1. (4.5.3) is equivalent with the equations
~ dx -~+~nd~
du
=
- ( 2 + x)u,
Ux-- 1
-
-
e -1,
~ ~: 0.
(4.5.4)
Expanding
= ~ + ~I~ (r + o ( ~ ) , = ~0(~) + ~u, (~) + o ( ~ ) ,
(4.5.5) (4.5.6)
56
Chapter 4. The Method of the Strained Coordinate
we get the following differential equations
dYl
~-d-~- -- f l ( ~ ) + u0(~),
(4.5.7)
duo : ~-d~-
- ( 2 + ~)u0,
(4.5.8)
~dul -~=
- ( 2 -]-~)tt 1 -- fl (~)u0(~).
(4.5.9)
The solution of (4.5.8) with the condition uo(1) = e -1 gives 1
~0(~) :
~2
(4.5.10)
~-~"
Inserting this expression into (4.5.7) we pick for fl(() the solution
fl (~) -- ~
dr/= ~ 1
~dr/.
(4.5.11)
1
The results (4.5.10) and (4.5.11) determine now the equation (4.5.9) with the boundary value ~U 1(~*) : e -1 -- ItO(~*),
(4.5.12)
1 = (* + ell (~*) + O(e2).
(4.5.13)
where ~* satisfies the equation
The result is u:(:) =
- ~1 e- ( f
(77)dr/+
fl
e - I __ UO(~*)
However c -1 --
--
e
)(=1(~
--
1) , ~
-3e-
e/l(~*),
and so with the aid of (4.5.11), (4.5.13) and ~* = 1 + O(e) 1 _~ f fl(r/) Ul(~) = --~-~e r/ dr/+ O(e). 1 The final result we have obtained reads now as follows: x -- ~ A- ~ f l (~) -f- O(~2),
1 _~ (1 -- e J
U = ~--~e
1
f l (r/) r/
dr/) -~- O(g'2),
(4.5.14)
6. Limitations of the Lindstedt-Poincar~ method
57
with
.f~ (r = ~ f --~-dn, e-~
1 or
x - ~(1 + e~(~)) + O(e 2)
(4.5.15)
1 -~(1 - e / u = ~-se
(4.5.16)
O(~)drl) + O(e 2) 1
with ~(~) -
f --~e--rbdr/.
(4.5.17)
1
(4.5.15)-(4.5.16) gives a parameter representation of the solution u(x) of the initial value problem (4.5.3) for all values of ~ _> 5 > 0, with 5 arbitrarily small, but independent of e. The approximation contains the secular terms eO(~) and e f : ~)(r/)dr/. 6.
L i m i t a t i o n s of t h e L i n d s t e d t - P o i n c a r 6
method
The Lindstedt-Poincar6 method treated in this chapter has the advantage that secular terms may be eliminated from the expansion of the solution and the method has been successfully applied to vibration and wave problems described by differential equations of the type
d2x
02u
dt 2 + w2x -- ~ f ( x ) or - ~
_ c 2 02u
~ x 2 -- ~ f ( u ) .
However, there may arise difficulties, whenever a small damping term is introduced into the equations, even in the linear case. Let us consider as an illustration the simple initial value problem
d2x dx + 2e-~. + x = O , dt----~ dx x(O) - O, --~(0) = a
t > 0
(4.6.1) (4.6.2)
Its solution is readily obtained and it reads e
--et
x(t) = a-~/1 _ e 2 sin (V/1 - s2t),
(4.6.3)
valid for arbitrarily longtime intervals. Expanding this expression into powers of r leads to secular terms and we cannot expect that the naive regular perturbation method of Chapter 3 yields a satisfactory solution
Chapter 4. The Method of the Strained Coordinate
58
of the initial value problem (4.6.1)-(4.6.2). Also the modification by Lindstedt-Poincard does not give any help as is shown by the following small calculation. We expand the coordinate t as t = (1 + r
+ e2ca2 + . . . ) T ,
and the initial value problem for x as function of T becomes
d2x dT 2 -[- 2e(1 + r
9(o)
=
o,
dx + g2ca2 -~-" "")dT ~- (1 + gcal -~- g2ca2 -~-'" ") 2x = 0
dx = a(1 + ewl + r
-f-'" ")
Putting
~(~) = ~0(~) + ~ 1 ( ~ ) + ~ ( ~ )
+.-
leads to the system
d2xo dT----~ + x0 = 0, d2xl
dxo x0(0) -- 0,
-~T (0) -- a,
~dxo
d~---~ + ~
= -z-jV
dxl (0) - 2~1~0,
~(0)
= 0,
~
= ~1.
So we obtain
Xo(T) -- a sin T and
d2xl + x 1 -- - 2cala sin r - 2a cos r. dT 2 Both secular terms can only be eleminated by taking a = 0 which leads to the trivial solution x - 0. The cause of the trouble is easily understood and it lies in the appearance ax in the original of the t e r m - 2 a cos T, which originates from the damping t e r m 2e 2? differential equation (4.6.1). In the next two chapters we deal with the m e t h o d s of averaging and multiple scales and it will appear t h a t the addition of a damping term is not an obstacle for the construction of an asymptotic approximation, even valid in an unbounded time interval.
Exercises 1. Let x(t)
be the solution of the initial value problem
d2x +x+ex dt 2
3=eAcoswt,
t>0,
0<e<
with the initial conditions x ( 0 ) = a, ~dx- ( 0 ) = 0. A and ca # 1 are constants independent of v. Show with the aid of the Lindstedt-Poincard m e t h o d t h a t formally x(t) = a cos { (1 + 3 e a u ) t } + O(r
59
Exercises uniformly in bounded t-intervals. Compare this result with that of section 2.1. 2. Consider the same initial value problem as in the preceding exercise, but now w = 1 + O(r Show, again with the L.P. method, that
x(t)
= a cos {1 + e ( ~ a 2 - ~A) t }
+ O(~),
uniformly in bounded t-intervals. 3. Let u(t) be the solution of the initial value problem
d2u d~ 2
+ u = a + cu 2 + r
with
u(O) -- cr # a,
0<~<<1,
du -~(0) = O.
Give formal approximations of u(tg) in bounded v~ intervals up to O(e) for the cases w # l and independent of e and w = l + O ( e ) . Compare the results with those of section 2.2. Hint: take u0 = a (1 + a - -aa COS ~). 4. Let u(x) be the solution of the problem
(x + eu) ~dux x + ( 2 + x ) u = 0 ,
0<x_
1 , 0 < e _ ~ 1,
with u(1) = A. Use Lighthill's technique to obtain a formal first order approximation uo(x) of the solution u(x), valid in the segment 0 < x < 1, together with an order estimate of eul u0
Show that Lighthill's technique to continue the solution into the right neighbourhood of x = 0 fails in the case A < 0; see Ref. [108, pp.79-82].
This Page Intentionally Left Blank
Chapter 5 THE METHOD 1.
OF AVERAGING
Introduction In Chapter 3 we considered initial value problems of the type
dx d---[ = f (t' x; e)'
t > O,
x(O) = xo,
(5.1.1)
with x E ] R n and oo
f (t, x; e) ~ E eJ f j(t, x), j=o
(5.1.2)
for e small positive. Examples are the equation for the oscillations of a weakly nonlinear spring (3.3.1) and the equation for the motion of a planet moving around the sun (3.3.13). The method of regular perturbations provided an asymptotic expansion of the solution in powers of c, which is, however, due to the appearance of secular terms only valid in a finite time interval, say 0 _< t <_ to, with to independent of E. In this chapter we deal with a method, which is appropriate for the construction of asymptotic approximations uniformly valid in much larger time intervals [0, t1~-1], with t z again some number independent of r The method is called the method of averaging. In the next section we formulate the principle of averaging and give a proof of its validity, and in consecutive sections we treat a number of examples and applications, among which the nonlinear spring with and without damping and the perihelium precession. Before we enter into the details of the formulation of the principle, the conditions under which it is valid and its proof, we give already here a simple example taken from [3] that clarifies the principle of averaging. We consider a very simple Hamiltonian system in so-called action-angle variables I C R n, ~ E l ~ n "
dI dt = 0 '
d~
~=w-7(:0,
I(0)=I0,
~(0)=~0,
(5.1.3)
with w a constant vector. This system is slightly perturbed by adding a perturbation term as follows"
dI
d~
d--t- = eg(~)'
~ - = w,
(5.1.4)
62
Chapter 5. The Method of Averaging
with g(~o) continuous and periodic 21r and with the same initial conditions as in (5.1.3). The solution of this initial value problem is given by t A r m
~o = ~Oo+ wt and I(t) - Io + ~ / g(~oo + w'r)dT,
(5.1.5)
0
and so I(t) = Io + O ( e ) , uniformly in any bounded time interval, which is in accordance with the theory of Chapter 3. The function g(~o) can be splitted into two parts, viz
1/
1/
21r
g(~o) = ~
21r
g(r162 + {g(~o)- ~
0
g(r162
0
:= 0 + 0(qo),
(5.1.6)
where ~ is the average value of g(qo) and ~(~o) periodic 2~r with average value equal to zero. The solution of the perturbation problem (5.1.4) may now be written as ~ o - qOo + wt
and t
t
I ( t ) = Io + e / ~dv + e / ~(~O + wr)dT 0
0 cot
= Io + etO + -
~(~oo+ r')dr'
O.) 0
= Io + etO + ~h(wt), OJ
(5.1.7)
~o
with h(~o) =
f ~(~0 +
r162
which is periodic 2~r. It follows that
h(wt) is bounded
0
in [0, e~), and therefore we obtain
I(t) = Io + etO + O(e),
(5.1.8)
uniformly ~ l i d in [0, oo). It appears that the time evolution of I(t) consists of two parts: a "systematic" part J(t) with time derivative eO and an "oscillating" part R(t), given by the periodic function ~h(wt). The part J(t) is O(1) uniformly in any bounded time interval, the part R(t) is O(e) uniformly in [0, oo), while the whole I(t) is approximated up to O(e) by Io + etO, uniformly in any time interval [0, ~] with tl any arbitrary number. Hence we have obtained the result that the solution of the initial value problem (5.1.4) may be approximated up to O(e) uniformly in the time scale !E by the solution of the initial value problem dJ dt = e~'
d~o - ~ = w =/=O
(5.1.9)
2. The Krilov-Bogoliubov-Mitropolski Theorem
63
with J ( 0 ) = I0 and ~ ( 0 ) = ~ 0 and where ~0 is the average value of g(~). The secular term eta0 has been isolated and has been given a clear physical meaning: it gives the "systematic" motion of the system. This simple, however, important result is generalized in the next section. This method of averaging was already used by Lagrange [91], who averaged certain quantities varying slowly with time. Also Gausz applied an averaging principle in his study of the mutual influence of the planets during their motion; he distributed the mass of each planet over its orbit proportional to time and replaced the attracting force of each planet by that of a ring. Also van der Pol [117] applied, as early as in 1920 in his study of triode oscillations, an averaging principle neglecting terms with zero average over the oscillation period. The method of averaging is known as the method of Krilov, Bogoliubov and Mitropolski who proved the averaging principle and used it in many applications; we refer the reader to the book by Bogoliubov and Mitropolski [14], where one can find many references to mathematicians and physicists who developed the asymptotic theory of nonlinear oscillations. More recent contributions to the theory have been given by a.o. Stoker [131], Hale [60], Eckhaus [31], Sanders-Verhulst [123] and Sari [124]. In particular, we mention the work by Eckhaus who introduced the concept of local averaging; he generalized with the aid of this concept the conditions to be satisfied by the differential equations in order that the method is justified.
2.
The Krilov-Bogoliubov-Mitropolski Theorem
The Krilov-Bogoliubov-Mitropolski theorem concerns approximations of solutions of initial value problems of the type dx dt = f ( t , x ; e ) ,
x(O) = Xo,
t > O,
(5.2.1)
where x and f are vector valued and e a small positive parameter. According to the properties of f, for instance, periodicity or nonperiodicity in t, there exist in the literature several variants of this theorem, see e.g. the references [14, 131, 60, 31, 123, 124]. In order to understand the principle of averaging and some variants of the K.B.M. theorem, we discuss first in the next subsection a rather simple case, where we impose some restricting conditions on the function f. In the following subsections we treat some more general cases relaxing the conditions on f. Before stating and proving various theorems we give first the basic theorem on existence and uniqueness of initial value problems of the type (5.2.1). THEOREM 1 Suppose f is defined in the domain G - [0, t*] • 19 • [0, e0] with 1 9 {x e 1~; [ x - x0[ _< d} and continuous in (t,x,e) and Lipschitz continuous with respect to x in G, with Lipschitz constant independent of t and e. Under these conditions the initial value problem (5.2.1) possesses for each value of e E [0, e0] a unique solution for 0 <_ t <_ inf .[t*, d ] with M = suPG [f[. Since our considerations concern vector valued functions in ]~n, the norm ]x I denotes (Ei=Xn and similarly [f[ = ( ~-~-i=1~f2)1/2
Xi2)1/2
64
Chapter 5. The Method of Averaging
For the proof of this well known theorem, we refer the reader to standard books on differential equations, e.g. Ince [74, Ch.III] or Hale [60, Ch.II]. The appearance of the p a r a m e t e r e does not give any complication. 2.1.
Introduction
to First Order Averaging
We consider the following initial value problem for the vector valued function 9 (t) e R ~
dx dt = e f ( t ' x ) '
t>0,
x(0)=xo,
0<e_<eo,
(5.2.2)
with e a small positive parameter. We make the following assumptions i) f is T-periodic in t with T independent of e. ii) The vector valued function f is defined in the domain G = [0, c~) • l:) with D closed and connected in Rn; it is continuous in G and continuously differentiable with respect to x in G; without loss of generalization, we take henceforth for l:) the ball I x - x 0 1 _< d. iii) The vector f and the matrix v x f = ( ~ . ) have in G norms bounded by constants, independent of e; we take a common upper absolute bound M. iv) xo E T~. W i t h these assumptions we have
THEOREM 2 (K.B.M. theorem-first variant) Let x(t) be the solution of the initial value problem (5.2.2) and y(t) the initial value problem dy d-t = efo(y), with fo the average of f ,
y(0) = x0,
the solution of
t > 0,
(5.2.3)
i.e.
1/
T
fo(Y) = ~
f ( t , y)dt,
(5.2.4)
o
where the integration is performed as if y were a constant. Under these conditions the function y(t) belongs on the time scale e -1 independent interior subset f) of l:) and we have in f) ~(t) - y(t) = o(~),
to an
e-
(5.2.5)
uniformly for e $ 0 on the time scale e-1. The interior subset ~ is needed in order t h a t x(t) - y(t) + O(e) stays in the domain D. Before proving this theorem we need first a definition. DEFINITION 1 Suppose 6(e) is an order function of e; ~(t) E 1:) for e $ 0 on the time scale 6(e) -1,
2.1. Introduction to First Order Averaging
65
whenever ~(t) E T) on a segment 0 < 6(e)t < ~ with I a constant independent of e. ~(t) = O(e) for e $ 0 on the time scale 5(e) -1, whenever this estimate holds on a segment 0 < 5(e)t <_i with I a constant independent of e. P r o o f of Theorem 2 After scaling ~- = et one obtains by Theorem 1 the existence and uniqueness of x(t) in the interval 0 < t _~ ~-~, with M = maxG If(t, x)l. From the definition of /0 it follows that I0 is continuously differentiable, ih ~D and so f0 is also uniquely determined in a neighbourhood of t = 0. Further we have that m a x 9 If01 is smaller t h a n M and so y(t) is defined in the time interval 0 _ t < ~--~. On the T scale 0 < T ~ ~ the vector valued function y stays in 2) and due to the boundedness of f0 the vector y stays in an e independent subset 2~ C 2) for 0 _< 7" _< ~ with 6 arbitrarily small positive, but independent of e. Therefore y(t) stays in ~ on the time scale e -1. The four equations (5.1.6)-(5.1.9) of the preceding introductory section with w - 1 and ~0 -- 0 suggest to try as an approximation of the solution of our initial value problem (5.2.2) the expression z(t) = y(t) + ~,(t, ~) (5.2.6) with
/
1/ f(t,x)dt}dT
t
u(t,x)--
T
{f(T, X ) - ~
0
0 t
-= / {f(T,x)- fo(x)}dT, 0
2MT,
which is T-periodic in t and bounded by Because we try as approximation
compare also (5.1.7).
9 (t) .~ z(t) = y(t) + o(~), we may replace the expression for
u(t, x)
as well by
t
u(t, y)
= /
{f(z',
y) - fo(Y) }dT,
(5.2.7)
0
and we take as "Ansatz" for the approximation of
x(t)
the function
~(t) = y(t) + ~ ( t , y).
(5.2.8)
In order to check this approximation we use
/(dx t
x(t) - ~ ( t ) =
dt
o
dw ) dt dt
(5.2.9)
66 and
Chapter 5. The Method of Averaging
dx dt
dy Ou dw = e f ( t x) - e f o ( y ) - e V u . -d-[ - e-if[ dt = e { f ( t , x ) - f ( t , y ) } - e 2 V u . fo(Y) = e{f(t,x) - r
- f ( t , w)} + e { f ( t , w) - f ( t , y ) }
V u. fo(Y).
We use now the Lipschitz property of f and the boundedness of u, This gives dx dw I-d-[ - -~-[ I <- eL[x - w[ + Ce 2,
v f
and V u. (5.2.10)
where [xl 2 = E i =nI x i2, etc. while L and C are constants independent of e. Inserting (5.2.10)into (5.2.9) we get
/dx dw / w(t)[ <_ 1-~- - --~-[Idr <_ e L t
Ix(t)-
0
t
I x ( t ) - w ( t ) l d t + Ce2t
0
or
t
Ce / Ix(t) - w(t)[ + -~- ___e L
Ce Ce { Ix(t) - w(t)[ + -~-}dt + --ft.
0
Application of Gronwall's lemma (Oh.3, Lemma 3) with qo(t) = r - 1 51 -- eL and 52 "-- -c~ gives finally L--
Ix(t)-
w(t)[ +
-~,
< Cr exp[eLt] I~(t)- ~(t)l + - Cr L - L or since u(t, y) is bounded we have also the result the time scale ~, 1 by which the theorem is proved.
Ix(t)- y(t)]-
O(e) uniformly on [::]
Remark The replacement of the approximation (5.2.6) by (5.2.8) is very useful, because the proof becomes otherwise more complicated. In that case one needs a.o. the "implicit function theorem", see [3, pp.294-296].
2.2.
Generalization of Theorem
2; K . B . M . T h e o r e m - S e c o n d
Variant
In spite of the rather mild conditions of Theorem 2 it is still possible to relax these conditions; it will appear that the condition of the continuity of v x f ( t , x ) may be omitted. Because of Theorem 2 it makes sense to compare the solutions of dx
d--t = e f (t, x),
t > O,
x(O) = xo
(5.2.2)
and dy _ e f o ( y ) , dt
t > O,
y(O)
xo
(5.2.3)
2.2. Generalization of Theorem 2; K.B.M. Theorem-Second Variant
67
with f (t + T, x) = f (t, x) and fo(Y) = -~ foT f (~-, y)dT. It follows from (5.2.2) and (5.2.3) that t
~(t) -
y(t) = ~ f {f(~,
~(~)) - f0(y(~))}a~.
(5.2.11)
0
To estimate the difference between x(t) and y(t) one should approximate f(T, X(T)), which is in general not periodic in T; moreover, we need this function for all values of T on the time scale e-1. Averaging over [0, T] makes no sense, but instead we take a so-called local average, introduced by Eckhaus, [31]. DEFINITION 2
The local average over the time interval To of the vector valued function defined by To 1/
fT~
= Too
f ( t + T,x)dT,
f(t, x) is
(5.2.12)
0 where x is again taken as a constant and where To is an arbitrary positive number. Viewing other applications of this definition, f(t, x) does not need to be periodic in t. It follows that this local average is an average depending on the coordinate t, and it is a generalization of the average, whenever f ( t , x ) is periodic in t with period To; indeed in this case fTo (t, x) = fTo (x) = f0(x), and we leave the proof as an easy exercise for the reader. This local average has two useful properties, which we give in the next two lemmas; applications of these lemmas yield finally a nice refinement of Theorem 2; see also Sanders-Verhulst [123, pp.36-39]. LEMMA 1
If 99(t) is Lipschitz continuous with Lipschitz constant A then ~(t) = ~ o (t) + O(To). Proof
1/
(5.2.13)
To {~(t) -- 9o(t + T)}dTI
I~(t) - ~To(t)l--IT0
0
To
TdT =
<
-To
1
-~
ATo
o
As to be expected the local average approaches Lipschitz continuous functions up to O(T0), whenever To is small.
68
Chapter 5. The Method of Averaging
To estimate the first term in the right-hand side of (5.2.11) with the aid of the local average of f(T, X(T)) we have LEMMA
2
Suppose x(t) satisfies (5.2.2) with f bounded and continuous in the domain G = [0, oo) x/9 with 7) a closed ball in 1Rn and also f Lipschitz continuous in x E 1) with uniform Lipschitz constant A, independent of t. If t
~(t) = / S(~, ~(~))~,
(5.2.14)
0
then t
~,% (t)
= i f% (% ~(r))a~- + O(To)
(5.2.15)
0 on
the time scale }.
Proof By the definition of the local average we have To
t+r
1
~%(')-
~of~': i 0
To
To
:(" ~('))~')
o
i
0
t+r
I
'I
"r
To
To 0
~-
i
0
t
: '-~oi ~ i,(~ 0
+ ~, ~(~
~>>~~ + o(~o>
0
To
t
: '-~oI~~ I,(~ o
0
To
t
: '-,oi~:'~ i,( ~ 0
0
1i
=(~
To
t
Too
+-, ~(~+,>>~,} + o(~o>
da'[
o
i
{ f ( a ' + r, x(a' + T)) -- f ( a ' + T, x(a'))}dT] + O(To)
o t
=
i fTo (% X(v))aT + O(To)+ 0
t
To
d~'[ o
{s(<,' + -, ~(~' + -)) - s(~' + - , ~(~'))}d,] o
(5.2.16)
69
2.2. Generalization of Theorem 2; K.B.M. Theorem-Second Variant
The last term is estimated with the aid of the Lipschitz assumption, the differential equation (5.2.2) and the boundedness of f, say with bound M:
1/ / t
To
0
0
t
.of fff
To
A
0
t
da'
A O
0
0
To
0
a'+~"
dT[
t
ef(Tt'
X(Tt))dTt[ "
To
eAff da'
< TOO --
a~
0
M T d T = ~1 M T o A e t = O(To)
'
0
on the time scale e -1. Substitution of this estimate into (5.2.16) yields the validity of (5.2.15). By now we have made sufficient preparations for the proof of the following generalization of Theorem 2. 3 (K.B.M. theorem, second variant) Suppose x(t) satisfies the initial value problem
THEOREM
dx
dt = 6 f ( t , x ) ,
t > O,
x(O) = Xo
(5.2.2)
with f periodic in t with period T. Further f is bounded and continuous in G = [0, cx~) x 79, with 79 a closed ball in R '~, and f is Lipschitz continuous in x with Lipschitz constant A independent of t. Further it is assumed that y(t) is the solution of the initial value problem dy = cfo(y), dt with
t > O,
1/
y(O) = xo
(5.2.3)
T
fo(Y) = ~
f(T, y)dT.
(5.2.4)
0
Under these conditions the function y(t) belongs on the time scale e -1 dependent interior subset f) C 79 and we have in f) 9 (t) - y ( t ) - o ( ~ ) ,
to an e in-
(5.2.5)
uniformly for e $ 0 on the time scale e-1. Proof The function fo(Y) inherits the Lipschitz property from the function f ( t , x ) and the existence of y(t) in an e-independent interior subset ~ C 79 on the time scale e -1 is
Chapter 5. The Method of Averaging
7O
proved in the same way as in T h e o r e m 2. W i t h the aid of (5.2.2) and (5.2.3) we have on the time scale e -1 t
t
x(t) - y(t) - r / f(T, X(T))dT -- e / fo(y(T))dT 0
(5.2.17)
0
Applying local averaging with To - T, using L e m m a s 1 and 2, and remembering t h a t due to the periodicity of f ( t , x ) the local and the ordinary average of f ( t , x ) coincide, we get from (5.2.17) on the time scale e -1 t
t
f
-:/fo
0
+
0 t
= e / { fo (x(r)) - fo (y(r)) }dr + O(eT). o The Lipschitz condition for f induces a Lipschitz condition for Lipschitz constant, and so we obtain
f0
with the same
t
Ix(t) - y(t)[ < EA / I x ( T ) - y(T)ldT + O(eT). 0
Using Gronwall's l e m m a (Ch.3, L e m m a 3) once again we have on the time scale e -1
Ix(t)- y ( t ) l - O ( e T e ~ t ) ,
(5.2.18)
which proves the theorem.
2.3.
The K r i l o v - B o g o l i u b o v - M i t r o p o l s k i T h e o r e m for Nonperiodic Fields; K.B.M. Theorem, Third Variant In the preceding subsections we treated the system
dx d---t = e f (t' x)
t>0,
x(0)=x0
with f periodic in t. We consider now the more general case where f is no longer periodic in t. In this case the K.B.M. theorem reads as follows, see also [31], [123]. THEOREM 4 (K.B.M. theorem, third variant) Suppose
x(t)
satisfies the initial value
problem d_ff = e f ( t x) t > 0, x(0) = x0, (5.2.19) dt ' ' where f is bounded and continuous in G = [0, oo) x T), with 1) a ball I x - x 0 l ~_ d in ]R~, and further f is Lipschitz continuous in G with respect to x, while the Lipschitz constant A is independent of t.
2.3. The Krilov-Bogoliubov-Mitropolski Theorem for Nonperiodic Fields
71
Further it is assumed that the average T
(5.2.20)
f o ( x ) = Tlim - ~ T1 f f (t, x)dt 0
exists for all x C 79. Let y(t) be the solution of the initial value problem dy d---[ = efo(y),
t > O,
y(O) = xo,
(5.2.21)
belonging to an e-independent interior subset f) C 79 on the time scale e - 1 . Under these conditions the solution of the initial value problem (5.2.19) satisfies in f) -
y(t)
(5.2.22)
=
uniformly on the time scale e -1 with t
5(e) =
sup
sup
x(t)ET)
tE[O,Lr -1]
~1
f
fo(x))dTI,
(5.2.23)
0
where L is some constant independent of e. Proof According to Theorem 1 the solution x(t) of the initial value problem (5.2.19) exists on a time interval d 0 _< t _< inf [t2 e ' e M ( t l ) ]' with tl arbitrarily large positive and M(tl) :
Taking tl 0 ( t (
w
max If(t,x)l
=ET~ O
<_
sup =E~D O_
x(t)
sufficiently large we know that
If(t,x)l
: M
certainly exists in the time interval
d eM"
Because T
I f o ( v ) l - [ Tlim - ~ T1 f f (T' x)dTI <- M 0
we have also due to the Lipschitz property of fo(Y), inherited from f ( t , x ) , that y(t) exists in the same interval. Further, due to the boundedness of fo(Y) the vector y(t) belongs to an e-independent interior subset ~ of 79 for 0 < t < c~?-4MM with a independent of e and 0 < a < 1. Defining now L - -~ we have the result that as well x(t) as y(t) exist for 0 _ < t < - gL, while y (t) stays in the interior e-independent subset 7} C 79.
72
Chapter 5. The Method of Averaging
After this precaution we may write for 0 < t _ ~, O
with L independent of ~ and
t
9 (t) - y(t) = ~ f {s(~, ~(~)) - fo ( y ( ~ ) ) } ~ 0 t
= ~ f {s~(~, ~(,)1 - So(y(,))}d, 0 t
(5,2.24)
+ ~ f {s(,, ~(,)) - s~(,, ~(,))}e,, o
where the local average parameter T is to be determined later on. We have
1/ T
1/
0
T+r
1]
{s(~', ~(,)) - s0(~(,))}e~,- ~
0
{s(~', ~(,)) - s0(~(~))}e~,.
0
For ~ T - o(1) and 0 < T < L_-~ it follows with the aid of (5.2.23)
Inserting this into the first term of the right-hand side of (5.2.24), we get for the first term t
f {s~(., ~(.)) - s0(y(.))}e. = 0 t ~o
c
0
However, T
fo (X(T)) -- fO (y(T)) -- ~-~oo lim ~1 / {s(r,~)- SO-,y)}dT, 0
and on account of the Lipschitz condition for f we have T
I~if {s(r,=)- so-,y)}~rl o
< AIx-yl,
2.3. The Krilov-Bogoliubov-Mitropolski Theorem for Nonperiodic Fields
73
and therefore t
t
0
0
valid for 0 < t _ L__-.The estimate of the second term in the right-hand side of (5.2.24) follows immediately from the Lemmas 1 and 2 of the preceding section 2.2' t
/{s(~, ~(~)) - s~(~,
~(~)) }d~ = ~(t) - ~ ( t ) + O(T)
0
= O(T). Hence we obtain finally t
0
on the time scale Lemma 3, gives
r
,
s (0 _ t _~ -j). Application of Gronwall's lemma, Chapter 3,
] x ( t ) - y(t)[ = { 0 ( 5 r 1 6 2
= O(~T))+O(r
on the time scale r
Making an optimal choice for T, viz. eT = 5(r cT we get
Ix(t)- y(t)luniformly on the time scale r
0(5 89(~)),
which proves the theorem.
(5.2.22) D
Remarks
1. An immediate application is given by an example of an almost periodic function f(t,x), see e.g. [60, Appendix]~ We consider N
f(t,x) = E an(x)eiVnt' n--O
(5.2.25)
Chapter 5. The Method of Averaging
74
with un real valued, v0 = 0, and where at least two frequencies do not have a rational ratio, so t h a t f ( t , x) is not periodic in t. In this case T oo
fo(X) = E an(x) T-.oolim ~1 / ei~n~dT _ ao(x) n=O
(5.2.26)
0
and 5(e ) = sup sup eI a.(x)eW"~dT] xED tE[O,Le-1] 0 n=l = o(~).
2. Under certain restricting conditions for f ( t , x ) it is possible to obtain higher order approximations as well in the periodic as in the nonperiodic case. Assuming these conditions one obtains the extra information t h a t the estimate (5.2.22) may be improved, namely x(t) = y(t) + O(5(e)) (5.2.27) uniformly on the time scale e -1. We refer the reader to Sanders-Verhulst [123]. 3. Another interesting question is, whether it is possible to extend the range of t for which the approximation by the solution of the averaged equation is valid. This is indeed the case for initial value problems, which involve attraction to an asymptotically stable critical point of the averaged equation. Eckhaus [31], Sanchez-Palencia [122] and Sanders-Verhulst [123] have shown t h a t the range of validity may be extended to 0 _< t < o0. One has the following theorem THEOREM 5 (Eckhaus//Sanches-PMencia) Let x(t) be the solution of the initial value problem
dx = ~f(t, ~), ~(0) = ~0, t > 0, (5.2.2s) dt where f satisfies in G = [0, oo) x D the conditions of Theorem 4 of this section; /9 = {x; x - Xo[ _< d}. Further it is assumed that the averaged equation
dy dt = efo(y) with y(O) = xo,
t > 0
(5.2.29)
has the property that fo(c) = 0 and that the point y = c is an asymptotically stable critical point in the linear approximation of (5.2.29). If c belongs to the ~ independent interior subset Z) C 7), fo is continuously differentiable in l), and xo belongs to the domain of attraction 7)o of y - c with 7)o C I), then x(t) = y(t) + 0(5 89(e)) (5.2.30)
for O < t < c~. Hence also x(t) approaches for e sufficiently small the critical point x = c. For the proof, we refer the reader to the references cited above. In the next sections we apply the theory of this section to a number of oscillation problems.
75
3. Weakly Nonlinear Free Oscillations 3. 3.1.
Weakly Nonlinear Free Oscillations The General Case We consider weakly nonlinear free oscillations governed by the differential equation (5.3.1)
d2u (du) d t---~ -t- u = ~ f u , - - ~
,
with f ( u , v ) continuously differentiable with respect to u and v and e is a small positive parameter. The initial conditions are given by u(0) = a, Substituting u -
du - ~ ( 0 ) - / 3 , with C~2 -~-/~2 r 0.
(5.3.2)
zl and -~ du = z2 we may write (5.3.1) as the system dzl dt = 22,
dz2 _ dt - - z l + ~ f ( z l , z2).
(5.3.3)
For r = 0 the general solution reads Zl -- a cos(t + 0),
(5.3.4)
z2 = - a s i n ( t + 0)
with the amplitude a and the phase 0 as arbitrary constants of integration. We use now Lagrange's method of variation of constants and consider (5.3.4) as a socalled phase-amplitude transformation from the functions (zl (t), z2(t)) to the functions (a(t), O(t)), which is clearly nonsingular as long as the Jacobi determinant, equal to - a , does not vanish. After a straightforward calculation we get for (a(t), O(t)) the equations da2 d d-t or
-- d-t (z12 -~- z22) -- 2ZlZ2 "~- 2 z 2 (
da dt
= -r
-
-
z l -~-
~/(z~, z2))
+ O ) , - a sin(t + 0 ) ) s i n ( t + 0)
(5.3.5)
and dO _ dt -
d (t + arctan dt
~11)-
-
~zlf(zl, z2) z2 + z2
or
dO e = --f(acos(t + 0), - a sin(t + O)) cos(t + 0). (5.3.6) dt a The initial conditions for the system (5.3.5)-(5.3.6) follow easily from (5.3.2) and (5.3.4); we get
a(0) -- V/a 2 +/32 and 0(0) - - arctan -~. a
(5.3.7)
The function f as function of t is periodic with period 2~, and so we have according to the Krylov, Bogoliubov, Mitropolsky theorem (first variant) tl (a, 0) - (5, ~ ) + O(E), uniformly for 0 _ t < -c
(5.3.8)
76
Chapter 5. The Method of Averaging
with tl some constant independent of e and with equations
(a, 0)
satisfying the averaged
21r
da d--t- =
r f 27r j f ( ~ c o s ( t + tg), - ~ s i n ( t + tg)) sin(t +
~)dt
0
and
dr9 dt
271-
a27r
f (a cos(t + tg), - a sin(t + tg)) cos(t +
~)dt
0 or 271"
da- = d-t
r f f (a cos ~0, - a sin ~o) sin ~od~o 21r
(5.3.9)
0
and 211"
dr9 = d-t
a 2e r f f ( a cos ~o, - a sin ~) cos ~ad~a,
(5.3.10)
0
with a(0) = V/a 2 +/32 and zg(0) - - a r c t a n / 3 .
(5.3.11)
The solution of this autonomous system gives a first order approximation for (a(t), O(t)) and hence also for u(t) = a(t)cos (t + ~(t)) and the approximation is uniformly valid for 0 _~ t < tl/r with tl some number independent of e. For practical applications it is useful to have an estimate for the value of tl. Suppose that the greatest of the upper bounds of ]f{acos(t + ~), - a s i n ( t + zg)} I and I~f{ac~ in the domain (D) 0 < t < c ~ , 0_~ l a - v / c ~ 2+/32 I _ d , 0_~ I z g + a r c t a n ~ l < c ~ is Md (in case fT(=0 for a = 0 , we should take d < V/c~2 +/32 ), then the solution of the initial value problem (5.3.5), (5.3.6), (5.3.7) exists for any positive value of e and for d
0
-
~Md"
This follows from a modification of the formulation of the existence Theorem 1 (see e.g. [74], p.71); the value of t* appearing in Theorem 1 does not play any role since t* = cx:). Because the solution (~, zg) of the averaged equation should stay in an interior domain 7) C D the time interval [0, ~Md] should be shortened to [0 , 7-M--~] d-~ with ~ arbitrarily small positive, but independent of e. Hence the value of tl is ( d - ~)/Md. We remark that in general Md increases with d and so the value of tl may decrease, whenever d increases. Further it should be noted that in case Md is uniformly bounded by M for all value of d, then t l - - ~ with d arbitrarily large. This may happen, for instance, in oscillation problems with damping, see subsection 3.4.
77
3.3. The Perihelium Precession 3.2.
The During Equation
We consider the initial value problem (4.2.1)-(4.2.2) of Chapter 4:
d2u dt 2 ~ u +
eU 3 =
(5.3.12)
0
d~t with the initial conditions u(0) = a =/= 0, -du = 0. Applying the K.B.M. theorem and the results of the last subsection we get tl
u ( t ) = a(t)cos (t + tg(t)) + O(e), uniformly for 0 < t < --
with (a, 0) satisfying the initial value problem 27I"
da e /~3 d--t- = ~ COS3(p sin cpd~,
a(O) = a
0 2~"
_ e J a3 COS4~dcp, -dt - 2~ra
0(0) = O.
dO
0
It follows immediately that a(t) -- a and ~(t) = 3za2t, and therefore
3
u ( t ) = c~cos{(1 + gec~2)t} + O(e)
(5.3.13)
uniformly for 0 < t < t~ with tl some constant. This result is in agreement with the formula (4.2.18) of Chapter 4. 3.3.
The Perihelium Precession
The trajectory of a planet around the sun is according to section (4.2.1) given by a solution of the differential equation d2u d~ 2
where u= r
,
a=
(4.2.19)
+ u -- a + e u 2
GMm~
h2
and e =
3GMm c2r
<< 1.
(r, 0) are the polar coordinates of the center of the planet with respect to the center of the sun and ~ is a characteristic length. As to the other physical constants one is referred to Chapter 3, section 3.2. As initial conditions, we choose again
u(o)
=
~,
du
-~
=
~.
78
Chapter 5. The Method of Averaging
We put u = a + av and v satisfies the initial value problem d2v
dO 2 ~ - v = c a ( l + v )
2,
v(O)=
o[. -
dv dO (0) = a
a
The K.B.M. theorem yields again
v(o) = A(o)cos (o + ~(o)) +
o(~),
uniformly for 0 < 0 < ~-~ with 01 some positive constant while A and (~ satisfy the differential equations 2~"
dA 1 / dO = -ea-~-~ (1 + A c o s ~ ) 2 s i n ~ d ~ -
0
0
and 291"
d~)_ dO-
eaA=21rl J ( 1 + Acos ~) 2 cos ~d~ = - ~ a . 0
So it follows from (5.3.11) ft. = constant = _1V/(C~ _ a) 2 + ~2 a
and = - ~ a 0 - arctan (c~ ~ - )a . So we obtain
v ( 0 ) - a1 V/-ic~ _ a) 2 +/32 cos { (1 - ~a)0 - arctan ( a _ a )} + o(~) or
u(O) : a + V / i a - a) 2 + ~2 cos { (1 - ca)O - arctan ( a fl
-a
) } + 0(r
(5.3.14)
uniformly for 0 < 0 < ~-~ with 01 some positive number, independent of ~. This result is in agreement with formula (4.2.29) of Chapter 4. 3.4.
The Linear Oscillator with Small
Damping
We consider a linear oscillator with small damping, represented by the initial value problem d2u du dt---~ + 2~-~- + u = O,
t > O,
(5.3.15)
with the initial conditions u(O) -- O,
du
-~/-(0) = 1.
(5.3.16)
79
3.5. The Free Van der Pol Equation
Illustrating the m e t h o d we apply the p h a s e - a m p l i t u d e t r a n s f o r m a t i o n and we get for the a m p l i t u d e and the phase the initial value problems da = r
cos 2(t + tO) - 1},
dO = - e s i n 2(t + tO),
dt
(5.3.17)
dt
with
71"
a(0)=1,
tO(0)=
2
Averaging yields d~z dt
dtO --~
-e~z,
0;
a(0)
Hence
1,
-
~r
tO(0)=
2
71"
a(t) = e -a + O(e),
tO(t) = - - ~ + O ( e ) ,
uniformly valid for 0 < t < ~ with tl an a r b i t r a r i l y large positive n u m b e r , i n d e p e n d e n t of e. T h e l a t t e r follows from the fact t h a t the r i g h t - h a n d sides of (5.3.17) are uniformly b o u n d e d for all values of t, a and tO with a > 0 , ( da < 0 and so a ( t ) < 1 ) see subsection 3.1. Hence du -~t u ( t ) - e -~t sin t + O(~) and - ~ = e cos t + O(e),
(5.3.18)
uniformly valid for all values of t, 0 _< t < oo. 3.5.
The Free Van der Pol Equation This e q u a t i o n reads (5.3.19)
d 2 x + x -- e(1 - x 2 ) -d~x-. dt---g
The general solution m a y be a p p r o x i m a t e d as x ( t ) -- ~ ( t ) c o s (t + 0(t)) + O(e)
uniformly for 0 < t <
tl
with 27F
dd dt
2rr
(1 -
cos 2 qo)sin 2 qodqo = T
1 - -~-
,
0 271"
d~ dt
21r . / (1 - ~2 cos 2 ~o) sin ~ocos ~odqo - O. 0
We leave it to the reader to show t h a t one m a y take e l e m e n t a r y calculation we get ~ 2 .__
1 - Ae -et '
tl
a r b i t r a r i l y large.
= c o n s t a n t = ~(0) "= too,
After an
80
Chapter 5. The Method of Averaging
where A and
t~0 are d e t e r m i n e d by initial conditions. Hence x(t) =
2
V/1
Ae-~t
cos(t + ~o) + 0 ( ~ ) ,
(5.3.20)
uniformly for 0 < t < t_x with tl an a r b i t r a r y positive number, i n d e p e n d e n t of ~. E In case x ( 0 ) = c ~ > 0 and -aT(0) dx = 0 we have according to (5.3.7) and (5.3.11) = c~ and Oo = 0 V/1-
A
or
A = 1
4 c~~ and ~ 0 -
0.
(5.3.21)
In contrast to the foregoing examples, it is not the phase t h a t changes slowly with time, b u t it is now the amplitude; this is, of course, due to the d a m p i n g t e r m e(1 - x 2) ~-. dx T h e a m p l i t u d e increases for c~ < 2, and decreases for c~ > 2; for c ~ - 2 the m o t i o n differs from a periodic m o t i o n only in a t e r m O(e) as long as 0 < t < 4.
4.1.
Weakly
F o r c e d N o n l i n e a r Oscillations
T h e Case w i t h o u t D a m p i n g We consider the n o n h o m o g e n e o u s Duffing e q u a t i o n
d2u dt 2
4- u 4- ~u 3 - eA coswt,
0<~<<1,
A>0,
(5.4.1)
with the initial conditions
u(O) = ~,
du ~(o)
= o.
T h e frequency w of the driving force is nearly equal to the frequency of the oscillation in case ~ = 0; w is defined by w2 = 1 4- ~# with # some positive or negative n u m b e r i n d e p e n d e n t of ~. We want to investigate the influence of the forcing t e r m on the m o t i o n as described in section 3.2 of this chapter. P u t t i n g again Zl = u and z2 = -~ the e q u a t i o n (5.4.1) is equivalent with the s y s t e m
dzl - z2, dt _
dz2 - - Z l - cz 3 4- EA cos wt. dt
I n s t e a d of the nonsingular t r a n s f o r m a t i o n Zl ---- a cos(t 4- tg),
(5.4.2)
(zl, z2) -4 (a, v~) given by z2 -- - a sin(t 4 i 0),
(5.3.4)
we use now the nonsingular t r a n s f o r m a t i o n zl -- a cos(wt + v~),
z2 = - a w sin(wt + ~),
(5.4.3)
81
4.1. The Case without Damping with
(5.4.4)
to=to(t) a n d a ( 0 ) = c ~ a n d t o ( 0 ) = 0 .
a = a(t),
We insert this transformation into (5.4.2), and we get for a2(t) the equation da2_ d-~
~d (212 + z~ ~. , : 2 z l z 2 ( 1 -
~1 ) - ~2~z2 (z~ - A cos ~t)
_ _ _
= e[
a 4
#a2 sin 2(cot + to) + - - s i n 2(cot + to)cos 2 (cot + to) co
co
_ 2a____AAsin(cot + to)cos cot]. co
After an elementary calculation we get finally for the amplitude a(t) the equation da a a2 a3 dt =e[~co (-~- - #)sin2(cot + to)+ ~--cosinn(cot + to)
A -- sin(cot + to)cos cot] =
-
co a 2
a
a 3
e [~co (-~- - #) sin2(cot + to)+ ~--cosin 4(cot + to) A A sin(2cot + to) - 2---wsin to]. 2co
(5.4.5)
Since wt + to = - arctan ~wz1 we get for to(t) the differential equation dto
d
d---t- = - w
~ arctan-
z2
~z~
= -w-
~1
wz 2 _
CO : - --CO - - CO2Z 2 -+- Z 2 { Z l ( - - Z x
-
~
- ~2z~ + z] =
~ {-,z~
a 2 co
We have
- - ~ Z 3 -~-
+ zi
fzl'dZ-~--z2dzl~ ~ dt
zf
cA cos cot) - z 2 }
(-~z~ + z~- Az~cos~t} + z 4 - Az~ cos~t}.
a2 z 2 = a 2 cos2(wt + to) = --~-{1 + cos 2(wt + to)}
and
a 4
z 4 = -~- { 3 + 4 cos 2 (wt + to) + cos 4 (wt + to) } and substitution gives for to(t) the differential equation dto dt
a4
--f~a~ {1 + c o s 2(~t + ~)} + -g { 3 + 4 cos 2(wt + to) + cos 4(wt + to)}
~[a2w -
Aa cos(wt + to)cos wt]
--#a
~[
a2w
2
2
a4
{1 + c o s 2 ( w t + t o ) } + - ~ { 3 + 4 c o s 2 ( w t + ~ ) + c o s 4 ( w t + t o ) }
Aa 2 cos(2wt + to)
Aa 2 cos tO].
(5.4.6)
Chapter 5. The Method of Averaging
82
The right-hand sides of (5.4.5) and (5.4.6) are periodic in t with period -~-, 2~ and so by using Theorem 3 of section 2.2 of this chapter the functions a(t) and 0(t) may be approximated uniformly in 0 < t < k by the solutions of the equations
d~
d~ = ~
w f
(
__A s i n 0 ) d t = 2w
_
e A sin0,
(5.4.7)
2w
0
and
dt~ _ r w f dt - a 2 w 2 r
a2 3a4 _ ~Aa ~)dt (-tt-~-+~ 2 cos
0 =
--r 2w
3r
t
2
cA
8 w
-
2gw
(5.4.s)
cos O,
with the initial conditions a(0) - a,
t~(0) = 0.
(5.4.9)
The constant tl is again some positive number, independent of ~. The critical point of the system (5.4.7)-(5.4.8) is given by (ac, 0) with 3_ 3
~a~ - a ~ # - A - 0. Therefore, in case u(0) = a satisfies the condition 3a3 4
a#
A
0,
3 2w 2 = 1 + -~ea
A e--,a
or with w 2 - 1 + r (5.4.10)
the solution of the nonhomogeneous Duffing equation (5.4.1) with du d-y(0) = 0 may be approximated as
u(0) -
a
and
(5.4.11)
u ( t ) = a cos wt + O(e),
uniformly for 0 < t < ~ . We encounter here a remarkable phenomenon: for r = 0 the solution of the initial value problem (5.4.1) reads u ( t ) = a cos t,
while a small perturbation, resulting into (5.4.1), gives an approximate solution u(t) =a coswt,
tl
0~t
E
83
4.1. The Case without Damping
whenever a satisfies the relation (5.4.10); it is in this case just the forcing t e r m r cos wt t h a t determines the frequency of the oscillation. This p h e n o m e n o n is called "entrainment of frequency" or "locking in phenomenon". The equation (5.4.10) is called the "equation of the frequency response curve". This curve is sketched in Figure 1, while we give in Figure 2 the sketch of this curve for the case of the linear equation
d2u + u = r
dt 2
cos wt.
(5.4.12)
The solution of this equation with the initial conditions u(0) - a and -3)du (0) -- 0 reads
CA u(t)= (a
CA
1-w 2)c~
1
w2COSWt,
and so the frequency response curve is given by the relation w2_l
~A
(5.4.13)
! s'
s
s
I I I
t
I
t
!
I
|
I
I
I t i I I !
Frequency response curves Fig. 5.1 Nonlinear A > 0
Fig. 5.2 Linear A > 0
f
Chapter 5. The Method of Averaging
84 4.2.
The
Case
with
Damping
The results of the last subsection are so intriguing, that it is worthwhile to consider also the case with a little bit of damping. So we study the differential equation
d2u du + eS-=. + u + eu 3 = eA coswt, dt---V dt
t > 0,
with 5 positive and independent of e, 0 < e << 1, w 2 = l + e # introduce again the transformations (u, -+ (Zl, Z2) ~ (a, @)"
(5.4.14) and
A > 0. We
u = Zl -- a cos(wt + O) du = z2 = - a w sin(wt + @), dt
(5.4.3)
and we repeat the procedure of Section 4.1. The results are:
da [ a a2 5a {cos 2(wt + ~) _ l } d--t- = e ~ww(-2- - ~u) sin 2(wt + t~) + -~a3 A sin(2wt + @) A sin v~] + ~wwsin 4(wt + tg) - ~ - 2---w
(5.4.15)
and
dO dt
e a2w
--#a
a4
{ 1 + cos 2(wt + 0) } + -~- { 3 + 4 cos 2(wt + O) + cos 4(wt + O) }
2
--
.
a2~w 2
. sin 2(wt . + tg) .
. cos(2wt . Aa + @) 2
Aa cos@ ] 2 J
(5.4.16)
The solution of this set of equations with certain initial conditions is approximated up to order O(e), uniformly in [0, ~ ) with tl arbitrary and positive by the solution of the set = -e dr9 dt
-e
(5.4.17)
zwsint~+ [ # 2w
3 a2 8 w
A - - cos tg) 2~tw
(5.4.18)
where ~ and t~ satisfy the same initial conditions as the functions a and 0. The critical point of the system (5.4.17)-(5.4.18) is given by A 5a~ + - - s i n ~ = 0 w
and 3_2 A - - - cos ~ = #, ac ac
(5.4.19)
85
4.2. The Case with Damping or with w 2 = 1 + 6 #
36~ ~
w 2 -- 1 + ~ Fixing w and picking initial conditions values satisfy (5.4.19)-(5.4.20)we obtain
6A
- Ja,,: cos0c
(5.4.20)
a(0) = ao > 0, 0(0) = 00,
a(t) -= a0,
such t h a t these
tg(t) = ~o.
Hence (5.4.21)
u(t) = ao cos(wt + ~)0) approximates the solution of (5.4.12) with initial conditions du
-~- (0) -- - w a o sin ~o
u(0) -- ao cos ~o,
up to O(6), uniformly in [0, ~] with tl an a r b i t r a r y positive number, independent of 6. Substitution of the first equation of (5.4.19) into (5.4.20) yields the equation of the frequency response curve with damping, namely
w 2 = 1 + ~ 3ca2 T 6
- 52w 2.
(5.4.22)
A sketch of this response curve is given in Figure 3; in order to illustrate our result more clearly the w-scale has been stretched around w - 1
f .,....,
~
...
J
..
.~t 9
. . . . .
/
. ' Io"
,,
/
t !
.
I
i
,',(
r ~ r
a I
t
:,
~ '
', ! I ! I
tO
Fig. 5.3 Frequency response curve with damping stable critical points; . . . . . unstable critical points
86
Chapter 5. The Method of Averaging
3 2 The part of the curve left to the hyperbola ~O2 : 1 § ~r corresponds with the - sign and the part to the right with the + sign in (5.4.22). For a value of w with w2 < w < w3 there exist three values of a~, and so we have in principle three solutions (5.4.21) with entrainment of frequency and the question arises, which branch might represent a stable solution. To answer this question we return to the averaged equations (5.4.17)-(5.4.18) and make a stability analysis.
at = - r
~wwsin ~ +
d~ (# d--t = - r 2w
3a2
A ) t 2~w cos z~ =
8 w
w 2 - 1 - 3ea2 + ~--cos t~ 4 a
2w
(5.4.23)
. - F ( a , ~, w)
:= G(a, t~,w)
(5.4.24)
For the investigation of the stability of the equilibrium points lying on the frequency response curve, it is sufficient to study the linear approximation of (5.4.23)-(5.4.24) in the neighbourhood of these equilibrium points. According to the Poincar~-Lyapunov theorem (see e.g. [123], or [60, p.190]) an equilibrium point is asymptotically stable, whenever the eigenvalues of the matrix
OF O_..~F) oa o~ OG OG
M
-0--d
~
a=ac, ~ = ~
have negative real part and the equilibrium point is unstable, whenever at least one eigenvalue has positive real part; ( f i ( t ) - ac and ~ ( t ) - tgc decay exponentially to zero, whenever the real parts of the eigenvalues are negative). It is now easily derived that a sufficient and necessary condition for the stability is given by
OF
OG)
< 0
(5.4.25)
and det M -
OF OG 0a 0~
OF=
0~ ~a
a . . . . ~=~c
> 0
We have
OF 0a (a~, ~ ) = - ~ ,
5
OF A b-~ (a~' ~ ) = - ~ 2 ~ cos~
and
OG
3
OG
A sin 0~
o~ (a~, ~ ) = ~-2~ a~
A cos0~
-- - ~ '
(5.4.26)
87
5. A Linear Oscillator with Increasing Damping
see (5.4.19). The first condition (5.4.25) is trivially satisfied, because we assumed ~ and 5 positive and there remains only to establish the sign of the determinant of M along the response curve given by F(a~, ~ , 02) - O, G(ac, 0~, 02) = 0. Differentiating with respect to 02 we get along this curve
OF OF da~ 002 = Oa------~d----w+ OG OG da~ 002 cOa~ dw and elimination of ~dw
OF i)~ OG 00~
d~)~ dw' d~c dw '
yields __
( oa oF
da~ . det M d02 \ 002 0 ~
002 00~
"
Hence the sign of det M is determined by the sign of
d02
002 00~
002 ~
dac [ e A d02 ~ 2 0 2 c o s ~ - ( e
A
sinO~)(e
A sin ~)~) } 202 ac '
which by (5.4.19) and (5.4.20)equals
1 dac 3 2 202 d02 {ac(1 + 4 e a c - 022)_ le252ac } Neglecting the term of O(e 2) we get for positive values of ac determinant M and hence stability, whenever
dac(l + 43
2
022 ) > 0 .
(5.4.27) and
02 a positive
(5.4.28)
The stable and unstable equilibrium points are indicated in Figure 3. We do not investigate the stability in a neighbourhood of the point S of intersection of the hyperbola 1 2 c02 _~c may not be neglected in this w 2 = l+~3ea 2 and the response curve; the term ~e neighbourhood. 5.
A L i n e a r O s c i l l a t o r w i t h Increasing Damping
To illustrate the K.B.M. Theorem 4, concerning a nonperiodic field, we take an interesting example from Ref. [123]. Consider a linear oscillator with damping increasing in time:
d2u du dt 2 + ~{2 - y ( t ) ~ =
+ ~ = 0,
t > 0
(5.5.1)
88
Chapter 5. The Method of Averaging
d~ with the initial conditions u(0) = c~ > 0, -du = 0. We take for F(t) e -t and so the damping factor increases from e to 2e. Pu t t i n g again
du
u(t) = a cos(t + O),
dt
the function
- a sin(t + 0),
and variation of constants yields according to (5.3.5) the equations
da (2 F ( t ) ) s i n 2 ( t + ~ ) dt - - e a . . . dO _
dt - -~
(2
-~2{2
.
F(t)) sin(t + 0)cos(t + 0) . . .
.
F(t)}{1 e {2
cos2(t+0)}
F(t)} sin2(t + 0)
(5.5.2) (5.5.3)
with the initial conditions a(0) = a, 0(0) = 0. After averaging, we obtain due to F(t) = e -t the result da
dt
dr9 -~- = O,
-ea;
a(O) = c~,
0(0)
(5.5.4)
O,
and it follows from Theorem 4 that
a(t) = ~e -Et + 0(5 89(e)),
O(t) = 0(5 89(~))
or
u ( t ) - c~e-~` cost and
+
0(5 89(e))
du (t) = -c~e -et sint + 0 ( 5 } (e)), dt
(5.5.5)
uniformly for 0 -<- t -<- ~r with t 1 some positive number independent of e. The error 0(5 89(e)) is estimated with the aid of (5.2.23). Because the damping is positive, we know 0 < a(t) < c~, and so we have only to estimate the following two expressions t
sup 0
-1
(5.5.6)
~1f [-~a { 2 _ F ( r ) } { l _ c o s 2 ( r + O ) } + a ] d r l 0
and t
sup O
~1f [-~1 { 2 -
(5.5.7)
F(T)}sin2(T + O)]dT I
0
The second integral is bounded for all values of t, and so the expression (5.5.7) gives a contribution of O(e) to 5(e). The first integral may be written as t
t
a / cos 2(T + O)dT + l2a / F ( T ) { 1 - - C O S 2 ( T + O ) } 0
0
ST"
Exercises
89
Using F ( t ) = e-' we see t h a t also the expression (5.5.6) gives a contribution of 0 ( 6 ) to 5(6). Therefore 5(6) = O(6) and we finally obtain the result
~(t) = ~-~' ~ost + O(v~) and du dt (t) = - a e -Et sin t + O(v/~),
(5.5.8)
uniformly for 0 _< t < _ t_x with t l some positive number independent of 6. Because it follows from i5.5.2) t h a t a is uniformly bounded (damping), the same is true for the right-hand sides of (5.5.2) and (5.5.3) and so we may take for tl any positive number independent of 6; see subsection 3.1. Exercises
1. Give the exact solution of the initial value problem d2u du dt 2 + 6 - d T + U = O ,
t>o,
with u ( 0 ) = a and ~(0)-0. d~ Compare this result with the first order result obtained by averaging up to 0(6). 2. Give a first order approximation up to O(6) and valid for all t >_ 0 of the initial value problem d2u du dt--T + 65-d-[ + u + eu 3 = 0 , t>O, with u ( 0 ) = c~ and
d= -~-(0) -- 0; 5 is a positive number independent of 6.
3. Investigate the system of two weakly coupled electric L - C circuits, where the charge on each capacitor is given by ql(t) and q2(t) respectively.
ql and q2 satisfy the coupled equations d2ql d2 q2 k2 d2 q2 d2 ql dt---5- + 6 ~ + q~ = O; - ~ + 6--d--~ + k2q2 = O, t > 0 with k 2 - (LC) -1 and 6 the coefficient of m u t u a l inductance. Take as initial = q2(O) -- d-~(O) = O. conditions ql (0) = q, d-~-(O) dt dt Give an exact solution and compare the result with the first order result to be obtained by averaging up to 0(6). See Ref. [127], p.54.
90
Chapter 5. The Method of Averaging
4. Investigate as in section 4.1 the weakly forced nonlinear oscillations of a "soft" spring, represented by the equation
d2u dt 2 with
CO 2 =
-Jr U - - ~ U 3 =
r
cos wt,
A>0
1 + ~#.
5. Investigate as in section 4.2 the weakly forced nonlinear oscillations of a "soft" spring with damping, represented by the equation
d2u dt 2 with w 2 = 1 + ~ #
+ r
du
+ u - r 3= r
cos~t,
A>0
dt
and ~ > 0 ,
independent of ~.
6. Investigate with the aid of averaging the forced nonlinear oscillations of a "hard" spring with damping, represented by the equation
d2u dr---~
+ r
du dt
+ u + cu 3 - A cos wt,
with all three p a r a m e t e r s ~ > 0, A > 0 and w independent of c. 1 i Take w ~ 0 A w ~ • 1 7 7 ii W h a t happens whenever w - 0 V w = + 3 V w - + ! ? 3" iii Consider also the case w = ~#. Warning: this exercise takes much time.
Chapter 6 THE
1.
METHOD
OF
MULTIPLE
SCALES
Introduction
In the previous chapter we have investigated the approximation up to the first order of the Duffing equation d2u dt 2
-~- U -~- E U 3 - - 0 ,
(6.1.1)
with the initial conditions u(0) = a and -d~ ~ ( 0 ) - 0. The result is given by (5.3.13)"
~(t) = ~ cos(t + 3a2et) + O(r
(6.1.2)
tl uniformly valid for 0 <__t < 7-, with tl some positive number independent of r
This result exhibits two so-called time scales: a "fast" time t and a "slow" time 7- - ct. The same phenomenon appears in the study of the perihelium precession determined by the equation d2u
dye---5 + u = a + cu 2, with the "initial" conditions u ( 0 ) = a and Formula (5.3.14) yields
(6.1.3)
d~ ~(0) = ~.
u(O) -- a + V i a - a) 2 + 2~2 COS {0 - - acO - arctan ( a - - ~ ) } + O(~),
uniformly valid for 0 _< ~ < t~l, c
where 01 is some positive number independent of r Again we have two scales: a "fast" scale 0 and a "slow" one ~ = ~0. It is already a century ago that Lindstedt [98, 99] in his studies of celestial mechanics introduced explicitly different time scales in order to eliminate secular terms such as we have met in Chapter 3 in the construction of expansions of the solutions of (6.1.1) and (6.1.3). Another interesting equation with solutions exhibiting two time scales is given by the linear oscilator with weak damping. d2u du dt 2 + 2~-~- + u -
0,
0 < e << 1,
(6.1.5)
92
Chapter 6. The Method of Multiple Scales
with initial conditions, e.g. u(0) = 0 and ~-(0) d~ = 1. The exact solution is given by --~t
e
u(t) = v/1 Expanding v/1
-
e 2
_ ~
sin (V/1 - e2t)
(6.1.6)
into a power series of e we obtain 62
62
u(t) = e -6t(1 + -~- + . - . ) s i n ( t - --fit + - . . ) = e -et sin (t
-
~2
-2 t + " " " ) + -g.2e - s t
~g.2t + . . . ) + " " .
sin (t
(6.1.7)
Also here two different time scales appear T 1 - - et and T2 = t ( 1 - 5e 1 2 +...), the former being the slow varying and the latter the fast varying variable. One can find in the literature several methods where different time scales have been used explicitly in order to construct power series expansions of solutions of equations of oscillatory type, uniformly valid in time intervals 0 < t < ~ . We mention here the two-variables expansion method of Kevorkian and Cole [85, Ch.3], applied to equations of the type
dt---~+u=a+ef
u,-~
,
with the time scales T 1 --
et and T2 = t(1 +
C l C -[- C2 g'2 - [ - ' " ").
(6.1.9)
In this chapter we use a simplification introduced by Reiss [120] with t = t and T = et.
(6.1.10)
Solutions of initial value problems for differential equations of the type
d2u dt---~ + u = a + e f
(
du) e,t,u,-~ ,
(6.1.11)
T)C n + cN+IRN+I(t; C).
(6.1.12)
are formally approximated by the "Ansatz" N
u(t; C) = E u n ( t , n-~O
A generalization of this method consists in the introduction of multiple scales t =
t,
T1 -- ~t,
T2 = c2t,...
TN
-- ~ N t ,
(6.1.13)
2.1. The Duffing Equation
93
and one constructs formal solutions of the form N-1
u(t; ~) = E
u~(t, T1, T2,... TN)E~ + eNRN(t; ~).
(6.1.14)
n--O
Kollett [87] and Smith [130] have shown that under suitable conditions to be imposed on the function f the remainder term sN+IRN+I(t; C) in (6.1.12)is uniformly O(c N+l) in the interval 0 < t < k with t l again some constant independent of ~. Greenlee and Snow [54] have even proved that the expansion (6.1.12) for solutions of certain subclasses of (6.1.11) involving damping are uniformly valid for the whole time interval t _ 0; see also [127, Ch.4]. Expansions of the type (6.1.14) with the multiple scales (t, T1,..., TN) are used in case one requires uniform validity in a n interval 0 ~ t < ~ with t l some constant independent of ~. In this chapter we treat only the two-time scaling (t,~t), because the other methods are, due to many but trivial calculations, rather complicated. It would be worthwhile to develop software packages with formula manipulation such t h a t these calculations can be performed with the aid of a computer. The method of two scales will be illustrated in the sections 2 and 3 where we study once again the examples of the weakly nonlinear spring, the perihelium precession and the linear oscillator with weak damping. In section 4 we apply the method to the wellknown Mathieu-equation and in section 5 we give a general proof of the validity of the method for the class of equations of the type (6.1.11). The chapter is concluded with a generalization of the method, such t h a t it may also be applied to partial differential equations of wave type. For more information on the theory and the applications of the method of multiple scales the reader is referred to the book by Kevorkian and Cole [85, Ch.3] and to that by Smith [127, Chs 3 and 4]. The first reference gives many applications, while in the second one the mathematical correctness has been stressed. 2. 2.1.
W e a k l y N o n l i n e a r F r e e Oscillations T h e D u f f i n g Equation We consider the initial value problem
d2u + u + cu 3 = 0, dt 2
(6.2.1)
with the initial conditions du
u(0) = a and -d-/(0) = 0.
(6.2.2)
We introduce the two variables t = t and w = st and try to construct a formal approximation of u(t) by means of the two variable expansion N
u(t; ~) =
~-~un(t, r)e" + eN+~RN+~(t; ~). r~--O
(6.2.3)
94
Chapter 6. The Method of Multiple Scales
In order to determine the coefficients u~ we should substitute (6.2.3) into (6.2.1) and put equal powers of e together. The chain rule gives dun Our Oun dt = Ot + e--~T "= un,t + eu,~,r,
(6.2.4)
and d2 u n
C~2?~n
02Un "C2 0272n 0~_2 := Un,tt nu 2eUn,tr + e 2 U n , r r . Ot 2 + 2eO--~T +
dt 2
(6.2.5)
Further we have ~
(6.2.6)
n=0
m=0 t=0
Inserting these relations into (6.2.1) we get formally
(~o,. + ~o) + ~(2~o,~. + ~ , . + u~) + ~(uo,.. + 2~,~. + ~ , . + ~ ) + e3(Ul,~-~ - Jr- 2Zt2,t~- -Jr- U3,tt Jr- Zt3) - ~ - . . . - ~ - ~ g . n + l n=O
Ug'ttmltn_g_ m
--
t=O
)
-- O.
Putting equal powers of e together we obtain the following set of coupled differential equations UO,tt nc Uo = O,
(6.2.7)o
Ul,tt -~- Ztl ~ --2Uo,tr -- U 3,
(6.2.7)1
U2,tt nc U 2 = --UO,rr -- 2Ul,tr -- 3U2Ul~
(6.2.7)2
n--1 n - - l - - m Un,tt nt-ttn = --Un-2,r~- -- 2 l t n - l , t r -- ~
m=0
~
UlUmUn-l-f.-m,
n = 3,4,....
(6.2.7)n
l=0
The initial conditions (6.2.2) give the following initial conditions for the coefficients
~.(t, ~) ~o(0, 0) = ~ ~o,~(0, 0) = 0, ~ ( 0 , 0) = 0, u.,~(0, 0) = - ~ . _ ~ , . ( 0 , 0),
(6.2.8)0 n = 1,2,....
(6.2.8)n
We have arrived at a set of initial value problems (6.2.7)-(6.2.8) which, in principle, can be solved one after the other. From (6.2.7)0 and (6.2.8)0 we get u o ( t , T) = A O ( T ) c o s ( t + v~0(~'))
(6.2.9)
with Ao(0) = a and ~o(0) = 0.
(6.2.10)
95
2.1. The Duffing Equation
The amplitude A0 and the phase zg0 are now obtained by the requirement that the next term ul(t, T) in the expansion (6.2.3) does not exhibit secular behaviour. By means of (6.2.9) we have from (6.2.7)1 the equation Ul,tt
+
?.t 1 - -
2 ~dA~ sin(t + ~0) + ( 2 A 0dig~ ~ - - A 3) cos(t + tgo)- ~A ] cos3(t + t9o) (6.2.11)
In order to eliminate secular terms in ul(t,v) it is necessary that the coefficients of sin(t + tg0) and cos(t + ~ It follows that Ao(T) and ~0 (T) satisfy the differential equations
dAo d~
= 0 and
Ao d~~ - -8A].
(6.2.12)
Using the initial conditions (6.2.10) we get 3
Ao(T)--~ and ~)0(T)= ~a2T
(6.2.13)
~0(t, ~) - ~0(t, ~t) = ~0 cos(t + ~3a2et)
(6.2.14)
and hence
This term is the first formal approximation of the initial value problem (6.2.1)-(6.2.2) and by (6.2.3) we have .2
u(t; ~) - a0 cos(t + 8sa2t) + ~Ri(t; ~),
(6.2.15)
If we want to make a higher order approximation, we have to solve according to (6.2.11), (6.2.7)1 and (6.2.8)1 the initial value problem ?-tl,tt Jr- Ul = ---~Ol 0 cos3(t
+ a2T),
(6.2.16)
with U 1 (0, 0) -- 0,
U l , t ( 0 , 0) = 0.
(6.2.17)
The result is
Ul(t,T)=AI(T)COS(t+~)I(T))+-~aoCOS3(t+
a2T).
(6.2.18)
Similarly as above the amplitude A1 (T) and the phase ~91(T) are obtained from substitution of (6.2.14) and (6.2.18) into (6.2.7)2 and the requirement that u2(t, T) should not exhibit secular behaviour. Repeating this proces one might obtain the formal asymptotic expansion (6.2.3) for the solution of the problem (6.2.1)-(6.2.2). In order to show that this procedure is justified, it should be shown that at least CRl(t; r = o(1) or more generally t h a t ~ N + I R N + I ( t ; ~) = o(r uniformly in some interval of t. The equation (6.2.12) with the initial conditions (6.2.13) are in complete agreement with the equations for the amplitude ~(t) and the phase ~(t) in section 3.2 of the
96
Chapter 6. The Method of Multiple Scales
preceding chapter, where we dealt with the averaging principle of Krilov, Bogoliubov and Mitropolski. So it follows already from this principle that the remainder term eRl(t; e) in (6.2.15) is indeed O(e) uniformly in the with e expanding segment 0 < t < ~ with t l some positive number independent of e. This justifies the application of the two scale method in the Duffing equation. It is also possible to give a direct proof for the uniform boundedness of Rl(t; e) in the segment [0, ~]. This is done by the substitution of (6.2.15) into (6.2.1)-(6.2.2) which yields an initial value problem for the remainder R1 (t; e); recasting this initial value problem into an integral equation and applying finally Gronwall's inequality one obtains the desired estimate. For the details of this procedure we refer the reader to D. Smith, [127, pp.106-116]. In section 5 of this chapter we give a general proof of the validity of the two-scale method for equations of the type (6.1.11), which contains the Duffing equation as a very special case. 2.2.
The Perihelium Precession
We consider again the initialvalue problem of section (5.3.3) d2u -4- u = a + CU2, dO 2 with
(6.2.19)
du u(O) = cr and ~-~(0) = ~.
(6.2.20)
~z = a ff-all ,
(6.2.21)
We put and the initial value problem for v becomes d2v dO---~ -4- v = ea(1 + v) 2,
v ( O) = ~ -a a
(6.2.22)
__d~gd v (0) = -'/~a
(6.2.23)
According to the two-scale theory we introduce the coordinates t9 = t9 and et9 = tgl and the formal expansion e~
N
v(tg) ~ ~-'~v,~(~,tgl)e" = Z v , . , ( O , ~ l ) e " n=O
+eN+IRN+I(~;
e).
(6.2.24)
n=O
Substitution of this expansion into (6.2.22)-(6.2.23) gives a set of initial value problems for the coefficients vn (vg, 01). The result is 02vo ~ - a ~, 0t92 + vo = O; vo(O,O) = ~ ' a Vo,~(O,O) = -a (6.2.25)0
02vl 0~9----~ -4- Vl = a(1 + 2v0 + v2) - 2v0,~1, vx(O, O) = O, 02Vn
Vl,o(O, 0) = -v0,~l (0, 0)
(6.2.25)1
rt--1
+ Vn --a(2Vn--1 -4- y ~ VmVn--l--m)- 2Vn--l,tg#,- Vn--2,tg,tg,; tn--O
vn(O, O) = O,
vn,9(O, O) = -v,~-l,9~ (0, 0),
(6.2.25)2
97
2.2. The Perihelium Precession where Vn,z9 :
0
- ~ Vn ( t~ , t91),
0
~,,~ = b-~l v,(O, ~1), 0~
~v~(~,~l),
etc.
From (6.2.25)0 we obtain immediately (6.2.26)
vo(~, ~1) = Ao(~l)cos (~ + qOo(t91)) with Ao(0) cos (r
= a - a a
(6.2.27)
and Ao(0) sin ~o(0) = -fl--
(6.2.28)
a
and therefore A o ( 0 ) = 1V/(a a
a)2+f12
~o(0) = - arctan ( a
a ).
(6.2.29)
The next term GV1 (Vq, ~ql) in our expansion is obtained from (6.2.25)1 with the requirement that r ~1) should not contain a secular term. By means of (6.2.25)1 and (6.2.26) we have for vl (zg, ~1) the differential equation
02vl
0v~-----5- + Vl = a(1 4- ~Ao2) + 2Ao(a 4- ~10 ) cos(vq + r 1 2 cos 2(t~ + ~oo). dAo~ sin(O+ ~o) + -~aAo + 2-j5~-
(6.2.30)
In order to eliminate secular terms in Vl (v~, ~1) it is necessary that the coefficients of cos(~+qo0) and s i n ( t g + ~ o ) v a n i s h and therefore Ao(~l) and qoO(~l)shouldsatisfy the following differential equations
dAo dr91
=0andAo(a+~
d~o)
(6.2.31)
=0.
It follows now from (6.2.29) that do(z91) = -1 V/(a - a) 2 nt- t32 and ~oo(vQ1) = - arctan ( a a
and the first term of the
a )-a~l
formal approximation reads
,o(~, ~1) = , o ( ~ , ~ ) = 1 V/(a _ a) 2 + Z2 cos (t9 - arctan ( a fl a) - e e l ) . a
(6.2.32)
98
Chapter 6. The Method of Multiple Scales
The differential equations (6.2.31) together with the initial conditions (6.2.29) are again in complete agreement with the initial value problems for the amplitude A and the phase ~ as given in section 5.3.3 resulting from the m e t h o d of averaging. So it follows from C h a p t e r 5 t h a t we have
,(o) - ~o(O, o,) + o(~), uniformly for 0 < 0 < ~
(6.2.33)
with 01 some positive number, independent of e.
This result may also be proved directly by substituting v(0) = v0(0, 01) + 8Rx (0; E) into (6.2.22)-(6.2.23) and by estimating R1(0; e) with the aid of Gronwall's lemma, see [127, pp.106-116]. The result (6.2.33) will also follow from the general t h e o r e m in section 5. Finally, we obtain from (6.2.21) u(0) = a + V/(a - a) 2 +/32 cos (0 - arctan ~
- aeO) + O(e),
(6.2.34)
Cl~--a
uniformly for 0 < 0 < ol 3.
T h e L i n e a r Oscillator with Damping
As a third example for the application of multiple scale technique we treat the initial value problem for the linear oscillator with damping. The initial value problem reads as follows
d2u du dt----~ + 26-~- + u = O,
0 < ~ << 1,
(6.3.1)
with the initial conditions du
~(0) = ~,
~ ( 0 ) = b.
(6.3.2)
The exact solution is readily given and it reads b + ea -~t + ~,/y_z~_~ ~ si~ ( v q -
u(t) -= ae -~t cos (V/1 -- r
~t)
(6.3.3)
We have a fast time t and a slow time T = r and as a illustration of the two-time scale m e t h o d we construct an approximation of u(t) up to O(~), uniformly valid in an expanding t-interval 0 < t < t~ with t l independent of r We make again the formal expansion oo
u(t) ~ y~u,,(t, T)e n, n--O
(6.3.4)
99
3. The Linear Oscillator with Damping and we obtain for the coefficients u,~ (t, T) the following set of initial value problems 02Uo + u0 = 0; Ot 2 02ul Ot 2
uo(0,0) = a;
-~- ~tl = --2U0,t -- 2U0,tr;
u0,t(0,0) = b, Ul (0, 0) "-- 0;
(6.3.5)0 Ul,t(0, 0) = --U0,r(0, 0),
(6.3.5)1
02 ?Jtn Ot----5- + u,~ = - 2 u n - l , t
- 2 U n - 2 , r - 2 U n - l , t r - Un--2,TT;
u~(0, 0) = 0,
~,~(0, 0) = - ~ _ ~ , ~ ( 0 , 0),
n -"- 2, 3 , . . . .
(6.3.5)n
The equation (6.3.5)o gives immediately uo(t, T) = AO(T) cos t + Bo(~') sin t,
(6.3.6)
A0(0) = a and Bo(0) = b.
(6.3.7)
with Consecutively, we construct the next term u l ( t , T) such that again no secular terms appear; this procedure yields differential equations for Ao(~-) and B0(~-) which, together with (6.3.7), determine the amplitude A0 and B0. We obtain from (6.3.5)1 and (6.3.6) 02ul Ot----5- +
OAo)
U l
----
2 (Ao + ~
OBo
sin t - 2 (Bo + ~ ) c o s
t,
(6.3.8)
and the requirement to avoid secular terms gives AO(T) + A~o(T) -- 0 and B o ( r ) + B~o(T) -- 0;
using the initial conditions (6.3.7) we get A o ( T ) = a e - " = ae - s t and B0(~') = be - r = be - s t ,
and we have obtained the result uo(t, r) = ae - ~ cost + be - ~ sint = ae - s t cost + be - s t sint
(6.3.9)
u ( t ) = ae - s t cos t + be - s t sin t + oR1 (t; ~)
(6.3.10)
and therefore (6.3.10) is up till now only a formal approximation, but because the two scale method leads here again to the same result as the method of averaging, (see subsection 5.3.4) the remainder term ~R1 (t; e) is uniformly O(e) in any expanding segment [0, t~] with tl independent of e. Hence the two scale method is justified and we have u ( t ) = ae -Et cos t + be - s t sin t + O(e),
(6.3.11)
100
Chapter 6. The Method of Multiple Scales
uniformly for 0 < t < Lt with t l an arbitrary number independent of 6. It is also possible to prove directly that R1 (t; 6) is uniformly bounded for all t _ 0; since this direct proof is relatively simple for our linear initial value problem, we give it right here. Inserting (6.3.10)into (6.3.1)-(6.3.2)we get for Rl(t; 6) the initial value problem d2R1 dR1 dr----if- + 2 6 ~ + R1 = 6(ae -~t cos t + be -~t sin t),
(6.3.12)
RI(0; 6) = 0 and dR1 (0; 6) = a.
(6.3.13)
with
Since u(t) exists for all values of t the same is also true for Rl(t; 6). To simplify the calculations we put R1 (t; e) = e-erR1 (t; 6), (6.3.14) and /~1 satsfies the initial value problem d2/~1
dt 2
+ R1 = 6(acost + bsint) + 62kl,
(6.3.15)
with
Rl(O;
and -d/~l ~ - (0; 6) = a.
6) = 0
Variation of constants gives for 1)1 the integral equation t
/~1 (t; e) = a sin t + e / ( a cos r + b sin T) sin(t -- ~')dT 0 t
+ 62 / 1)l(r; 6)sin(t -- r)dr. 0
From this equation it follows that t
Ik~(t; ~)1 < I~1 + ~(1~1 + Ibl)t + ~ f Ih~(~; ~)ld~ 0 or
I/)l (t; 6)1 + (lal + Ibl) _< lal + (lal + Ibl)+ t
+ e / { I/~I(T, 6)1 + (lal + Ibl)}dT. 0 Applying Gronwall's lemma (Lemma 3, Chapter 3) we get I/~l(t; e)J + (]aJ + ]bl) < { la] + (la] + ]bl) } exp[6t],
(6.3.16)
4.1. Introduction and hence by n i ( t ; e) =
er
101
~)
IRl(t; c)l < {lal + (lal + Ibl)} = O(1), uniformly for all t _> 0. 4. 4.1.
T h e Equation of M a t h i e u Introduction
We consider an inverse pendulum with a prescribed force at its lower end S which is permitted to slide freely along a vertical slit. This vertical slit is represented by the y-axis and the prescribed force in the y-direction by the function Y(t) with t the time, see Figure 6.1.
Y
M
Fig. 6.1 The inverted pendulum We assume that the mass of the rod with length I is negligible in comparison with the mass m at the upper end M of the rod and so we may assume that the total mass of the pendulum is concentrated in the point M. Further we suppose that the motion of the point M takes place in the (x, y) plane under the influence of the weight mg, the applied force Y(t) and the reaction force X(t) in the support S, due to the constraint of the rod. The dynamics of the point M is determined by Newton's law and the angular momentum law resulting into the equations
d2x r n ~ = X(t),
(6.4.1)
Y ( t ) t ~ s i n ~ - X(t)~ cos z9 = 0.
(6.4.2)
and
102
Chapter 6. The Method of Multiple Scales
The latter equation results from the fact that the moment of inertia about the centre of gravity M is zero. Assuming that the deviation ~ from equilibrium is small, we get after linearization
m i ~ - Y(t)t9 = O. Putting the applied force Y(t)
equal to
Y(t) = mg-
rap(t)
we get finally the equation g
~
(6.4.3)
In case p(t) is periodic the linear equation (6.4.3) is known as Hill's equation, usually written in the form d2w + q(t)w = 0 (6.4.4) dt 2 with q(t) periodic. If p(t) =_ 0 the solution of (6.4.3) is unbounded and the equilibrium solution 0 = 0 is unstable. Nevertheless, it is possible to choose the periodic function p(t), such that a bounded stable solution exists. Considering a normal pendulum, not upside down, the term -}v~ in equation (6.4.3) changes its sign and the equilibrium solution 0 = 0 is stable. Mutatis mutandis it is again possible to choose a periodic p(t), such that an unbounded unstable solution exists. In this section we consider in particular the equation
d2u dt---$ + (5 + e cos t)u = O,
(6.4.5)
where 5 and e are parameters with 0 < e << 1. The equation is Mathieu's equation and it has, as we will see, very interesting properties. As to the periodicity and the stability of its solutions, we have at our disposal the theory of Floquet, concerning linear ordinary differential equations with periodic coefficients. Since this section is only meant to illustrate the method of two time scales, it is outside the scope of this chapter to give a detailed treatment of Floquet's theory. Instead we give in the sections 4.2 and 4.3 the results and we refer the reader to lit.J131], where Floquet's theory for second order equations is treated in detail and where the stability of solutions of Hill's equation is discussed. 4.2.
Floquet's Theory for Linear Equations with P e r i o d i c C o e f f i c i e n t s We consider the equation
d2w dw dz---~ + p(z)-~z + q(z)w = 0
(6.4.6)
with p and q periodic with real period w and analytic in a strip containing the real axis, (z - x + iy). Although the equation (6.4.6) is periodic, a solution does not need to
103
4.3. Application to Hill's Equation
be periodic. Suppose the functions wl(z), w2(z) form a fundamental set of solutions of (6.4.6); because wl (z + w) and w2(z + w) are also solutions, we have the relations
Wl(Z -~- w) z allWl(Z) -~- a12w2(z),
w2(z + w) = a21wl (z) + a22w2(z), with aij constant. The problem to determine a set such t h a t
wi(z + w) = orwi(z),
i=1,2
leads to the necessary condition all
- - or a21
a12 a22
z
0,
(6.4.7)
- - or
which is called the characteristic equation. There are now two possibilities i) The equation (6.4.7) has two different roots ai=e ~,
i=1,2.
There exist two independent so-called normal solutions, which have the form
~,(z) = ~ , ~ , ( z )
(6.4.s)
with ~,(z + w) = ~,(z) and c~1 ~ c~2. ii) The equation (6.4.7) has a double root. There exists only one normal solution wl = e ~ l z ~ l ( z ) and the other solution independent of wl has the form
w2(z) = e ~lz { ~
a
z
w ~ l (z) + r (z) }
(6.4.9)
with r (z + w) = r The solutions of (6.4.6) are called stab/e, if all solutions remain positive values of z, and unstable otherwise, i.e. if there exists a bounded for x --+ +oo. In case i) we have stability for Re c~i < 0, i - 1, 2 and instability we have stability for Re c~1 < 0, a ~ 0 or Re c~1 < 0, a - 0 and
4.3.
bounded for all real solution which is not otherwise; in case ii) instability otherwise.
A p p l i c a t i o n to Hill's Equation The equation (6.4.6) takes the particular form
d2w dz----~ + q(z)w - 0 with q(z -F w) -- q(z).
(6.4.10)
104
Chapter 6. The Method of Multiple Scales
The two cases of the preceding section reduce now to the following two possibilities i) There exist two independent normal solutions of (6.4.10), which have the form (6.4.8). Moreover, ala2 - e (~1+~2)~ = 1; it follows t h a t stability occurs only when Re O~ 1 = Re a2 = 0. ii) Because the roots a l and a2 coincide one has a 2 = 1 and so a = :kl. Hence the normal solution has the property
w(z + w) = w(z) or w(z + w) = - w ( z ) ,
(6.4.11)
and therefore there exists a periodic solution, which has either period w or 2w. The other independent solution has again the form (6.4.9) with e al~ - =kl, i.e. a l = i mr ~ , n C Z . It is unstable for a ~ 0 . In practical problems the coefficient q(z) may depend on some real parameters, e.g. q(z) = 5 + cr(~z) and we have the equation d2w
dz 2
+ {5 + cr(z)}w = O.
(6.4.12)
The n a t u r a l question arises for which values of the parameters 5 and r are all solutions stable and for which values of 5 and r do we get a transition from stability to instability, so-called transition values. W h e n the pair (5, r has a transition value there exists, in general, one periodic solution with period w or 2w , (see (6.4.11)) and these values of 5 and r lead, in general, to unstable solutions, because the other independent solution has the form (6.4.9); it follows t h a t transition values for 5 and r belong to the set of unstable values of (5, r It has been shown by H a u p t [65] t h a t the set of transition values (5, r consists of curves Ci i - 1, 2, 3 . . . in the 5 - r plane, separating this plane into regions in which the (5, r values have alternately stable and unstable values (see Fig. 6.2). For each fixed r =p- 0 there exists an infinite set 5i (i - 1, 2, 3 , . . . ) of isolated 5 values bounded at the negative side of the horizontal 5-axis. The curves Ci, containing the points (5i, r are separated from each other and upon moving from left to right in the horizontal direction, these curves fall into pairs of adjacent curves - with the exception of the first curve at the extreme left - in such a way, t h a t there exist along these pairs solutions, which have alternately periods w and 2w; along the transition curve C1 at the extreme left, we have a solution with period w, along the next pair C2 and C3 we have a solution of period 2w, along the next pair C4 and C5 again a solution of period w, and so on. The solutions of (6.4.12) corresponding to (5,r values at the left-hand side of the extreme left transition curve C1 are unstable solutions, and so in the next region b o u n d e d by the extreme left curve C1 and the next curve C2 we have stable solutions, and in the next region bounded by C2 and C3 we have again unstable solutions, etc.etc. In the next Section 4.4 we carry these considerations over to the Mathieu equation (6.4.5); consecutively we calculate in 4.5 the transition curves for small values of r and finally in 4.6 we give approximations of the solutions at both sides of a transition curve. In this final step we use the multiple scale technique.
105
4.4. Application to Mathieu's equation
C,
C~
Fig. 6.2 Regions of stability and unstability, shaded regions denote stability 4.4.
A p p l i c a t i o n to M a t h i e u ' s equation The equation of Mathieu reads d2w
dz 2
+ (5 + e cos
z)w = 0,
(6.4.13)
where for the m o m e n t 5 and e are arbitrary real numbers. It is not difficult to show t h a t this equation possesses a fundamental set of solutions Wl and w2, one of which is even and the other is an odd function of z. For transition values of (5, e) there exists only one solution having either the period 2 r or 4r. In case w(z) has period 2~ we may write oo
~(z) = ~a0 +
a~ r
(a)
n--1
or
(6.4.14) oo
(b)
w(z) = E b n s i n n z n--'l
and in case
w(z) has period 4Ir oo
a 0
n
+ n--1
(a)
106
Chapter 6. The Method of Multiple Scales
or
(6.4.15)
w(z) = E b n sin ( 2 z ) .
(b)
n--1
Substitution of (6.4.14) into (6.4.13) gives sets of recurrent relatiops for the coefficients an and bn ; w e get 5no + gal = O,
(~ - ~ ) a . + ~(a._~ + a.+l) - O,
n = 1,2,....
(6.4.16)
n = 2, 3 , . . .
(6.4.17)
respectively ( 6 - 1)bl + ~b2 = 0 ( 5 - n2)b,~ + ~ (b._ 1 + b . + 11 - 0 ,
Similar results are obtained after substitution of (6.4.15) into (6.4.13). It can be shown t h a t along a transition curve only one of the cases (a) or (b) occurs, and so we may obtain in principle the transition curves with a periodic 2 r solution by elimination of the coefficients an or bn from (6.4.16) respectively (6.4.17). This elimination may be carried by truncating the system (6.4.16)-(6.4.17) and by requiring t h a t the coefficient determinant (Hill's determinant) should vanish. A similar procedure can be used to obtain the transition curves with a periodic 47r solution. The result is sketched in Figure 6.3, see also Ref. [131]. n2
The stable regions are connected at the points 5 - -~-, ~ = 0, with n integer. For these values of (6, e) the equation (6.4.13) has the solutions cos (2 z) and sin (~z) which form a fundamental set with period 21r for n even and with period 47r for n odd. Finally, we remark t h a t (5, ~) values in the stable regions give rise to normal solutions of the type w(z) = e ~ ( z ) with ~(z) periodic 27r and a purely imaginary. 4.5.
The Transition
Curves
for t h e M a t h i e u
Equation
In this section we calculate approximations for some transition curves under the assumption t h a t ~ is positive and very small. The transition curves 6 = 5(~) are characterized by the fact t h a t along these curves there exist a periodic solution with period 21r or 4~r. We assume quite formally, that w and 5(~) may be expanded in an asymptotic series of powers of ~ as follows: w = w 0 -~ ~Wl -~- c 2w2 -~-'-" 5 "-- 5 0 + C 5 1 -~- C 2 5 2 -~-- " " "
(6.4.18)
4.5. The Transition Curves for the Mathieu Equation
107
\
Fig. 6.3 Regions of stability for the Mathieu equation and so the quantities by should be determined in such a way that the above characterization is satisfied, i.e. the solution w is periodic with period 21r or 47r, which means that also each function w~ ~hauld be periodic w~tk the sam~ period. Substitution o~ (6.4~18) into (6.4.13) yields the fbllowiug set of differential equations for the functions wj: //
Wo +50w0 = 0 l/
w l + 6oWl -- - 6 1 w o - wo
cos z
l/
(6.4.19)
The firsteclu~t~ou[~ads to
n 2
J0=--~-, n = 0 , 1 , 2 , . . . ~
~ o = ~o~ ( ~ z )
o~ ~ 0 = ~
(~z),
~ = 0, ~, ~ , . . . .
1) T h e case n = 0 . We get 6 0 = 0 and we take w 0 ~ 1 and the equ~t~o~ for wl becomes W // I ~
--(~I -- C O S Z.
(6.4.20)
108
Chapter 6. The Method of Multiple Scales Because wl should be periodic, we have to eliminate the secular t e r m and so we are b r o u g h t to the r e q u i r e m e n t t h a t 51 = 0 and we get ~ 1 ( z ) = c o s z + ~,
where
c is the c o n s t a n t of integration. T h e e q u a t i o n for w2 b e c o m e s now W 2" ---- - ~ 2
-
= -62
coos z -
- c cos z -
z) 2
(r
( 1 + ~1 c o s 2z),
from which we o b t a i n due to the required periodicity of w2 t h a t have got up to t e r m s of the second order in c.
5=-~
le2 +...
62 = - 8 9 So we
(6.4.21)
and along the t r a n s i t i o n curve c o r r e s p o n d i n g w i t h this a p p r o x i m a t i o n we have as a p p r o x i m a t i o n of a periodic solution w = 1 + e cos z, where we have t a k e n z)
The
c = 0.
1. 1 We have 5 0 - - 4 a n d case
(6.4.22)
n=
w0-cos~
If we take
we o b t a i n for wl
Z
or w 0 - s i n
2;
Z W 0 = COS 2 '
the e q u a t i o n ,,
1
z
z
W 1 q- ~ W 1 ---- --61 COS ~ -- (COS ~ ) ( C O S Z )
z_l
----- ( -- 61 -- 1 ) COS ~
z ~ COS 3 2 .
Avoiding the occurrence of a n o n p e r i o d i c t e r m we should require 51 - - ~ 1 and we get as a p p r o x i m a t i o n for the t r a n s i t i o n curve 1 1 5 = ~ - ~e + . . .
along which we have a 4 r
(6.4.23)
periodic solution
W :
z (1 cos z)2
C O S -2 - t-e
4----.
(6.4.24)
109
4.5. The Transition Curves for the Mathieu Equation If we take w0 - sin ~z we obtain similarly 1
1
~= ~+~+... and
z
(6.4.25)
z)
w--sin~+~
sin3~
(6.4.26)
+....
3) T h e case n = 2. We now have 5 0 - 1 and again the two cases w0 - c o s z and w 0 - sin z. Taking w0 = cos z we obtain for Wl the equation W l ,, ~t_ W l
_
(cos z) 2
--(~1 COS Z --
=
---
1 2
- - (~1 C O S Z
1 - - cos 2z. 2
Elimination of the secular t e r m gives 51 - 0 and wl = -~
1
1
-~ 6 cos2z.
Inserting these results into the equation for w2 we get !!
w~ + ~
= -~
cos z - (
= (-52)
cos z -
1
1
~ + g cos 2z) cos z 1 1--2cos 3z.
It follows t h a t the formal approximation for the transition curve is given by 5 = 1+ 5e 2 +...
(6.4.27)
with the 21r periodic solution ~ ( z ) = ~os z + ~ ( -
~1 + ~1 cos 2z) + . . . .
(6.4.28)
Similarly, we obtain for w0 = sin z the approximations 5 = 1with
1E2 +.-. 12 1
w ( z ) -- s i n z + e ( ~ sin2z) + . - . .
(6.4.29)
(6.4.30)
All these results are in agreement with the results of H a u p t as s t a t e d in the preceding section and a sketch of the a p p r o x i m a t e transition curves is given in Figure 6.4.
110
Chapter 6. The Method of Multiple Scales
/
\
Fig. 6.4 Approximate transition curves 4.6.
The Approximation
of the Solution outside the Transition
Curves
We consider the initial value p r o b l e m
d2we + (5 + E cos z)w~ = 0 dz 2
(6.4.31)
with the initial conditions
dw~
we(0) = a and --~z (0) = b.
(6.4.32)
We intend to investigate the behaviour of the solution at b o t h sides of the two transition curves corresponding w i t h 50 = ~1
(n = 1) ; these curves are according to (6.4.23) and
4.6. The Approximation of the Solution outside the Transition Curves 1
111
1
(6.4.25) approximated by 5 - ~ T ~e. The calculations for other transition curves are omitted, because these run along similar lines. Introduction of the two scales = ez and 7/= z transforms the initial value problem into the following initial value problem for the function w(~, 7/) = w(ez, z) := we(z)
e2 0 2w
02w
02w
+ 2e OfOrj + ~
+ (5 + e cos ~)w - 0
(6.4.33)
with the initial conditions w(O, O ) = a and
Ow
OW)(o,o) -- b.
(6.4.34)
Taking 5 = -1 q- 851 q- e252 q - . . . 4
,
and assuming W(~, 7/) :
WO(~, 7/) -~" ~ W
1 (~,
7/) -~- 9" " ,
we get the following recursive set of initial value problems for w0, Wl,...
02 wo 1 0712 + -~wo = O,
wo(O, O) -- a,
~_~
(O, O) - b,
(6.4.35)
02Wl 1 02w~ - (51 + cos r/)wo, Oo?,]----~---- + ~w I -- -20~Or/ with (6.4.36)
o~ W l ( 0 , 0 ) ---- 0 and ~
(0, 0) = - - o~0 - ~ (0, 0) ,
etc. Solving (6.4.35) we obtain wo(~, 7/)= A0(~)cos (~rl) + B0(~)sin ( ~ )
(6.4.37)
with A0(0) = a and B 0 ( 0 ) = 2b.
(6.4.38)
112
Chapter 6. The Method of Multiple Scales
Substitution of these results into (6.4.36) gives
t --(~IBo)
02Wl + 1 Wl --" iOAo
1 sin ~'7 + (
OBo
1 6z Ao) cos ~ 7/
1 1 - A0 cos ~ 77cos 77 - B0 sin ~ 77cos 77 - ( -00_~~0 + 21Bo - (~lBo)
+ ,
sin 217/
1 1 3 1 3 1 Ao - 61Ao) cos ~ r / - ~ Ao cos ~ r / - ~ Bo sin ~ 77. 2
OBo 0~
According to the method of multiple scales we have to eliminate secular terms in the expression for wz and so we have to put
OAo 1 O---~ + (2 -- ~I)Bo = 0 and
(6.4.39)
OBo
1
O~ -[- (2 -[- ~I)A~ -- 0.
The solution A0(~), B0(~) with initial conditions (6.4.38) is determined by the characteristic equation A 1/2-~1 _ 0 1/2 + 6 z A 1 or A2 - ( ~ - 6 1 2 ) = 0 . For [61[ < ~1 we obtain two real roots corresponding to the region of instability and for [6t[ > ~z we have two purely imaginary roots corresponding to the region of stability. The values 6z - =k 1 correspond with transition curves and this is in agreement with the results of the preceding section.
1) The case of instability [61[ < 51 If [~1[ < ~1 we obtain the fundamental set of solutions
(A(1)(~)/= /-r Bo(1) (~)
/
+ g / 1 + ~1 exp [ V / ~ - ~12 ~]
+lexpE
and
-
-
-
+1/ o
113
4.6. The Approximation of the Solution outside the Transition Curves Hence we have
~0/~/= ~111 ~1explll ~ ~1+~11-~ and
~0~) ~il+~ex~ ElI ~ ~l+~i~+~lexp E 1 4 ~1~~1 Inserting the initial conditions (6.4.38) we get a--Clll-
51 -Jr-C 2 1 ~ - 51
2 b = C l l l - + - 5 1 - t - C 2 ~ / 1 +51 and so A0(~)+
(
~a - b
~1 5+5i
( ~ 1 ~+ba
exp [
51) 2zexp[5+51
- 512. ~] il
-52.~]
(6.4.40)
and
( j1 ( i1
B0(~)= -~a ~~ +-5~1 + a
+
+~
g+51
b>exp[ )
-~.~]
1 51 + b exp [ ~_
_ 512 . ~c],
(6.4.41)
Finally substituting these results into (6.4.37) we obtain the unstable leading term of the expansion of the solution of the initial value problem (6.4.31)-(6.4.32) with 5 = ~1 -Jr-51g and
1 For 51 1"+1~ or 51 $ --51 we get the leading term of 151t < 2"
the expansion of the solution along the transition curves. After a small calculation we obtain for 51 1" +51. 1 1 w0(~, rl) = a cos ~ r / + (2b - a~) sin ~rl 1 = a cos-z + (2b2 and for 515
1 2
acz)sin-z
(6.4.42)
21 1
1
w0(~, ,/) = (a - 2b~) cos ~rl + 2bsin ~,/ 1 1 = (a - 2b~z) cos ~ z + 2b sin ~ z
(6.4.43)
"~
o
o
o ..~
~
.~
~
~176 "~
" "~~
o ~o .,-,
~
,_,~.~ ~
~.~
~'0 ~
,,,,~
~
~
I
II
' "~
+
~
I
~
I
~
Ii I
0
~ 9~
l
II
~ 0
~ 0
II
.,~
~'~
I
o
o c.,)
C + 9
'-'
C o,i,
: C ..., ~
8
r
r.
~
~
u
II
"I=I
,
0 C9
0
'-0
~I~
.,,.
.,d
o o
~
,,d
o o
4-
II
I
-l-
I
q
.-.....:>
II
o
.o,i,
,,~ i r 1 6 2
o
o.../,
5.1. Introduction
115
Substituting finally, these results into (6.4.37) we obtain the stable leading term of the expansion of the solution of the initial value problem (6.4.31)-(6.4.32) with 5-
1 ~ + 51e + . . .
and
I~1 >
1 ~.
We leave it to the reader to show that one obtains for 51 $ 1 the result (6.4.42) and for 51 ~'--~1 the result (6.4.43). Sumarizing the results of this section, we have obtained the following: The solution of the initial value problem (6.4.31)-(6.4.32) with 5
-
1
~
+ e51
is for e $ 0 approximated formally by 1 + Bo(ez) sin (~l z ) , w~(z) ,~ wo(~, 77) - Wo(eZ, z) = Ao(eZ) cos (~z)
(6.4.46)
where the functions Ao and Bo are given by (6.4.40)-(6.4.41)in c a s e stability) and by (6.4.44)-(6.4.45)in case 1511 > 1 (stability).
~11 < 1 (in-
From the general theorem in the next section it will appear that the formal approximaz* with Z* some tion (6.4.46) is uniformly valid in the expanding interval 0 < z < -7positive number independent of e. 5.
5.1.
The General Case and the Error Estimate
Introduction
In this section we deal with the two scale method applied to the differential equation of the general form
d2u dt---~ + u = e f ( e , t , u , - ~ ) , with the initial conditions u(O) -- a,
t>0,
0<e<
du
-~-(0) -- b.
(6.5.1)
(6.5.2)
d~ with We assume that f belongs to the class C 3 in the variables t, u and v "- -~the condition that f and all of its derivatives up to and including the third order are uniformly bounded for all values of t in bounded (u, v) domains. This condition can be weakened at the expense of a more elaborate formulation, but we need in any case that the derivatives of f with respect to u and v up to and including the second order are Lipschitz continuous uniformly for all values of t _~ 0 and 0 _~ E _~ 1. However the C 3-formulation will be sufficient in most cases, including the examples given in the preceding sections, and so we stick to this condition. We approximate the function u(t) formally by
u(t; e) = to(t, T) + Cit I (t, 7") d- R(t; e) :=
U(t, r)+ R(t; e)
(6.5.3)
116
Chapter 6. The Method of Multiple Scales
with T -- et. Using the regularity of the function f we shall prove that the remainder T where T is term R(t; e) is O(e), uniformly in an expanding segment 0 < t < -/, some positive number independent of e. Further, because it will appear that ut (t, ~-) is O(1) , uniformly in [0, -/], T we obtain also
u(t; e) = uo(t, ~') + O(e), uniformly on [0,--T].
(6.5.4)
After having proved this result it is clear that all the formal approximations for the solutions of the initial value problems of the preceding sections are fully justified. So, we start now with the construction of the formal approximation (6.5.3) and consecutively we give the required estimate for the remainder R(t; e). 5.2.
The Formal Approximation We insert
u(t; e) - uo(t, T) + eul (t, T) + R(t; e) := U(t, T) + n(t; e) into (6.5.1)-(6.5.2) and we obtain with the aid of dui
Oui
dt
Ot + e-O-TT := ui,t + eui
igui
and d2ui
02ui
dt 2
Ot 2
+
02Ui
2eo-~T +
e2 02Ui OT 2 "= Ui,tt "~- 2eui,t~ + e 2ui,Tr,
i=0,1
the equation Uo,tt -~- ~(2UO,tT ~- Ul,tt) -~- E2(tt0,rT "~- 2Ul,tr) "{- e3Ul,~-T -[- U0 -~- ~ttl -~- Rtt ~- R = ef(e, t, Uo -+- eul + R, uo,t + e(uo,~ + ~tl,t) -{- ~2~tl,v -{- Rt) : ef(0, t, Uo, Uo,t) + e[f (e, t, uo + eUl + R, uo,t + e(uo,r + Ul,t) + e2Ul,T ~- R t ) f(0, t, ~0, ~0,,)].
Using the two scale technique we get the set of initial value problems ~o,,, + uo = o
uo(O, o) = ~,
~o,,(o, o) = b,
(6.5.5)
Ul,tt + Ul = - 2 u o , t , + f(0, t, uo, uo,t), with
Ul(O, o) = o,
Ul,,(o, o) = -~o,~(o, o).
(6.5.6)
The remainder R(t; e) satisfies the initial value problem
n . + R = p(t; ~),
R(0, ~) = 0,
~dR (0, e) = - e 2ul ,~ (0, 0),
(6.5.7)
where the right-hand side p(t; e) is given by
p(t; e) = elf(e, t, uo + eUl + R, uo,t + e(uo,, + Ul,t) + e2Ul,r + nt) - f ( 0 , t, ~0, ~o,,)] - ~ ( ~ 0 , ~ + 2~1,,~) - ~ , ~ .
(6.5.s)
117
5.3. Estimate of the Remainder Term
The initial value problem (6.5.5) is solved with the prescription that no secular term will appear in the solution of the initial value problem (6.5.6). It does not follow from (6.5.7)-(6.5.8) that R(t; e) = O(e 2) uniformly for all t; we have to be much more modest, and indeed we shall show that
R(t;
T e) = O(e), uniformly in [0, ~-]
with T some positive number independent of e. 5.3.
E s t i m a t e of t h e R e m a i n d e r
Term
In order to get an estimate of the remainder term R(t; side (6.5.8) of the differential equation (6.5.7) as follows
e)
we write the right-hand
p(t; -C) : c[f(c, t, U 0 -~- CUl -~- R , Uo, t nL ~(~t0, r nt- ~tl,t) -~- C2Ul,r -~ R t )
-f(r
t, Uo +
CUl, ~to,t -~- C(Uo,r -1- Ul,t) -1- g'2Ul,r)]
-~-~[f(~, t, It 0 -~- g'Ul, Zto, ` -~- c(Zto, r -~- Ul,t) -Jr g'2Ul,r) - f(O, t, Zto, Uo,t) ] --C2(U0,1-T ~- 2Ul,t~-) -- e3Ul,r~-
or with the aid of the postulated regularity of the function f =
Of (e, t, U(t, 7"), ~dU (t, T))R(t; e) +-~vOf(e, t, U(t, T), dU_~(t, T)) -d~dR(t;
e)] +
eg(t, R, --~dR)+ eh(t)
(6.5.9)
with 1
g (t, R, --~) =
(1
- s) -~fis2(e, t, u + sR, -d-[ + s ---d~) ds
0 1
+2
+
(
ds
(6.5.10)
o
where o_~ ~ and ~~ f are evaluated at the argument Ou2, ouo, (e, t, U(t, T) + sR, ~dU(t, T) + S-x~)dR and
h(t) -
f(e, t, uo + eUl,
uo,t + e(uo,~ + Ul,t) + e2Ul,~)
- e ( u 0 , r r + 2Ul,tr) - e2Ul,~.
- f(O, t, uo, uo,t) (6.5.11)
Because u0(t, T), ul(t, T) and their derivatives up to the second order are uniformly bounded on [0, T] by some positive number independent of e ( u 0 and Ul have no secular behaviour!) and due to the regularity of f with respect to e, u and v we have
h(t)
= O(e), uniformly on [0,--T].
(6.5.12)
118
Chapter 6. The Method of Multiple Scales
Inserting all these results into (6.5.7) we may write our initial value problem for the remainder R(t; E) in the form
d2R dt 2
Of dU dR { Of (~,t,u, dU } dR e-~v(e,t,U,--~)-~+ 1-e-~u --~) R=eg(t,R,--~)+eh(t)
with
(6.5.13)
~dR (0;
n(0; ~) = o,
e) = -e2ul,~(0, 0)
The left-hand side of this equation is nothing else than the linearization of the differential equation (6.5.7). Using variation of constants it is not difficult to show that the initial value problem (6.5.13) is equivalent with the integral equation t
R(t; e) = k(t; e)+ e
K(t,t'; e)g(t', R(t'; e),--~-(t'; e))dt'
(6.5.14)
0
with t a t 4
k(t;
~) = -c2u1,~(0, 0)w2(t; ~) + e /
K(t,t'; 6)h(t')dt'
(6.5.15)
0
K(t,t'; e)
and where the kernel
is defined as
K(t, t'; e) = ~[~(t'; 1 ~)~(t; ~ ) - ~2(t'; ~)~(t; ~)] 9
exp[-e f Ov0f(~, t" ,
V(t"), -~dU(t"))dt"]
(6.5.16)
0
with wl and w2 the solutions of the homogeneous equation (6.5.13) with right-hand side identical zero and with the initial conditions Wl(0; e) =
dw2 (0;
~) = 1,
dwl (0;
e) = w2(0; E) : 0.
In the expression of the kernel K we have used the expression for the Wronski determinant t~
dw2 wl (t')~(t')
-
dwl w2(t')--~-(t')
= exp [ +
f Ov cgf (~, t", U(t"), --~(t"))dt"]. dU e 0
We investigate now the integral equation (6.5.14) by considering the right-hand side as a mapping in the Banach space B~ consisting of all functions R(t) of class C 1, defined on the segment 0 < t < T and with the norm
IIRIII:--max IIRII,II--~-II
119
5.3. Estimate of the Remainder Term where ][R][ and ]]dR [[ are defined as IIR[I =
dR dR max IR(t) l and [ l ~ l l -- max [--d-~-(t)[.
0
o
T
The equation (6.5.14) is now written as
R=MR with t
M R ( t ; c) = k(t; ~) + c
f
K ( t , t'; e)g(t', n(t'; ~),--~-(t';
(6.5.17)
0
and the solution of the integral equation (6.5.14) is the fixed point of the mapping (6.5.17). The solution may be found by an iterative procedure as given in Chapter 3, section 2, and an estimate of R(t; r on [0, T] can be obtained in this way. To show the reader a modification of this method for estimating the order of R and also, because this modification will be used again in the sequel we estimate R by applying now the well-known contraction mapping theorem.
Contraction Mapping Theorem Suppose f is a contraction mapping defined on the Banach space B, equipped with the norm [[. [[, i.e. there exists a number k with 0 <_ k < 1 such that [ [ f ( x ) - f(y)[[ <_ k[[x-y[I , Vx, y E B. Then there is one and only one point x* C B such that f ( x * ) = x*. Moreover, if x is any point of B and x , is defined inductively by x , = f ( x , - l ) xo = x, then lim x,~ = x* in 17.
with
For the proof we refer the reader to any introductory text on functional analysis, e.g. Naylor/Sell [109]. In the following we show that our operator M defines a contraction map in the ball Kr : : {R e B~ ; IIRII~ ~ r}, with r sufficiently small; it will appear that the value of r depends on r and this value gives an estimate for the remainder R. To apply the contraction mapping theorem we need to estimate the functions k(t; ~) and g ( t , t'; r on the segment [0, T]. So we interrupt our exposition for first proving the next lemma. LEMMA 6.1 The functions Wl (t; ~), w2(t; 8) and the kernel g ( t , t'; ~) satisfy the estimates
120
Chapter 6. The Method of Multiple Scales
Iwj(t; ~)1 <- exp[T(llf~,ll + IIL, II)],
(6.5.18)
[~wjd(t; e)l -< exp[T(llf~,ll + Ilfvll)], [K(t,t'; E)[ < 2exp[TIIf~ll]" exp[2T(llf,,ll + 1-~0K(t,t'; e)[ <_ 2 exp[TIIf,,ll]" exp[2T(llf~,ll
IIL, II)],
(6.5.19) (6.5.20)
+ IIf,,ll)],
(6.5.21)
T uniformly for O <_ t _< T~ and O _< t' _< -[,
0 <_~ < 1
Of (e, t, U, dU
I~ = ~
.
Of (e, t, U, dU
~).
7?) ~ d I . - ~
[1-[1 denotes the m a x i m u m norm taken over the segment
0 < t < T_.
Proof The functions wl and w2 are solutions of the homogeneous equation
d2w dU dw dU dt---5- - e f , (e, t, U,--d-~)--d-~ + {1 - e fu (e, t, u, --~ ) }w = o. The substitution, see also (5.3.1)-(5.3.6), w = r sin 0,
dw
dt
= r cos 0
and "variation of constants" give for r and 0 the first order system dr d--t = ~r cos tg(f,~ sin t9 + f~ cos O)
dO
d---t- -- 1 - e s i n 0 ( f ~ , s i n z9 + fv cos 0).
Integration of the first equation yields immediately t
r(t; r = r(O; e) exp[r J cos tg(f~, sin 0 + f,, cos O)dt]. 0
Using the initial conditions for Wl and w2 and the regularity of f(~, t, u, v) on T [0, ~-] we obtain the estimates (6.5.18), (6.5.19). The estimates (6.5.20) and (6.5.21) for the kernel K follow now directly from the definition (6.5.16). [] Now we return to the estimate of the remainder term operator M to the unit ball
R(t; ~). First, we restrict our
K1 "----{R e B~ ;IIRII1 _ 1}.
121
5.3. Estimate of the Remainder Term
It follows from the regularity of the function f and from the definition of the function g(t, R, ~tt), (6.5.10), t h a t there exists a constant C1, independent of e such
dR [g (t, R, -~-)[
(6.5.22)
< C111RII~,
T and for all e C (0, 1]. uniformly for 0 < t < -}From the l e m m a and the regularity of the function existence of a positive constant C2 such t h a t
f(e,t, U, ~t)
we have also the
IIK(t,t'; e)lll _< 2e 5c~T,
(6.5.23)
uniformly for 0 _ t , t ~_< ~-, T 0 < e _ < 1 and with C2 -- m~x{llf~ll IIf~ll}" The last function to be estimated is the function k(t; e) defined by (6.5.15), and consecutively we can make an estimate of the action of the operator M, (6.5.17), on the remainder R(t; e). First, we have from (6.5.15) and the estimates (6.5.18) and (6.5.23)
Ilk(t; ~)11~ <_ ~ 2 j c ~ , § 2T~SC.TIIhl I § 2e~SC.Tiihl I with a = ul,~(0, 0) and 0 < e _< 1. W i t h the aid of the definition h and its estimate (6.5.12) we m a y also write
Ilk(t; e)ll~ _< 5(e)
(6.5.24)
with
5(e) = e2c~T{~e 2 + 2C3e 3C"T (e 2 +
Te)},
(6.5.25)
where C3 is again a suitable constant independent of c. If we restrict the operator M to the ball K~ := {R e Be, IIRII1 < r} and we take
(6.5.22)
1 > r => 26(e),
then we get from the estimates (6.5.24), (6.5.23) and
[[MR[]I _< ~r + 2(T + r
2 _< ~l r + 2(T + 1)Cle5C~Tr 2
1 or for Hence the operator M maps the balls K r , with 2(T + 1)ClehC~Tr <_ -~ 1 r < - ( T + 1)-lClle -5C2T into itself. -4 This b o u n d for the radius r is certainly possible for e sufficiently small, because the only restriction imposed on r is t h a t it should be larger or equal to 25(e) = O(e). It remains now only to show t h a t M is a contraction in some ball K~ with r _< -1( T + 1) -1 C~le -5C2T . Therefore, we take two elements R1 and R2 out of 4 K r and we consider t
K(t,t'; e) g(t',R2(t'), ~ ( t ' ) )
M[n2](t) - M[n~](t) - e
0
- g(t',Rl(t'), ~ ( t ' )
dr'.
122
Chapter 6. The Method of Multiple Scales
From the Definition (6:5.10) of the function g and the Lipschitz continuity of the second derivatives of f with respect to u and v we obtain the existence of a positive constant C4, independent of e, such t h a t
Ig(t, R2(t), ~dR2 (t))
dR1 - g (t, n l (t) , ~
(t) I _<
C4rllR2 -
R~II~
T 0 < e < 1. Using finally the estimate for R1,R2 C Kr, uniformly for 0 < t _ -/, (6.5.23) for the function g ( t , t ' ; e) we get
IIM[R2]- M[R~]II~ <_ 2eeSC'TC4rllR2 - n~lll + 2TeSC~TC4rllR2 - R~II~ _< 2(T + 1)ehC'TC4rlln2 - R~II~ uniformly for 0 _ e _< 1. So it follows t h a t M is a contraction map on g r , when r > 25(e) - O(e), 1 e--hC2T r < - (T + 1)- 1C~ 1e-hC2T and r < 2(~+1) c4 and therefore we have a contraction --
4
in the ball K25(e ) for e sufficiently small, say 0 < e _< eo. Applying finally a modification of the contraction mapping theorem and starting with an approximation Ro(t; e), e.g. Ro(t; e) - k(t; e), we get after iteration the result t h a t the fixed point R(t; e) lies inside K25(e) and hence we have
IlR(t; ~)11~ < ~(~)
T = 0(~), uniformly in 0 _< t < --,
(6.5.26)
with T some number independent of e. P u t t i n g all this together we get the following result due to Mahoney [103], Kollett [87], Greenlee, Snow [54] and Smith [29]. THEOREM 6.2
Let the function f (e, t, u, v) belong to the class C 3 for O <_ e <_ 1, t >_ O, u E R, v E R, with the condition that f and its derivatives are uniformly bounded for all values of t >_ 0 in bounded (u, v) domains. Under these assumptions the initial value problem (6.5.1)-(6.5.2) is approximated by the first two terms of the two scale expansion u ( t , ~) = uo(t, ~) + ~u~ (t, ~-),
~ = ~t,
and we have
II~(t) - u ( t , ~)11~ = o(e), uniformly in an expanding time interval 0 ~_ t ~_ -~, T with T independent of e.
(6.5.27) some positive number
C o r o l l a r y 6.3. Under the same condition as stated in the theorem above, the solution of the initial value problem (6.5.1)-(6.5.2) satisfies also the approximation
Ilk(t) - u0(t, ~)11~ - 0(~),
(6.5.28)
6. Averaging and Multiple Scales for Perturbed Wave Equations
123
uniformly in any expanding time interval 0 ~ t ~ T_. Proof (6.5.28) follows immediately from the fact that the function bounded for 0 _ t _< -/ T (absence of secular behaviour).
ul(t, T)
is uniformly
We end this section by the remark that (6.5.28) does not give only an approximation for the solution of the initial value problem, but also for its derivative. Further, the theorem may be applied to all the examples in the preceding sections of this chapter. 6. 6.1.
A v e r a g i n g a n d M u l t i p l e Scales for P e r t u r b e d
Wave Equations
T h e A p p r o x i m a t i o n by C h i k w e n d u a n d K e v o r k i a n
In this and in the preceding chapter we dealt up till now only with ordinary differential equations and so it stands to reason to investigate whether it is possible to extend the theory to partial differential equations, in particular to hyperbolic equations representing wave propagation. It is impossible to present a general theory, because each class of nonlinear equations has its own particularities and requires its own method for the justification of an approximation of its solutions. Here we restrict ourselves to a method introduced by Chikwendu and Kevorkian [18] and its justification by Bckhaus
[31]. Chikwendu and Kevorkian studied the wave equation
utt-uxx+eH(ut,
ux)=O,
t_>0,
-cx~<x<+cx~,
0<~<<1
(6.6.1)
with the initial conditions
u ( x , O ) - p(x),
u t ( x , O ) = #(x),
-cxD < x < +exp.
(6.6.2)
Assuming p(x), It(x), u(x, t; ~) and its first order derivatives uniformly bounded ~nd H sufficiently regular we expand u into a series of the form
u(x, t; e) = uo(x, t, T) + ~U1(X, t, T)-~- e2U2(X, t', T) -[- 999
(6.6.3)
with the two scales t'-- (1 + e2w2 + e3w3 + . . . ) t
(6.6.4)
7 = et.
(6.6.5)
A term with. r in (6.6.4) has been omitted, because the dependence of the solution on et is accounted for through the variable ~-. To substitute (6.6.3) into (6.6.1) we have to use the straightforward expansions
O__U_U_ Ouo (1 + e2w2 + . . - ) + e Duo 0T
ot-
o~
OU1 (1 + e2 W~ + ' ' ' ) + e~ OU1
+ ~-j
Or
+ E20u2 (1 + e2w2 + . . . ) + e30u2
--j
o7-
......................................................... __ OUo .
-~ e
/ OUo
+
OQUl~
+e
2/
w2
OUo
+
(~Ul
+
OU2
+ ."
Chapter 6. The Method of Multiple Scales
124
02u Ot2
_ -
02U~(l + e2w2 + 0{ 2
..
02 0T 2
02U~(l + e2w2 + . ..) + e2 u0
.)~
+ 2 e ~
02Ul(1 + e2w2 + ... )2 + 2 e 2 002Ul ---~T(l+e2w2+..-)+'"
+e---~
+e2w2 + .- .)2 + . . . + e -202u2(1 ~
02uo . {- 02uo
: ~
O~ul'~
*~[~o3~ + - ~ ) + ~
and
H(ut, u x ) = H
2 {-
Ouo
O~uo
O~ul
02u2
[ 2 ~ ~ + - ~ +2o-~ + o~ ]
+ '
(o.oouo) iouoo..)o.(o oouo) Or' Ox + e \ OT +---~ ~ Oul OH ( Ouo Ouo )
+ ~ o---~ ou----:\ o~ ' o~
Or' Ox
+"
Inserting these expressions into (6.6.1) and taking terms with equal powers of e together, we get the following set of linear hyperbolic equations:
02uo 0{2 02ul 0{2 02u2
02uo --0 Ox2 02ul = _ 2 0 2 u 0 - H(OUo Ouo) Ox2 ot~r o~' o~
02u2 02uo -- - - 2 w 2 - Ox2 O~2
o~
-
(ouo ~+
o~l~
02uo Ov2
(6.6.6) (6.6.7)
02ul OtOv
2 _
ouo) o.(o.o o.o)
o~'o~
o~ o~,~
o~'o~
"
(6.6.8)
The initial conditions (6.6.2) imply
uo(~, o, o) = p(~),
-Ouo - ~ (~, o, o) = #(~),
(6.6.0)
u~ (~, o, o) = o,
Ou1 o, o) = - - gOuo -g~-(~, ; ( (x, 0, 0)
(6.6.10)
~,~(~,o,o) = o,
Ou2 o, o) = - - gOul --~(~, ; ( ( ~ , o, o) - ~ - ~Ouo (x, O,O)
(6.6.11)
The solution of the initial value problem (6.6.6), (6.6.9) is given by
uo(~, ~, ~) = f0(o, ~) + go(~, ~),
(6.6.12)
where f0 and g0 are functions of ~- and the characteristic variables
a--x--t,
~ -- x-k- t,
(6.6.13)
6. Averaging and Multiple Scales for Perturbed Wave Equations
125
while the initial conditions (6.6.9) yield
f0(~, 0)+ 90(~, 0)= o(~), and
-Ofo (x 0) + Ogo (x, 0) 0~
'
#(x)
--~
(6.6.14)
-~ < 9< +~
- o c < x < +cx).
(6.6.15)
These conditions do not determine the T dependence of the functions" fo and go; however, these are to be obtained from the requirement that Ul and its derivatives should be bounded. Therefore, we consider the equation (6.6.7) which reads in characteristic variables
02u: 02fo 029o (09o --4 00.O~ -- 2 00.OT -- 20~oT -- H --~
Ofo 09o Ofo'~ Oa ' O~ + M ]"
(6.6.16)
Integration of this equation with respect to ~ and a gives
OUl
-4--0~ a = 2~ and
.u: -4-0-~ -
02fO -
209of -
H
( Ogo
Ofo ~Ogo + Ofo ~. d~ + Of l (a, T)
0..o / (0.o Ofo Ogo Ofo ) Ogl - 2Oo~oT - H 0,~ Oa ' 0---(+ -~a da + - - ~ ( ~ , T)
(6.6.17)
o"
2
(6.6.18)
where f: and g: are as yet unknown functions to be determined from (6.6.7), (6.6.10) and the requirement that u2 and its derivatives are bounded. Because u: and its derivatives are bounded we must have after dividing (6.6.17) and (6.6.18) by ~ respectively a
02fO = lim
2 ~
o~o~
~-~
if
H
(Ogo o~
Ofo ~Ogo + Ofo'~d~, o~'
o~
~)
(6.6.19)
and
,.,0~ 20~0-----
o"
=-
,/(,,0 ,,0 ,,0 ol ~i m - a H O~ Oa O~ + ~
] da.
(6.6.20)
These equations together with (6.6.14) and (6.6.15) determine the functions f0 and go and so we obtain the first term u0 of the expansion (6.6.3). To calculate f0 and go we need an integration with respect to T and so we are led to an averaging with respect to T. In subsection 6.3. we show the approximate validity of (6.6.19) and (6.6.20) by using local averaging. Before dealing with the justification of these equations and the proof that uo(x, t, T) = fO(a, 7") + go(~, T), is indeed a good approximation of u(x, t), we give first two examples, taken from lit[18].
126
Chapter 6. The Method of Multiple Scales
6.2.
Examples
6.2.1.
Wave Equation with Linear
Damping
We consider the initial value problem
02u Ot 2
02u Ou Ox 2 = - e - fi i,
t >>_O,
- o 0 < x < +c~
(6.6.21)
with the initial conditions
u(x,O) = p(x),
Ou
dp
-~-(x, 0) = -~xx(X).
(6.6.22)
We take this simple example because we can solve this initial value problem exactly and make a comparison with the approximation. Further, we have chosen the initial conditions such that we get for e = 0 only progressive waves of the form u(x, t; O) = f i ( x - t); therefore the first term of the approximating expansion takes the form uo(x, t, T) = fo(a, T) and so a simplification of the method results. The function fo(a, T) satisfies according to (6.6.19) and (6.6.21) the equation 02f0
= lim 1 - - J H ( _ O f o ~ ~ ~o ~
-ffj '
O)d~=
Ofo - -ff-j ( ~' ~ ) '
(6.6.23)
while (6.6.14) and (6.6.15) give the initial conditions
dp (a). fo(a, O)= p(a) and ~Ofo (a, O) = ~--~a
(6.6.24)
These requirements are met by the function
f0(~, ~) = p(o)~- 89
(6.6.25)
and we expect that this expression is a first order approximation of the initial value problem (6.6.21) and (6.6.22). To verify this assertion we compare (6.2.25) with the exact solution of (6.6.21)-(6.6.22), where we take p(x) = sinpx and hence p(a) = sinpa. Using for instance Fourier transformation one obtains
u(x, t) =e- 89 sinpx cos ~p2 - ~e2 t e- 89 etp -
e
( cospx - k sinpx) sin 2 _ 162 2io 4
Ip
~/p
2 _ 162 t. 4
(6.6.26)
A simple asymptotic evaluation of this result yields
u(x,t) = e- 89~t sinp(x - t) + 0(6),
Vx,
Vt >_ O,
(6.6.27)
and so the Chikwendu-Kevorkian first order approximation is correct uniformly up to
o(~).
127
6.2.2. Wave Equation with Cubic Damping
6.2.2.
Wave Equation with Cubic D a m p i n g
In this section we present a first order approximation of the solution of the initial value problem
G~2U 02~t /0U) 3 Ot2 Ox2 = --e -~ , t > 0,
--cr < x < +cr
(6.6.28)
with the initial conditions
u(x, 0) The first approximation value problem
C~U
= 2 s i n x and - ~ ( x , 0) = 0.
uo(x, t, T)
(6.6.29)
satisfies according to (6.6.6) and (6.6.9) the initial
02uo O~2
02uo Ox2 = O,
with uo(x, 0, 0) = 2 sin x and - ~ (x, 0, 0) = 0. Hence
~o(~, ~,~) = fo(~, ~) + go(~, ~), where a = x - t ,
~=x+t
(~.6.3o)
and
fo(a, O) =
sin a,
(6.6.31)
go(~, O) = sin~.
It follows that f0 and go are periodic 21r in a respectively ~c. To calculate fo(a, T) and in a similar way g0(~c, T) we need according to (6.6.19) and (6.6.28) the equation
202f~ = lim 1 / ( Ogo OaOT ~o~ ~ H O~
Ofo ) On' 0 d~
:,im 1/(o o o,o) = lim l f { ( 0 g ~
~-+~
+lim
~-+~
Since go and
3
-0~-] - 3 \
~go(Of~176
\ o~
(Og~176
0~ ] ~-~
\~-J
3
Ofo are bounded and W( o90 is odd with respect to ~ = y we get
O~fo
2 o - j ~~ + 3b(~)
Ofo
+
(Ofo) ~ = o,
'~
2
(6.6.32)
with (6.6.33)
128
Chapter 6. The Method of Multiple Scales
Similarly one obtains for g0(~, r) the equations
02go
20~OT + 3 a ( r )
Ogo +
logo ~ ~ = ]
0
(6.6.34)
with
oaim -~ ~
~
do.
(6.6.35)
These differential equations are supplemented with the initial conditions (6.6.31)
fo(a, O) = sin a,
g0(~,0)=sin~,
~Ofo (a, 0) = cos a,
Ogo
- - ~ (~c, 0) = cos ~.
(6.6.36)
To get an idea how to construct a solution of (6.6.32) we take instead of b(r) a constant, 1 with c > 0 . say 5c The general solution reads
0o
v/F(o) - ~-~;'
where F(a) is an arbitrary function of a. Viewing this result we assume now that the general solution of (6.6.32) has the form
0fo =
0o
)~(r)
(6.6.37)
v~F(~) + r162
where the functions A and qa should be determined such that (6.6.37) satisfies (6.6.32). Substitution of this expression into (6.6.32) gives after a small calculation dqo d)~ + 3A(r)b(r) = 0, dr = A2 (r) and 2 -~r
(6.6.38)
and therefore f0 satisfies (6.6.37) with A and qo satisfying the equations (6.6.38). Similarly, we obtain as the general solution of (6.6.34)
Ogo = 0r
#(r)
(6.6.39)
~/a(r + r162
where G is an arbitrary function of ~ and where # and r d e = #2(r ) and 2 du dT d r + 3#(T)a(T) = O.
satisfy the equations (6.6.40)
Due to the symmetry of the initial conditions (6.6.36) it is clear that a(r) -- b(r), A(T) = #(T) and ~a(r) = r and so we may restrict our calculations to the function f0.
129
6.2.2. Wave Equation with Cubic Damping Using the initial condition (6.6.36) we have
Ofo (a, 0) = cos a =
0~ which is satisfied by F(a) - sec2a,
~(0) v/F(~) + ~(0)
A(0) = 1 and ~(0) - 0; therefore
o f 0 (~,.) =
0a
~ ( . ) cos
(6.6.41)
V/1 + ~(T)cos 2 a
and integration yields
fo(a,T)=
A(T) arc sin ( i ~ (~T ) sin a / . (T) 1 + ~(T)
(6.6.42)
Similarly, because f0 and go are identical,
gO(~, T) --
arc sin \ Y l
+ ~(T--------)s i n ~ / 9
To establish finally the first approximation (6.6.50), we still need the functions ~(T) and A(T). From (6.6.41) and (6.6.35)we get a
a(T) =
2 7r I i ~I- o - I
lim -if a
do" = lim 1 , ~ 2rn + a' 1 + ~(T) cos 2 a
o~
f
A2(T) cos 2 a do 1 + ~(T) cos 2 a
0
with n C N and O < a ~<2Ir. Using the periodicity of the integrand this expression is reduced further to 1 / a(T) -- ~
271-
A2(T) A2(T) cos 2a do = 1 + ~(T)COS 2 a ~p(T)
1 A2(T) / 2~- ~(7-)
271"
da 1 + ~(T)COS 2 a
0
0
A2(~-)
A2(T)
1
~(r)
~(r) V/1 + ~(r)"
(6.6.44)
It follows now from (6.6.38) and a(T) -- b(T) that
3
~
~
~lVT~'
or
X/I+w-- 1 log A-} = log ~ -- log V~ ~ ~ + 1 + const.
130
Chapter 6. The Method of Multiple Scales
On the other hand again by (6.6.38), 1
log A-] = log(~o')-g, and therefore ~o'= 26(1 + V/1 + ~ ) - 6 and
(6.6.45) A = V/~,
(the factor 26 follows from A ( 0 ) - 1 and ~0(0)-0). These equations give finally the functions A(T) and ~o(~-) and the first approximation of our initial value problem (6.6.28)-(6.6.29)is given by
u(x, t, T) -- X/~A(T){ arc sin (
1i +~0(T------~)~0(r) 1~O(T) sin~)}" (6"6"46) ) sin a + ( arcsin i ~ +~~
From this relatively simple example we infer that the two scale method applied to initial value problems of the type (6.6.1)-(6.6.2) leads easily to technical complications, because the equations for ~ and ~ are coupled and even a rather simple perturbation EH(ux, ut) may lead to difficulties for explicit solution. This difficulty is removed in case the first approximation yields a progressive wave so that either f0 or go vanishes (see 6.6.21). The method can be extended to nonlinear perturbations containing also second derivatives of the dependent variable. Chikwendu and Kevorkian give in [18] an interesting application to an example from gasdynamics. They consider the two dimensional supersonic flow of gas over a thin airfoil and they study the so-called "far field", away from the airfoil. Their method works nicely, because the equations for ~ and ~ are uncoupled and they represent the flow on the upper respectively the lower side of the airfoil. 6.3.
J u s t i f i c a t i o n of t h e C h i k w e n d u - K e v o r k i a n P r o c e d u r e In this final subsection we present the validity of the method as given by W. Eckhaus
[31]. We consider the perturbed wave equation
02u Ot2
02u (Ou Ou) Ox 2 = -ell -~'-~x ' t >_0,
- c ~ < x < +c~,
(6.6.47)
with H Lipschitz-continuous in both variables and 0 < e << 1. Our considerations are restricted to the (x,t) domain where u and its first and second derivatives are bounded. Similarly as in subsection 6.1 we introduce characteristic coordinates a=x-t, ~=x+t (6.6.48)
6.3. Justification of the Chikwendu-Kevorkian Procedure
131
and we put ~u
0--x -- f(cr, T) + g(~, T)
OU cot = --f(0~, 7) + g(~C,7-)
(6.6.49)
with T -- et. Note that we have written the partial derivatives of u in the form (6.6.49) and not the function u itself, as has been done in (6.6.12). It follows from (6.6.47) and from u~t = ut~ that the functions f and g satisfy the differential equations
]
Of
0~"
0_~g
= -H(-f+g,f+g) 2
'
OT
1
2H(-f
+ g, f + g).
(6.6.50)
Henceforth we write (6.6.51)
H ( - f + g, f + g) = G(f, g). Further, we introduce the initial conditions at t = O,
f(a, O) = f(x, 0) = ~(x) = ~(a) and g(~, 0) - g(x, 0) = r
-- r
(6.6.52)
which in their turn are determined by the values of u(x, 0) and ut(x, 0). The system (6.6.50) is, apart from the a and ~c dependence, of the form treated in the preceding chapter and therefore one expects that averaging may be useful for obtaining an approximation of f(a, T) and g(~, 7-) and consecutively also of u(x, t). We write (6.6.50) as a system of integral equations
1]
~'
G(f(a, T'),g(a + 2--,e r'))dT'
(6.6.53)
T' ~'), g(~, ~'))d~'. - ~1 f G(f(~ - 2--,
(6.6.54)
f(a, 7") = qo(o) + -~
0
and T
g(~, ~) = r
0
where we have used ~ = a + 2 t = a + 2 7 and a = ~ - 2 t = ~ - 2 7 To reduce (6.6.53) and in a similar way (6.6.54) we replace the functions f(a, T) and g(~, T) by their local averages, defined as T
if f(,~; ~-+ ~-')d~-'
f~(,~, ~-) = ~
0
and
1/ T
ar(~, ~) = y
o
a(~; ~ + ~')d~',
(6.6.55)
132
Chapter 6. The Method of Multiple Scales
see (5.2.11). Taking this average of f(a, T) we get from (6.6.53)
1: dT' {i T
r+~-'
:T(a, T) = :(a) + ~
o
.,,
)
T"
}
a(f(a, T'), g(a + 2--,e T"))dT"
0
1/
{/
T
TWr'
= ~,(.)+ ~
e~-'
0
a(.:(.,~-"),g(. + 2--,~ ~-"))e~-"
I"~
-4--~ 1: dT'{iG(f(a,T"),g(a+ 2--Z, )dT") T
"I"~
0
0
T
r+v'
1i (i
=r
+~
dT'
0
.,,
a(f(a, T"), g(a + 2--,e T"))dT"
1"~
}
+ {O(T)}I. The estimate {O(T)}t follows from the assumption that f and g are bounded and TIe so also G(f(a, T"), g(a A- -7-, T")) Reducing :T(a, T) further we have 2(T" + T '),T " + T'))dT" a(f(~,, ~-" + T'), g(~ + -~
It(o, ~-) = ~o(o) + ~1 i a~-'{ 0
0
]{]
+ {O(T)}I
= ~(a) + ~1
dr"
0
2 (r" + ~'')' r" + T'))dT' G(f(a, r" + T'), g(a + -e
0
)
+{O(T))I "i"
1 = ~o(o1 + -~
T
a(Ir(o,~-" -4- T') ,gT (~,+ -e
dr" o
+
1,
+
0
+ {O(T)}I + {O(T)}2, uniformly for finite values of T -- et, where we have used the Lipschitz-condition of G and Lemma 1 from Chapter 5, stating that for a uniformly Lipschitz continuous function f (t)
f(t) = fT(t) + O(T). Inspecting the T-dependence of both order terms we may safely differentiate with respect to T and we get the initial value problem T
CgfT(a,r ) = ~1 i C(fT(a, T + T' ),gT(a + -('re 2 + T'), T + T'))dT' + O(T) 0---7 o
6.3. Justification of the Chikwendu4Kevorkian Procedure
133
with
fT(o, O) ~ f (a, O) § O(T) ----~(a) + O(T).
(6.6.56)
Because T~ runs from 0 to T and because T is small we may write this equation a l s o as
T
OfT 1 i G(fT(a, 7"),gT(o" + -('re 2 + "P), T))dT' + O(T) -~f (a, T) ~. -j--~ 0
T 0
~+aT
~4T i
G(ST(a,'r), gT(~', T))d~' + O(T).
We suppose now that the initial values u(x, O) and ut(x, O) are periodic, with period independent of e, then also f(cr, T) and g(~, T) are periodic in a and ~ respectively, say with period w. Taking e - o(T), putting ~ = nw+5 with 0 ~ 5 < w, n E N, and using the periodicity of 9T(~~,z) we get ~w
0~ (~' ~) =
1 {/ 2 ( ~ + ~)
G(fr(~, ~), gr(~', ~))d~'
0
~+nw+5
+ f noJ
0
or
OfT 1 ~lim ~1 i G(fT(a, T), gT(~', T))d~' + 0(-~) e + O(T). OT (cr, ~-) -- -~ o In quite a similar way one obtains also o"
OgTOT(('
T) = -- 2~--~o~1 lim
la
i G(fT(a', T), gT(~, T))da' + 0(~) + O(T). 0
An optimal choice for T is now given by T - O(v/~) and we have finally obtained the result
OT ( a ,
OfT
T ) ---
lim ~ 1 f G(ST(U,"r),gT(~', r))d~' + R1 51 ~---,~ o
(6.6.57)
134
Chapter 6. The Method of Multiple Scales
and O
OgTOT(~C,T) = _12 a-~o~limla J G(fT(a', T), gT(~, T))da' + R2
(6.6.58)
o
with R1 = O(x/~) and R2 = O(v~) and the initial conditions
fT(a, O) = r
+ O(T) : ~o(a) + O(yr-c)
and
(6.6.59)
gT(,~, 0) = r
+
O(T) = r
+ O(v~).
In case the solution of these initial value problems depends continuously on the order terms O(x/~ ) in the vicinity of zero, one obtains a good approximation of the functions fT(a, "r) and gT(~, T) and thus also of f(a, "r) and g(~, r) by solving the initial value problems
Of (o, T) = 1
1/ o
1
1/
= -~ ~fim ~
H( - ](a, "r) + [7(~', T), ](a, T) + 0('~', "r))d~',
(6.6.60)
o with
f(a, 0) = ~(a), and O~
1
lim 1 ] o~- (~' ~) = --~ ~ -j
G(](a', T), ~(~, T))da'
o
_-
1 lim _1 ~fH( - ](a', T) + ~(~, r), ](a', T) + ~(~, T))da', 2 a--,~ a
(6.6.61)
o.i
o
with .~((, 0) = r The continuous dependence of the solution of the initial value problems (6.6.57), (6.6.58) and (6.6.59) depends on the function H and this continuity should be checked for every given problem. When we replace in the initial value problems (6.6.60) and (6.6.61) the functions f and ~ by ~ respectively ~ we get the same equations as given by Chikwendu and Kevorkian and presented in the formulae (6.6.19) and (6.6.20). Therefore,
0/0 (~, ~) Oa and
=
](~, ~) ~/(~, ~)
Ogo o~ (~' ") = ~(~' '9 ~ g(~, ~).
It follows that the solutions ~ and- o90 o~ of the equations (6.6.19) and (6.6.20) yield the functions f and g within the required accuracy and this justifies the Chikwendu and Kevorkian approach in the case that f(a, "r) and g(~, ~-) are periodic in a respectively
Exercises
135
The success of the multiple scale method has promoted the use of first order models for the description of several physical phenomena. Apart from the examples given before we mention here also the Korteweg-de Vries equation for long waves in a channel [94], [58] and the nonlinear SchrSdinger equation for the amplitude of modulated waves, [25], [58]. These examples have been a topic during the last twenty years in pure and applied research concerning the theory of solitons. Exercises
1. Give by using two time scales a first order approximation of the solution of the free van der Pol equation
d2u t--d ~+u=r
du~u2)-d-
(1
-
t>0
with du u(0) = cr > 0 and - ~ - ( 0 ) - 0.
2. Give by using two time scales a first order approximation of the solution of the initial value problem
d2u dt 2
du
+u=0,
t>0
with du u(0) = a and - ~ ( 0 ) = 0.
3. Give by using two time scales a first order approximation of the forced van der Pol equation d2u du dt--$ + w21u = r - u 2) --~ + 3" cos w2t with u(O)=a,
du -~-(O)=Oandw2-wl=O(1),
no resonance). Consider the following two cases: 3' is a constant with 3" = eft and 3" = O(1). See also [108, 6.2.3].
This Page Intentionally Left Blank
Chapter 7 SINGULAR ORDINARY
1. 1.1.
PERTURBATIONS DIFFERENTIAL
OF LINEAR EQUATIONS
T h e Initial V a l u e P r o b l e m Introduction
In this section we construct an a s y m p t o t i c approximatio~ of the solution of the initial value problem
d2ue due Le[ue] = ~ ~ + a(x) ~ + b(x)ue - f(x),
x > 0,
(7.1.1)
with 0 < r ~ 1 and the initial conditions ue(0) - a,
due
--~-x(0) - p,
(7.1.2)
where c~ and ~ are a r b i t r a r y constants independent of ~. T h e coefficients a and b and the right-hand side f are s u b m i t t e d to the condition t h a t they should belong to the space C~[0, c~) (continuously differentiable for 0 _< x < c~) and moreover t h a t
a(x)_>ao>O,
x>0,
(7.1.3)
with a0 a constant independent of c. The latter condition is very essential because o t h e r w i s e - as will a p p e a r l a t e r - the construction breaks down. First we construct in subsection 1.2 a formal approximation of ue, i.e. an a p p r o x i m a t i o n t h a t satisfies the differential equation and the initial conditions up to some order of ~. Consecutively we show in the next subsection 1.3 the validity of this formal approximation; the proof consists in an a priori estimate of the error. 1.2.
The Formal Approximation
We expect t h a t the solution ue will be a p p r o x i m a t e d to some extent by the solution w of the reduced problem with ~ = 0. So we consider the initial value problem
a(x) dw + b(~)~ = f(~),
(7.1.4)
138
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
with w(0) = a. The solution reads
eq(X){o~2r-e-q(')~(~)~} f(~)d
W(X)--
,
(7.1.5)
0 where q(x) = - y ~b(~) d~~.
0 However this function does not satisfy in general the second initial condition (7.1.2) and so we need a correction t e r m t h a t accounts for this lost second b o u n d a r y condition. Therefore we investigate the differential operator L~ in the neighbourhood of the point x = 0 more closely by using a "microscope" defined with the aid of the transformation x = r
v > 0.
(7.1.6)
The operator L~ expressed in the new coordinate ~ is d2
L~ = ~ 1 - 2 ~ , .
+~-~a(~V~)
d
. +b(~,~).
The most dominant terms are the first two whenever v > 1 and to get a proper balance between these terms we put v = 1. Using now the supposed regularity of the coefficients a and b we have L~ = E - I ~ _ 1 + ~0, (7.1.7) with
d2 s
and
d
= -7~,- +a(0)-7; ",
d f~o = ~--dx (zgE~)-~-~ . +b(e~) .,
0 < ~ < 1.
To satisfy b o t h initial conditions at x -- 0 and the differential equation up to some positive order of e we make the following a t t e m p t for the formal approximation of uE:
u,(~) = ~(~) + ~(~-) + R,(~) = ~ ( ~ ) + , ( ~ ) + R,(~),
(7.1.s)
where the correction t e r m v(() satisfies the b o u n d a r y walue problem
~_1 IV]----0, with the b o u n d a r y conditions
dv 1 dv dw dx (O) = -~-~(0) - ~ - -~-~x(O),
(7.1.9)
139
1.2. The Formal Approximation and
(7.1.10) lim v(~) = O.
~-+oo
The latter condition stems from the consideration that we want to represent the correction term v(~) as a boundary layer term, that has for r $ 0 only significant values in a right small neighbourhood of x - 0. It follows immediately from (7.1.9) and (7.1.10) that v(~) is given by dlu
v(~) = r
~(o)- Z a(O) exp [ - a(O)~-].
(7.1.11)
Putting w and v together we get
x Le[w(x) + v ( ~ ) ] -
f(x) + r
d2w
+ C0[v( )]
d2w
da
= f(x) + ~-j-fi~2 + r162162 with
w(o) + v(o) and
-
~ + ~
~'(o)
~ a(0)
dv
+ b(~r
(7.1.12)
-
dw dv dx (0)+ -~x (O) = ~.
Because the data of our initial value problem (7.1.1)-(7.1.2) are C1[0, cr it follows from (7.1.5) and (7.1.11) that the right-hand side of (7.1.12) is f ( x ) + O ( ~ ) , uniformly in any bounded segment [0,~] with I positive and independent of ~, (n.b. ~exp[-a(0)~] is uniformly bounded in [0, co) ). So w ( x ) + v(~) is a formal approximation of ue in [0, ~]. We call this composite expression a formal approximation, because it satisfies the differential equation and the initial conditions up to O(~), uniformly in [0, ~]. Finally to show that w ( x ) + v(-~) is indeed a good approximation we consider the remainder term Re(x) defined in (7.1.8). It satisfies the initial value problem
d2w da dv Le[Re] = Le[ue - w - v] = -6-~fix2 - ~-~x (O~)--~ - b(r = O(~), uniformly in [0, ~],
(7.1.13)
with the initial conditions
Re(O) = -v(O) = 0(r
dRe and -~-x (0) - 0.
(7.1.14)
Given this initial value problem it is possible to derive an a priori estimate of Re(x) without solving explicitly the initial value problem (7.1.13)-(7.1.14). This a priori estimate is constructed in the next subsection and it will appear that Re(x) = 0(r uniformly in any bounded segment [0, g].
140 1.3.
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations The A Priori Estimate of the Solution of a Singularly Perturbed Ordinary Differential Equation with Given Initial Data We consider the initial value problem
d2u du ~ x 2 + a(x; ~)-~x + b(x; ~ ) u - f(x; e)
0 < x < oo,
(7.1.15)
with the initial conditionS u(O) = a(6),
du ~x(O) = 13(6).
(7.1.16)
The coefficients a, b and the right-hand side f depend as well on x as on 6 and we assume that for .E sufficiently small a(x; E) is continuously differentiable with respect to x and that b(x; e) and f ( x , e ) are continuous in x for x _> 0. Moreover, there exist positive constants A , B and i, independent of E, such that [a(x; 6)[ + [dxxda(x; 6)[ < A, ]b(x; E)[ < B, for 0 _< x _< l,
(7.1.17)
uniformly for 6 sufficiently small, say 0 <. 6 < 60, where e0 will be used henceforth as a generic constant; this means that the value of e0 may be lowered in the sequel without violating earlier results. Further we assume that also the initial values a(6) and fl(r are bounded in [0, e0]. Finally, we make the crucial assumption
a(x; r _ ao > O, f o r 0 < x < g a n d 0 _ 6 < 6 0 ,
(7.1.18)
where a0 is some constant depending on ~ and e0, but independent of e. The estimate of the solution u(x) of (7.1.15)-(7.1.16) in the segment [0, i] is obtained by the method of energy integrals. Instead of estimating u we may as well estimate the function fi defined as ~t(x) = u(x) - 13x- oL. (7.1.19) It follows that fi(x) is a solution of the initial value problem
d2u dfi -C~x2 + a(x; e ) ~ x + b(x; e)~t = g(x; 6),
O<x<~
(7.1.20)
with the homogeneous initial conditions fi(0) = ~(0) dfi
=0.
(7.1.21)
The right-hand side is given by
g(x; e) --- f (x; e) -- fl(e)a(x; c) - (c~(e) +/3(e)x)b(x; e).
(7.1.22)
1.2. The Formal Approximation
141
We multiply (7.1.20) with 2fi(x) and with 2a(x; e)fi' (x); after addition of the results we get the inequality d ( a ~ 2 + 2e~,' + ea~ '2) dx
= (a I - 2b)~ 2 - 2 a b ~ t ' + (2s + a's - 2a2)u '2 + 2gfi + 2agft' < (1 + a' - 25 + b2)fi 2 + e(2 + a')fi '2 + 2g 2.
(7.1.23)
Moreover we have also a~ 2 + 2e~fi' + ~a~ '2 >_ ao ~2 -+- 2efifi' + ~ao ~'2 >_ (ao - V/e)~ 2 + e(ao - v/e)fi '2 > m(fi 2 + eft'2),
(7.1.24)
for e sufficiently small, say 0 < e < eo (so is generic!); m may be taken for instance 1 as m - - ~ao. Combining (7.1.23), (7.1.24) and (7.1.21) we get x
m ( ~ 2 + e~ '2) _< f ~ ( a f i 2 + 2 ~ '
+ eaft'2)d~
0
< ] { (1 + a' - 2b + b2)~ 2 + e(2 + a')fi '2 + 2g2}d~ 0 x
_< M { / ( f i 2 + efi,2)d~ + {Igtt~o,l]},
(7.1.25)
0
with M = max[]l + a ' - 2b + b2l, 12 + a'l, 2] [0,~] < 2 + A + 2B + B 2 independent of v, and with I]" [1[o,~] the L2-norm. The inequality (7.1.25) is uniformly valid in [0, g] for 0 __<E _< eo and we may apply Gronwall's lemma from Chapter 3, (3.2.23)-(3.2.24). With the aid of this lemma we obtain the estimate ~2(x ) + efi,2(x ) _< -M M M _< --]]g]l~o, t]e M m e, m- ilgll 2[o,l] exp [~-x] ?n
uniformly valid in [0, t] for 0 _< c < so. Returning to the original dependent variable u ( x ) f u ( x ) l <_
v/M
-~e
we have the results
~____~t {{[fl[ + If~l 9[lall + (ial + I~lg). llbll}
+ I~] + ]Zle,
(7.1.26)
142
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
and
/ - M M,
x/elu'(x)l ~- V -~e2.. {llfll -F Ifl[ " Ilall -b (Ic~l q-I/3[g). [Ibl]} + 1/31,
valid uniformly for 0 _ x _ s 0 < e < e0. It follows that we have obtained the following lemma. LEMMA 1 The solution of the initial value problem (7.1.15)-(7.1.16) satisfies the estimate
lu(~)l + v~l,~'(~)l < C m a x [maxlf(x)[ --
[O,e]
max In(e)[ '
O<e<eo
uniformly valid for 0 <_ e <_ eo in 0 <_ x <_ s C is a constant, depending on A, B, ao and s
1.4.
max [/3(e)[] '
O<e<eo
(7.1.27)
'
but independent of e.
T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d Final R e s u l t s
From the equations (7.1.13)-(7.1.14) for the remainder term Rc and Lemma 1 of the preceding section we get immediately the result dRe ]R~(x)l + v ~ l - ~ x (x)[ : O(e)
(7.1.28)
uniformly in any segment 0 < x < t~ for e sufficiently small and ~ independent of e. Remembering that v(~) is O(e), uniformly in 0 <_ x < oc we have the theorem THEOREM 1 Let u~(x) be the solution of the initial value problem d2ue du~ e~ + a(x)--~x + b(x)u, - f ( x ) with 0 < e << 1 and the initial conditions due ue(0) = a and --~--x(0) : / 3 . The coefficients a and b and the right-hand side satisfy the regularity condition to be of class Cl[0, oc) and a(x) >_ ao > O, Vx >_ O. Under these conditions one has for e $ 0 the approximations
[u~(x)- w ( x ) [ - O(e), uniformly in [0, ~]
(7.1.29)
and due dw dv x --~-x( x ) - -d--~x( x ) - ~xx(~) - O(vfe), uniformly in [0, s
(7.1.30)
where the functions w(x) and v(~) are given by the formulas (7.1.5) respectively (7.1.11)and where ~ is an arbitrary positive number independent of e. Moreover, for any 6 > 0 independent of e due dw --~-x (x) - -d-~x(x) - o(x/~), uniformly in [6, s
143
1.4. The Estimate of the Remainder Term and Final Results
Remarks 1. In the case t h a t the coefficient a(x) is strictly negative with a(x) < - a 0 < 0 for all values of x the whole construction breaks down. It can be shown t h a t in this case ue(x) diverges for ~ $ 0. The behaviour of ue(x) as function of x and c is described in Ref.[49], pp.12-17. If a(x) is zero in some point x - x0 we meet the difficulty t h a t the reduced equation (7.1.4) becomes singular in x - x0. In this case complications arise for i > x0. The point x = x0 is called a turning
point. 2. It is to be noted t h a t the b o u n d a r y layer t e r m v(~) has no influence in a first order pointwise approximation of the solution ue(x); its influence becomes only apparent in the first order pointwise approximation of the derivative of ue(x). 3. In the t r e a t m e n t above we assumed a, b and f to belong to the class C1[0, cx~). Whenever these d a t a are s u b m i t t e d to stronger regularity requirements sharper estimates of ue(x) may be obtained; the more regularity of a, b and f the higher the possible order of approximation. In case of C ~ (all d a t a have derivatives of any order) we have the following result. For any n C N U { O } n
n--i
k=O
k=O
X
(7.1.31)
and
due n dwk dx (x) - E e k ~ ( x )
n dvk x + Eek--d-X-X (--~) + O(e'~+l), uniformly in [0, ~], (7.1.32)
k=0
where
k=0
dwo a(x)--d-~x (X ) + b(x)wo(x) = f (x),
0(0)
=
and
a(x) dwk
+
-
d2Wk_l
dx 2 (x),
wk(O) -- --Vk-l (O),
k -- 1, 2 , . . . n .
The b o u n d a r y layer terms vk(~) = vk(~) ( V - I ( ~ ) ~ O) are obtained by expressing Le as a differential operator in ~; using Taylor-expansions of the coefficients a(r c) and b(r the differential operator Le may be written as n
j=--i
The b o u n d a r y layer terms vk(~) = vk(~) are solutions of the system /~_I[VO] = O,
dvo
I dvo
dwo
--d-~x(0) = ~---~-(0) = f l - ---dT(O),
r
vo(~) = O,
144
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
g-~[,~] = - g 0 [ , ~ - ~ ] with
dv~ dwk dx ( 0 ) - - - - - ~ - x (0) '
L~[~_:]...
lim
~---~ OO
-
vk(~)= 0;
~-~[~0],
k = 1 2,...n.
For details of the calculation the reader is referred to [49], pp.7-12. 4. To avoid in an introduction too m a n y calculations, not essential for the m e t h o d d2 of approximation we have taken the p e r t u r b a t i o n operator L2 as d--~" It is, of course, possible to replace
L2 -
d2
dx2"
by the more general operator
L2 -
422 9+c(x) -~x d . +e(x)" and a modification of T h e o r e m 1 is possible. dx 2.
The Boundary
Value Problem
Introduction
2.1.
In this section we study the singular p e r t u r b a t i o n problem
L~[u~] = eL2[u~] + Ll[u~] = e
~
+ al(X)-d-xx +
ao(x)ue
+ bl(X)--~x + bo(x)ue
= f(x),
(7.2.1)
valid for a < x < b and with the b o u n d a r y conditions
u~(a) = ~,
~ ( b ) = ~.
(7.2.2)
The coefficients ai(x), bi(x) (i - 0, 1) and the right-hand side f are independent of ~; ~ is again a small positive p a r a m e t e r and c~ and ~ are arbitrary constants independent of ~. So in contrast with the preceding section we deal now with a boundary
va]ue prob]em. The coefficients ai(x), bi(x) and the right-hand side f(x) are s u b m i t t e d to some mild regularity conditions to be specified later. An explicit solution of this b o u n d a r y value problem is, in general, not feasible and therefore we construct an asymptotic approximation of u~(x) valid in [a, b] for c $ 0. We may expect t h a t u~(x) is approximated to some extent by the solution w(x) of the reduced problem with c - 0, i.e. dw
bl (x)--~x + bo(x)w = f (x),
(7.2.3)
with
~(~)
= ~ o~ ~(b)
=
~.
It is clear t h a t only one of these b o u n d a r y conditions can Be satisfied by w and therefore w(x) will certainly not approximate ue(x) in the neighb~ourhood of the b o u n d a r y point where the b o u n d a r y value of us(x) is not satisfied by w(x). As in the preceding section we have again the loss of one of the b o u n d a r y conditions. We construct again a correction t e r m v~(x)such t h a t u~(x)is approximated by w(x)+ v~(x). To validate this result we should investigate the error R~(x) of this approximation and prove t h a t
2.2. The Maximum Principle for Ordinary Differential Operators
145
Re(x) approaches zero for e $ 0. This will be performed by formulating a b o u n d a r y value problem for Re(x) and by giving consecutively an a priori estimate for R~(x). This estimate is obtained with the aid of the maximum principle for ordinary differential equations of the second order, see lit.Ill9], Ch.1. Because this principle can also be used for the right choice of the b o u n d a r y value of w(x) we treat first this m a x i m u m principle and its consequences. 2.2.
The Maximum Principle for O r d i n a r y D i f f e r e n t i a l Operators
The M a x i m u m Principle may be formulated as follows: Let L be the differential operator d2 d . +c(x) 9"= M + c(x) L = a(x)-~x2.+b(x)-~x
(7.2.4)
with a(x) > 0 and a,b and c continuous in a bounded segment [71, 72] C JR. If a twice differentiable function O(x) attains a local m a x i m u m in an interior point P of (71,72) then M[(I,] < 0 in P. If on the contrary ~ satisfies in (71,72) the differential inequality M[O] > 0 then it is clear t h a t 9 cannot a t t a i n a local m a x i m u m in (71,72); if moreover ~ is also continuous in [71, 72] then the m a x i m u m of ~ can only be attained in one or in b o t h endpoints x = 71, x - 72. This is the m a x i m u m principle in its most simple form. If the m a x i m u m of ~ in an interior point P is positive and if c(P) < 0 then also L [ O ] < 0 in P. As a corollary of this principle we have the following useful lemma. LEMMA 2
If the twice continuously differentiable functions r interval (Tt, 72) C IR the relation
IL[~][ <
L[-r
and gJ(x) satisfy in the bounded
Vx e (71,72),
where L is the differential operator given by (7.2.4) with c(x) ~_ 0 in (71,72)
I~(~y~)l ~ ~(~,),
(7.2.5)
and if
i - 1,2
(7.2.6)
w: 9 [71, 72].
(7.2.7)
then also
Ir
_< r
Proof Suppose ~ - ~ possesses a positive m a x i m u m in an interior point P of (3'1,3'2) then we have according to the m a x i m u m principle
L[r
~ ] f . < o,
which contradicts the relation (7.2.5) and so ~ - ~ does not have a positive m a x i m u m in (71,3'2)- Since we have also ~ - ~ < 0 in x = 71 and x = 72 we must have r 9 _< 0 in [3'1, 72]. Similarly one proves - 4 9 _< 0 in [71, 72] and hence the relation (7.2.7) holds. K]
146
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
The function ~ ( x ) yields an upper bound for the absolute value of r in [')'1, ")'2], reason why ~ ( x ) is called a barrier function for the function r A priori estimates of solutions of b o u n d a r y value problems for second order ordinary differential equations may be obtained by the construction of suitable barrier functions t h a t satisfy the conditions of L e m m a 2. Remark The lemma can be generalized by replacing the strict inequality (7.2.5) by
[L[+][ < L [ - ~ ] ,
Vx 9 (71,')'2).
By this small generalization one obtains immediately as an interesting corollary the uniqueness of solutions of b o u n d a r y value problems. The proof is simple: suppose Ul(X) and u2(x) are two solutions of the boundary value problem with L[u] : f ( x ) , then the function ~ ( x ) - 0 is a barrier function for the difference u 2 ( x ) - Ul(X). For a proof of this generalization of L e m m a 2 we refer the reader to lit.Ill9], Ch.1. 2.3.
A n A Priori E s t i m a t e o f t h e S o l u t i o n o f t h e B o u n d a r y V a l u e P r o b l e m
We return now to our b o u n d a r y value problem (7.2.1)-(7.2.2). We assume for the moment t h a t the coefficients ai(x), bi(x) (i - 0, 1) and the right-hand side / ( x ) are continuous in [a, b]. Moreover, we suppose b l ( X ) r 0 in [a, b] and b o ( x ) < 0 in [a, b]. The first assumption bl (x) r 0 is in so far essential t h a t the reduced equation (7.2.3) does not become singular. In case bl(x0) - 0 for some point x0 E [a, b] the solution w(x) of (7.2.3) has a singularity in x = x0 and the theory becomes much more complicated. The point x = x0 is called a turning point; see section 3 of this chapter. The condition bo(x) < 0 is necessary in order to guarantee the uniqueness of the solution of our b o u n d a r y value problem (7.2.1)-(7.2.2). W i t h bo(x) < 0 we have also bo(x) + eao(x) < 0 for e sufficiently small, uniformly for all x E [a, b] and this guarantees according to L e m m a 2 the uniqueness of the solution of the b o u n d a r y value problem; (see also R e m a r k 3 at the end of this chapter). We take bl(X) > 0; this is not a restriction because the other case bl(X) < 0 can be treated in a similar way. The m a x i m u m principle provides a simple and useful a priori estimate for the solution u~(x) of (7.2.1)-(7.2.2). For this purpose we consider the function u*(x) : ue(x) - fl, t h a t satisfies the b o u n d a r y value problem
L~[u:] : f ( x ) -
t3bo(x)- et3ao(x)= f*(x; e),
a < x ( b,
with the b o u n d a r y conditions ue (a) = c~ - fl, The linear function inequality
(I)(x) = M ( b -
x)
with
u e (b) = 0. M
some positive constant satisfies the
L~[M(x - b)] = E { a l ( x ) M -4- a o ( x ) M ( x - b) } -f- b l ( x ) M -4- bo(x)M(x - b) > M { b l ( x ) A- ~al(x) -+-eao(x)(x - b)}, a < x < b.
147
2.2. The Maximum Principle for Ordinary Differential Operators Hence with 7 - min bl (x) we have the inequality [a,b]
L ~ [ M ( x - b)] > ~1 M ~,, valid for a < x < b and for c sufficiently small, say 0 < c < e0. Choosing
M = max
[~, b],o_<~_<~o
'
b- a
(7.2.8)
we get
IL~[u*~]l <
L ~ [ - M ( b - x)], a < x < b , M ( b - a) > I ~ - al = lu~(a)l, M(b - b) = 0 - ]u~ (b) l. It follows t h a t the function w(x) = M ( b - x) is a barrier function for the function u;(x) and so we get for the solution u~ of the b o u n d a r y value problem (7.2.1)-(7.2.2) the a priori estimate
lu~(x) - j3 I < M ( b - x),
VxE[a,b]and0<e_
with M given by (7.2.8) and M is independent of e. A similar a priori estimate gt(x) = M ( x - a ) for the function u e ( x ) - ~ because
L s [ - M ( x - a)] - e ( - M a l ( x )
(7.2.9)
is not possible,
- M a o ( x ) ( x - a)} - M b l ( x ) - Mbo(x)(x - a)
and the e-independent part of the right-hand side becomes negative in a right neighb o u r h o o d of x = a and so M ( x - a) is not a candidate for a barrier function of u ~ ( x ) - ~. The graph of the unknown function u~(x) lies in the region
t3 + M ( x - b) < u~(x) < j3 + M ( b - x). It follows t h a t
lim d_~ is b o u n d e d for all values of r
y---+O d x
with
(7.2.10) 0 < c < e0. As long as
and
dx ~d2ue are uniformly b o u n d e d in some part of (a, b) the differential equation (7.2.1) is well a p p r o x i m a t e d by its reduced part bl -g~ dw + bow = f ( x ) , but this needs not
to be true whenever
are not b o u n d e d for e sufficiently small. If for - d2 u~ in the instance and -- O ( ~ ) it is possible t h a t the t e r m ~d--~ perturbing part and the t e r m bl d_~ in the reduced part of of the equation may balance dx each other. Due to the result (7.2.10) this cannot h a p p e n in the left neighbourhood of the point x = b and therefore we guess t h a t the solution u~ of the b o u n d a r y value problem (7.2.1)-(7.2.2) is a p p r o x i m a t e d in this neighbourhood by the solution of the b o u n d a r y value problem ~due
1 O(~)
~d x
and
~d x 2
d2 u~ d--~
dw
bl(x)-~x + bo(x)w - f ( x ) with w(b) - / 3 .
148
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
In case bl (x) were negative in [a, b] we would make the guess t h a t the solution ue is approximated in a right neighbourhood of the point x = a by the solution of the b o u n d a r y value problem dw b 1 (x)-~x + bo(x)w = f ( x )
with
w(a)
--
o~.
It will appear in the next section t h a t this heuristic reasoning leads indeed to the construction of a formal approximation of u~, t h a t is to a function which satisfies the differential equation and the b o u n d a r y conditions up to some positive order of ~. 2.4.
The Formal Approximation
For the construction of the formal approximation we need some mild regularity conditions for the coefficients and the right-hand side of the differential equation. These conditions are:
bo(x) 9 Cl[a, b], bl(x) 9 C2[a, b], ao(x) 9 C[a, b], al(x) 9 Cl[a, b] and f(x) 9 e l [ a , b]. (7.2.11) Cm[a, b] is the class of all functions m times continuously differentiable in [a, b]; C[a, b] is the class of all functions continuous in [a, b]. Moreover, we suppose
bl(x) > O,
bo(x) < 0, Vx 9 [a, b].
(7.2.12)
Using the heuristic reasoning of the last subsection we consider first the reduced problem dw
bl (x)--~x + bo(x)w - f(x),
a < x < b,
(7.2.13)
with the b o u n d a r y condition
w(b) - ft. The solution reads
w(x)--eq('){'f e-q(~) bl(~)f(~)d~) ,
(7.2.14)
b
(7.2.15)
x
where b
q(x) = f ~bo(,~) d,~. x
This function is, in general, not a good approximation of u~ in the neighbourhood of the point x = a because w(a) =/=a. So we need a correcction t e r m t h a t accounts for the b o u n d a r y condition ue(a) - a . To investigate what happens in the neighbourhood of x - a we use a "microscope" defined by the transformation x = a 9- ~ ,
(7.2.16)
149
2.4. The Formal Approximation
where u is a positive number that will be used for the "focussing" of the "microscope". The differential operator Le expressed in the new coordinate ~ reads
Le=~
d _d~_ 2. + 61-Val(a + ~v~)__~ . +eao(a + ev~) . d + e-Vbl(a + ~'~)-7;," +bo(a + c~'~) 9 9
1--2u d2
a~
If v is larger than ~, 1 the most dominant terms are e 1-2v d~2" d2 and e - V b l ( a + e ~ @ ) ~ .. To get a balance in the differential equation between the perturbing part and the reduced part we put u = 1. Because of the supposed regularity (7.2.11) of the coefficients we have in [a, b] the expansions
al(a+~)--aj(a)+e~a~(a+~l(x-a)),
0
bl(a + ~ ) = bl(a) + efb~(a)+ 1~2~22 ~ bl''(a + ~ 2 ( X - a)) bo(a+cf)=bo(a)+e@b'o(a+z93(x-a)),
0<~2<1,
0 < ~ 3 < 1,
where primes denote differentiations with respect to x. Inserting these expansions into the differential operator LE we get
Le =
e-'Is
nt- s
(7.2.17)
-{- e s
where
d2 d = ~-~5" +bl (a)d~c .,
s
/:o = {al(a) + @ b ~ ( a ) } ~ . +bo(a)
s
e ,
-- {~a~(a -+-~01~ ) -~- ~2b~/(a + e02~)} d--~" + ao(a + e~). +C~b'o(a+ 6~3~) 9
To satisfy the differential equation up to O(e) and to fulfil the boundary condition at x -- a we make the following attempt for the formal approximation of ue
ue(x) = w(x) + vo( x - a) + evl ( x - a )
= ~(=) + v0(~) +
~(~)
and s
= -s
+ RE(x)
+ R~(~),
(7.2.18)
with s
0 < ~ < co,
(7.2.19)
and the boundary conditions v0(O) = a -
w(a),
v0(oo) = 0 a n d Vl(O) = vl (oo) = O.
(7.2.20)
150
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
The conditions at ~ = c~ provide the desired result that the correction terms v0 and Vl have only a significant value in an e-neighbourhood of the point x = a. From (7.2.19) and (7.2.20) we get d2vo dvo + bl(a)--7;~ -- 0, a~ d~---~
vo(a) -- c~ - w(a),
v0(cx)) = 0,
and hence __ v0(~) ---- { a -
vo(X-a) E
w(a)}exp[-bl(a)~]
= { a - w ( a ) } exp [ - bl (a) x - a].
(7.2.21)
The function vl (~) satisfies the differential equation d2vl d~ 2
dvl + bl(a)-~-
= (A + B,~) exp[-bl (a)~]
where A and B are some constants. With the aid of Vl(0) = vl (oc) = 0 we obtain v l ( X - - a) = Vl (~) = (Cl~ q- 62~ 2) exp [ x-a = {C1~+C2(
x-a c
bl(a)~] 2 ) }exp[-bl(a)
x-a] e 'J
(7.2.22)
where C1 and 02 are again some constants. Substitution of (7.2.18) into the differential equation (7.2.1) yields for the term Re the differential equation Le[Re] = Le[ue] - L~[w] - { e - I L _ I + s
+ es
+ evl]
= f ( x ) - { f ( x ) --k eL2[w]} - e/:0[vx] - eZ:l[v0 + evl] = -~
{
-- r
dw + a o ( x ) w ~d2w + al(x)--d--~x - r
}
- e'{al(a) + ~bi(a)} d~
+ CVl].
Due to the regularity of w (w e C2[a, b]), the exponential character of v0(() and Vl(() with bl(a) > 0, the behaviour of the coefficients in the operator s and the regularity of the coefficients ai and bi as stated in (7.2.11), we obtain L e [ R ~ ] - O(6), uniformly in [a, b],
(7.2.23)
for e sufficiently small, say 0 < e < co. As to the boundary conditions we have Re(a) - 0 and Re(b) -- ue(b) - w(b) - vo( b -_.____aa)_ CVl (b - a) ~ O.
(7.2.24)
151
2.4. The Formal Approximation ( ~ 0 means asymptotically zero). Hence we m a y conclude t h a t the composite expression
w ( x ) + vo (x - a) + evl ( x - a) C
satisfies the b o u n d a r y value problem (7.2.1)-(7.2.2) up to O ( e ) , uniformly in [a, b] and therefore we call it a formal approximation of ue. To prove t h a t it is really a good approximation of the solution we should give an estimate of Re and show t h a t Re vanishes as some positive order of e for e $ 0. This will be done in the next and final subsection. The construction of the formal approximation is visualized in Figure 7.1.
! ! ! ! !
~
-~
X
~
Fig. 7.1 The formal approximation w + v0 ue; w; . . . . . . vo 2.5.
The A Priori Estimate
of the Remainder
According to (7.2.23) and (7.2.24) the remainder small, 0 < e < e0, the inequality
ILe[Re]I < g e ,
Term and Final Results Re
satisfies for e sufficiently
for a < x _~ b,
(7.2.25)
with Re(a) = 0 and Re(b) ~ 0 and K is some suitable constant independent of x and e. To estimate Re we construct a barrier function. We define w(x) - { ~ g ( b - x ) / m i n b l ( x ) } + e - e g * ( b - x) + e, [a,b]
152
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
and we get due to bo(x) < 0
Le[-w] = - ~
and bl(x) > 0
d2w dw } do; ~ + a l ( x ) ~ x + ao(x)w - bl(x)-~x - bo(x)w
= ~ 2 g * a l ( x ) - ~ 2 K * ( b - x)ao(x) - 62a0(x) + {r
[~,b]
- e b o ( x ) { g * ( b - x) + 1} :> g o ,
for e sufficiently small, say 0 < e < e0, where ~0 is again a generic constant, t h a t is to say a value such t h a t earlier s t a t e m e n t s involving ~0 remain valid. Therefore [Le[Re]l
w(a) > 0 ~ R E ( a ) - - 0 , and w(b) - e > R E ( b ) ~ O. It follows t h a t obtain
w(x)
is a barrier function for
[Re(x)[ ~ w ( x ) < eK*(b - a) q- e : - eM,
Re
and with the aid of L e m m a 2 we
a ~ x ~ b,
O < e < eo,
(7.2.26)
where M is a number independent of r As an immediate result we now get
Ue(X ) : W(X) + VO( ~- -- a) -[- ~1)1 ( x -- a) -[- O(E), uniformly in a _< x _< b. However, because the function simplify this result to
vl (~--~) = Vl (~) is bounded for 0 < ( < c~ we may
(7.2.27)
ue(x) -- w(x) + Vo( x - a ) + o(E), uniformly in a < x < b. G Summarizing we have the following t h e o r e m
THEOREM 2 Let ue(x) be the solution of the boundary value problem
{ d2ue
due
}
due
a<x
with 0 < ~ << 1 and uE(a) -- a and ue(b) - ~. The coet~cients and the right-hand side satisfy the regularity conditions (7.2.11) with bl(x) > 0 and bo(x) < 0 on [a, b]. Under these conditions the following approximations are uniform!y valid for e sufficiently sma]l ue(x) = w(x) + vo( x -- a) + 0(~), G
a ~_ x ~_ b
153
2.4. The Formal Approximation
and u~(x) = w(x) + O(e),
a + 5 <_ x < b,
with 5 arbitrarily small positive, but independent of r The function w is a solution of the reduced boundary value problem dw bi(x)-d-~x +b0(x)w=f(x),
a<x
w(b)=
and vo(X= ---~) is a boundary layer term defined by (7.2.21). Remarks 1. A similar theorem can be given for the case bl (x) < 0 on [a, b]. In this case one takes w(a)= a and v0 is given by b ~
X
v0 ( - - - ~ ) = v0(~) = {/3- w(b)} exp [ b t ( b ) b - x].
(7.2.28)
2. Assuming more regularity of the coefficients ai(x), bi(x) and of the right-hand side f(x) one may obtain higher approximations of u~. One has the following result for sufficient regularity of ai, bi and f. There exist constants Cm, m = 1 , 2 , . . . m 0 , independent of e such that for sufficiently small m
m
ue(x) - E eJwj(x) - E eJvJ( x-- a) < Cmcm+ 1, j =0
(7.2.29)
j =0
uniformly in [a, b], with
dwo b l ( x ) - ~ z + bo(x)wo = f (x),
a < x < b,
wo(x) = 13,
bi(x) > 0
and
dwj { d2wj-1 dWj_ l ! bl(x)---~x + bo(x)wj = dx 2 + al(x) dx + ao(x)wj_l , wj(b) = 0 , The functions vj(~)= v j ( ~ ) operator in ~ ,
j=l,2,...m.
are obtained after expanding L~ as a differential m
L~ = E
ejLj"
j---1
They satisfy the boundary value problems L-i[v0] = 0, ~_l[Vj]--
a<x
v0(0) -- a - - wo(a),
--~0[Vj_l]-
~l[Vj_2] . . . . .
Vo(OO) = 0 ~j_l[V0],
154
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations with v j ( a ) = - w j ( a )
and
lim v j ( ~ ) = 0 ,
j=l,2,...m.
The value of m depends on the degree of regularity of the coefficients and the right-hand side of the differential equation. If they are C~176 b] we have the asymptotic expansion oo
oo
~ (~) ~ ~ ~J~j (~) + ~ ~J,j j =0
j =0
C
).
(7.2.30)
3. To guarantee the uniqueness of the solution u~ we made the assumption bo(x) < 0 on [a, b]. This assumption may be omitted. The conditions bo(x) E Cl[a, b] and bl (x) E C2[a, b] imply the existence of two numbers t0 and 5 such that tobl(x) + bo(x) < 5 < O, V x E [ a , b ] . Instead of the function we(x) we consider the function
~(~) = ~ - ~ ( ~ ) and the equation for fie(x) becomes
d2 ztr
d fie + { a l ( x ) + 20}-d-~x + { a 0 ( x ) + toal(x)+ to2}fi~ dfi~
-[- b l ( x ) - - - ~ x --[-- {tobl(X ) -[- b o ( x ) } u e --
f(x)e -tgx,
and herewith we are in the same situation as before. 4. The fundamental Lemma 2 may be generalized as follows. Suppose r the boundary value problem L[r
=
d2r
dR
+ b ( ~ ) ~ + ~(~)~ = f(~),
3'1 < x < "Y2
satisfies
(7.2.31)
with the boundary conditions -
7r
r
0 + 4(71)sin 0 = a, 0 some constant with 0 < 0 < ~,
+ r
cosy + r
sin~o = fl, ~0 some constant with 0 _< ~o _< 2 '
(7.2.32) (7.2.33)
where the primes denote differentiation with respect to x . The coetlicients b, c and the right-hand side f are continuous in ['YI,72)] and
c(~) <_ o
i.
[~, ~2].
Finally, suppose that not all equalities 0 = O, ~ = O, c - 0 hold. Under these conditions the following is true. If the functions r and ql(x) satisfy the inequalities IL[r < L [ - ~ ] , Vx E (~1, ~2)
2.4. The Formal Approximation
and
155
[~[ <_ - ~ ' ( ~ ) ~ose + ~(v~) sinO, I/31 < 9 ' (')2) cos ~o+ ~ (~/2) sin ~o,
then For the proof see [119], pp.14-18. A consequence of this generalization is that the theory of this subsection can be generalized to Neumann and mixed boundary value problems. 5. The maximum principle is a powerful tool for estimating solutions of elliptic differential equations. Dhrr, Parter and Shampine [24] used this principle in great depth to singular perturbation problems connected with two-point boundary value problems for second order ordinary differential equations as well linear as nonlinear. With regard to nonlinear problems we refer the reader to [24] and to Chapter 11. 6. The remainder term Re(x) satisfies the equation
d2Re dRe dRe Le[Rs] = e{ dx 2 +al(x)-~x +aO(x)Re}+{bl(x)-~x +bo(x)Re} = r(x) (7.2.34) with r(x)= O(e), uniformly in [a, b] and with the boundary values Re(a)= 0 and Re(b) ..~ O. We derived from this boundary value problem that Re(x) = O(e), uniformly in In, hi. From this result one may obtain estimates of d-~R (x) and ~d x 2 (x) and hence also dx u~ of ~due and d2dx2" We show this for the case no(x) = el (x) - O, which is not an essential restriction. It follows from (7.2.34) that
d2Rs dR~ dx 2 = e - l { r ( x ) - bl(x)--~x - bo(x)Re}.
(7.2.35)
We introduce the notation
dpRe Io, bl I ~(~)l
max
=
IR~lp,
p = 0, 1, 2 , . . .
and similarly for r(x), bl (x) and bo(x). With this notation it follows from (7.2.35) that
IR~I2 < c-1{I,'1o + IbllolR~ll + IbololR~lo.
(7.2.36)
We now use the well known interpolation formula: for each pair of natural numbers p and q with 0 < p < q the following inequality holds
Ivlp < ~1",-,I,~+ M~-~/cq-~)l~lo with M depending on p, q and [a, b] and with # arbitrarily positive (see for instance Miranda [107]). Taking p = l , q = 2 and # = ~eIbl[o 1 we get 1
IR~]I _< -~eIbllolIRel2 + 2Me-llbllolRslo
156
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations or
1 IR~I2 §
2ME-2[b~I~[R~Io.
(7.2.37)
Substitution of this inequality into (7.2.36) gives
IR~I2 ~ 26-11~1o +
(4ME-21bll 2 +
and therefore with the aid of the estimate
2~-~lbolo)lR~lo,
O(IR~10)-O(Irl0)
IR~12 _~ C2~-2lr[0
(7.2.38)
with C2 a constant independent of e. Using once again (7.2.37) and the estimate (7.2.38) we get IRell _~ C16-11rlo
(7.2.39)
with C1 independent of e. The moral of this remark is that a formal approximation of u~ up to 0(62) gives an approximation of ~ up to O(e) and that a formal approxmation of ue up to 0(63) gives an approximation of ~dx2 up to 0(6) 7. The boundary layer term (7.2.21) has the form characteristic in the so-called W.K.B. approximation, well known from the SchrSdinger equation, (Wentzel, Kramers, Brillouin). In fact the method of composite expansion as treated above and this W.K.B. approximation are related. This is illustrated by the following example. Suppose u(x) is a solution of the boundary value problem
d2u
~
du + b ~ ( ~ ) ~ + bo(~)u = o,
0 < x < 1,
(7.2.40)
with u(0) = a and u(1) = fl
(7.2.41)
while bl (x) > 0 and 0 < 6 << 1. According to the W.K.B. method we substitute
~(~) = ~xp [lq(~)] and q(x) satisfies the differential equation
~+bo(~) Expanding q(x) as
) + ~dq(dq ~+b~(~) ) =o. oo
q(~) ~ ~ ~"q~(~) n--0
(7.2.42)
157
2.4. The Formal Approximation we get the following two approximate solutions 1
q(1) (~) = c~ (~) + ~
/
b0(() d(,
bl(~)
x
q(~)(~) - -
bo(() b-~ar
bl(r162 + ~ [ - logbl(~) + o
+ C~(~),
o
and so the general solution of (7.2.40) reads in first approximation 1
I ~,q)
x
1 [/~ + A2(e) bl(x) exp
1/ - bl(()d(],
x
1 ~,q)
x
d(] .exp [ -
E
0
0
or 1
u(x) ~ A1 (e) exp [ +
f
x
b0(() bl (~) d(] + A3(r
[- e
x
0
Using the boundary conditions (7.2.41) we obtain finally 1
~(~) ~ ~exp[+
f b-~~] b0(~) x
1
+ { a - - ~exp [-b
x
b - ~ d ~ ] } exp [ - c j 0
0
In the first term we recognize the solution (7.2.15) of the reduced equation (f - 0) and in the second term the boundary layer correction (7.2.21). For more information about the W.K.B. approximation and composite expansions we refer the reader to [30, pp.133-139] and to [112, pp.92-99], where also inhomogeneous equations have been treated. Readers interested in the application of the W.K.B. method in quantum mechanics may consult e.g. "Quantum Mechanics" I by A. Messiah North-Holland, 1964.
158 3. 3.1.
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations Boundary
Value Problems
w i t h Turning Points
Introduction
In the preceding section we made the essential assumption that the coefficient bl (x) of dw in the reduced equation (7.2.13) has a fixed sign in the segment [a, b]. In case bl (x) > 0 we have a b o u n d a r y layer at the left-hand side x : a and in case bl (x) < 0 there is a b o u n d a r y layer at the right-hand side x : b. In this section we drop the assumption bl(x) =/: 0 in [a, b] and we discuss again the asymptotic approximation of the b o u n d a r y value problem and the appearance of b o u n d a r y layers and eventually also an interior layer in the neighbourhood of a point where bt (x) changes its sign. In order not to complicate our t r e a t m e n t we consider the following simplification of the b o u n d a r y value problem (7.2.1)-(7.2.2)
d2ye dye e - ~ x2 -t- f(x)-ff-ffx -I- g(x)ye -- O,
- 1 < x < 1,
(7.3.1)
with the b o u n d a r y conditions y ~ ( - 1 ) = a,
y e ( + l ) = b.
(7.3.2)
Further we assume t h a t f(x) has only one zero at x = 0 and t h a t dx dr(x) is either smaller or larger t h a n zero in [-1, +1]. Moreover it will appear useful to assume also t h a t f and g, considered as functions of a complex variable, are analytic for a _ x _ b. In the next subsection we address the most interesting case ddl(x) < 0 and in the following one the other case ~ ( x ) > 0. The case ~ ( x ) < 0 is therefore of particular interest because the value of the p a r a m e t e r _~_=_
g(O)
(7.3.3)
2
is decisive for the approximation of ye(x) in [-1, +1]; in case /3 :/: 2m, m = 0, 1, 2 , . . . the so-called outer approximation in ( - 1 , +1) is identically zero while this approximation is not zero for /9 - 2m. This interesting phenomenon has received the name of "resonance". We finish with some references for further study of differential equations with turning points.
3.2.
T h e Turning P o i n t P r o b l e m w i t h
.f'(x) < 0
To analyze the behaviour of the solution ye(x) of our b o u n d a r y value problem (7.3.1)-(7.3.2) we use a variation of the multiple scale technique as treated in Chapter 6. We consider Ye as a function of the two independent variables ~r
-lv(x),
7/-=--x,
C
where v(x) will be chosen later in an appropriate way. W i t h the aid of
dye dx
10ye dv e O~ dx
OyE Or;
(7.3.4)
159
3.2. The Turning Point Problem with It(x) < 0 and
d2ye _ 1 0 2 y e dx 2 - - ~ 0 ~ 2
(dv~ 2 "[,
+
lOyed2v dx . 2
.e .O~ . .
202y~ ~ -eO~Or/ ~ .
dv dx
02ye Or~2 '
we get after substitution into (7.3.1) (Ko + eK1 + e2K2)ye = 0, with
go-
~
-~ +
~
f o--~'
K1-2
-~x O~Or/ + \ dx 2]--~ + 02 K2 = 0772.
Or/+ g'
After a formal expansion of ye into an asymptotic series Yr : Y0(~, 7]) -3t- Cyl (~, 7]) -Jr E2y2(~, 7]) -~- 9" " ,
(7.3.5)
we obtain the following recursive system of differential equations
K0[Y0]= KO[Yl]
~
----
--~+
(7.3.6)
~xx f - - ~ - = O ,
(7.3.7) (7.3.8)
-KI[Yo]
Ko[yi] = -K2[yi-2] - Kl[yi-1],
i = 2, 3, . . . .
It is sensible to choose v(x) as
v(x) - + / f (x')dx', ~t where x0 become
is still free to be chosen in
[-1, +1];
(7.3.9)
the operators
Ko
and
K1
now
and K1
-
2f
02 0~0r/
+
df
0
0
+ f-x- + g. or/
The zero order approximation yo(x) of ye(x) follows now from 02
0)
b7 ~ + b~ yo(~, ~1 - o, and therefore
yo(C~,r/) = Ao(r/) + Bo(r/)e -~
(7.3.11)
160
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
yo(x)=Ao(x)+Bo(x)exp[If Y(~')d~']
or
(7.3.12)
~2 The coefficients A0 and B0 are determined as usual from the requirement that the next term eyl (x) in the asymptotic expansion (7.3.5) does not become singular. For Yl (x) we have the equation (7.3.7) which after a trivial calculation reads
f~(~ --~
~ "-[- - ~
{ ~176 #~176 ~176 Y l "-- --
{dBo
2f O~OrI + -d--xx---~-( + f ~
df
f ( r t ) - ~ (r/) + ~xx(r/)B0(r/) - g(rl)Bo(rl)
}
}
+ g Yo =
e -~ -
{
dA0 } f ( r / ) - ~ (r/) + g(rl)Ao(rl) 9
Therefore the functions A0(r/) and B0 (r/) satisfy the equations
dAo
(7.3.13)
f (r/) ---;-- + g(r/)A0(r/) = 0 at/ and
d d-~(f(n)Bo(n)) - g(n)Bo(n)
= O,
(7.3.14)
and integration of these equations gives with the aid of (7.3.12) the following expression for the zero approximation yo(x) x
x
y0 x/= Clexp[ / gf(x') x',dx I + ~C2 exp -; i fi x,) x,+f f(~' g/x,/dx, ) ~2
XO
, (7.3.15)
Xo
where x # 0 and where the path of integration avoids the singular point x - 0 by a small semicircle in the upper complex halfplane; it is noted that we use here the analyticity of the functions f(x) and g(x). Introducing the important parameter
we may also write instead of (7.3.15)
vo(~) = C~F~(~.xo) + C~F~(~..o). with
/ (g(x')
f~
FI(X, xo) = x~" exp
9 # o, /31)
\ f ( x ' ) + 2-~
-
dx'
(7.3.16)
(7.3.17)
f~ and
1 ~exp[- ~if f(x')dx' +
F2(x, xo) = - ~ x -
ill) dx' ] ;
f(x') + ' 2 ~
(7.3.18)
3.3 The Asymptotic Approximation around the Turning Point and the Case
161
the integration constants C1 and C2 in (7.3.16) are not the same as those in (7.3.15). The first term in (7.3.16) is essentially the function Ao(x) which satisfies the reduced equation (7.3.13) and therefore C1Fl(x, xo) corresponds with the outer solution outside boundary layers which are represented by the second term C2F2(x, xo). However we have to make at this stage an i m p o r t a n t remark. The point z = 0 is an ordinary point of the differential equation
d2y~ f ( z ) dy~ + g(Z) yE = 0 § dz 2 e dz and by Fuchs' theorem [22, pp.233-236] there exists for e ~ h value of ~ # 0 a unique regular function y~(z) which satisfies the differential equation in a certain neighbourhood of z - 0 and which satisfies initial conditions, say y(0) = h0, y'(0) = hi. This follows formally from the analyticity of the coefficients f ( z ) and g(z). Substituting oo
f (z) = ~ f n z n and g(z) = ~ gnz ~ one obtains for ye(z) the analytic expansion n----1
n--O
oo
n--O
with the recurrent relations
2r
= -goho
2.3eh3 = -flhl
- gohl - glho
(n - 1)nehn = - ( n - 2 ) f l h n - 2 . . . .
f,~-2hl - gohn-2 . . . .
gn-2ho, n > 4.
The coefficients h0 and hi determine uniquely the coefficients hn, n >_ 2, and the radius of convergence of the analytic expansion of y~(z) is the same as t h a t for the expansions of f and g, see [22, pp.233-236]. In case h0 and hi are both zero, we have y~(z) - 0 and hence we have either y~(z) - 0 or (h0, hi) # (0, 0). It follows t h a t the solution C1F1 (x, xo) of the reduced equation can only be a satisfactory approximation of the solution y~(x) of our boundary value problem when either C 1 - 0 or fl assumes the values 0 or 2. In case this is not true we need certainly an extra asymptotic expansion to represent the solution near x - 0.
3.3 The Asymptotic Approximation around the Turning Point and the Case
~=/=2m, m = 0 , 1 , 2 , . . .
We introduce the coordinate stretching --
X
- c r < ( < +cr
(7.3.19)
and we obtain for the first approximation v(~) of y~(x/~ ~) the differential equation
d2v dv + fl(-;-; + gov = O. d~ 2 at,
(7.3.20)
162
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
We introduce the notation
g(o)
fl = f'(O) = -27) ' < O, go = g(O) = f,(O i y'(o) = / 3 7 ,
and as before
g(o)
(7.3.21)
(7.3.3)
/3= - 2 ~ f'(O)" The differential equation (7.3.20) becomes d2~) d~ 2
- 2.~=
dv
a(,
(7.3.22)
+ Z . y v = o,
and application of the transformation ~2 v(C) = w(~) exp[+7-~-]
yields Weber's equation [148, pp.347-351]
d2w 1 lt2)w = 0 dt 2 + (n + -~ - -~ with t = ( v / ~
(7.3.23)
and n = 89 This equation has the general solution
w(t) = E:Dn(t) + E2D_n_:(it), where E: and E2 are integration constants and D,~ and D-n-1 cylinder functions [148, pp.347-351]. Hence we have for v(() ~ y~(v/e() the result (j-2 v(r = exp [7-~-] { E : D 8 9
) + E2D_ 89
}.
the parabolic
(7.3.24)
This solution should match with the solution C:F1 (x, x0) of the reduced equation and therefore the constants E1 and E2 depend on the value of C1 in the following way lim v ( ( ) = limC:Fl(x, xo),
lim v(~) = limCiFl(X, Xo),
(7.3.25)
when ~ is taken as ~ = x/x/~ with x fixed and nonzero and r -+ 0. Using the asymptotic expansions of the parabolic cylinder functions (see [148]) we get '
I )
v(:) : E: ( X / ~ : ) : t 3 (1 + O(~-ff ) + E2 ( i x / ~ : )
- 89
exp[7C2](1 + O ( , _ )) for : -+ +cr
(7.3.26)
3.3 The Asymptotic Approximation around the Turning Point and the Case
163
and v(~) = E I ( ~ - ~ )
1
1
5/3(1 + O(~-~))
- E1F(-I~/3) e x p [ 7 ~ 2 ] e + 8 9 1 8 9 1
+ E2 ( i x / ~ ) - 5 / 3 - 1
+ O()) 1
exp[~2] (1 + O(~-~))
- E2 P(~/3"x/c~+1) e-( 89
(iVan)
89 (1 + O(~-~ff)) , for ~ --+-00,
(7.3.27)
where the continuation of 4 from 4 > 0 to 4 < 0 is along a path in the upper complex half plane. Mathching with the outer solution
C1Fl(x, xo) - C1x~/2 exp -
/ g(x') /3 1 ( I ( x ' ) + 2 x'---))dx' f~
is due to the exponential factor exp[-y~ 2] only possible with E1 -- E2 = 61 = 0, unless 89 = 0, 1, 2, . . . . In the latter case the second term of (7.3.26) vanishes and we obtain with E2 = 0 , 4
v(~) = E I ( X / ~ )
89
+ O(~-~)) for ~ ~ •
and this function can be matched with the outer solution C1Fl(X, Xo); note that both expressions behave as x ~12. Hence it is clear that we have two cases whether /3 is an even nonnegative integer or not. If /3 ~- 2m, m = 0, 1 , 2 , . . . , the only possibility is E1 = E2 = C1 -- 0 and the outer solution represented by the solution of the reduced equation is the zero solution. This means that the solution y~(x) of our initial value problem is approximated by ye(x) = 0 in ( - 1 , +1). In the endpoints x = +1 we have boundary layers taking into account boundary conditions. For the point x = - 1 we take x0 = - 1 and we have by (7.3.16) in a first approximation in the neighbourhood of x = - 1 X
ys(x) ~ a e x p [ - - e1 / f(x')dx']. M
(7.3.28)
-1 For the point x - +1 we take x0 - +1 and we get similarly in the neighbourhood of
x=+l X
b exp[- 71 / f(x')dx'] +1
(7.3.29)
164
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
Under the assumption that these formal approximations are correct we have THEOREM
3 Let y~ = ye(x)
be the solution of the boundary value problem
d2ys dye e-~x 2 + f(x)-d-~x + g(x)y~ = O,
-l<x<+l,
with f and g analytic in [-1, +1], f ( 0 ) = 0
y~(-1)=a,
ye(+l)=b,
and ~ < 0 in [-1, +1]. Whenever
g(0) r 0 , - 1 , - 2 , . . .
y,(0) we have approximately
y~(x) =_0 for - 1 + 5 <__x < 1 - 5 for any small positive 5 independent of e. The solution exhibits boundary layers at x = 4-1, given by 1
f(x')dx'], x < 0 t a r
--1 x
1
f
y(~')d~'],
9 > o.
1
The case 13 - 2m, m = 0, 1, 2 , . . . requires further investigation and it will appear that in this case the outer approximation in ( - 1 , +1) does not vanish in general, reason why this is called the "case of resonance".
3.4
T h e A s y m p t o t i c A p p r o x i m a t i o n in the Case of R e s o n a n c e According to (7.3.16) we have in apparoximation x ~= 0
ye(x) -- CiFl(X, Xo) --b C2F2(x, xo),
(7.3.16)
with
g(x')dx, F1 (x, xo) = exp - ] )'(x') ago
and
F2(x, xo) = ~ - ~ exp
- -e
f (xt)dxt +
f (x') dxt , xo
which may also be written in the form (7.3.17)-(7.3.18). The location of the boundary layers is determined by the sign of the integral +1
I = f f(x')dx'. --1
(7.3.30)
3.4
165
The Asymptotic Approximation in the Case of Resonance
T h e r e are three possibilities: 1. I > 0 We take x0 = - 1 (7.3.16)-(7.3.18)
I > 0, I < 0 and
I-
0. I
and we have due to the e x p o n e n t i a l l y small factor exp [ - ~ ]
from
= b= C1FI(1,-1),
ys(+l) and therefore
-I-1
b C1 = F 1 ( 1 , - 1 )
=b
dx,]
f(x')
exp
"
-1
F u r t h e r we have y~(-1) = a
-- C1
+ C2F2(-1,-1)
=
C1
1 nt- j~, )~':--1' C2~
and therefore
C2-:(-1)(a-Cl)-
f(-1){a-bexp[-t-
+~Jg(xt)dx'l} f(x')
--I
S u b s t i t u t i o n into (7.3.16) yields t h a t y~(x) has in a first formal a p p r o x i m a t i o n a b o u n d a r y layer only at x = - 1 . In the intervals - 1 + 5 < x < 0 and 0 < x _ 1 we have, again due to the small e x p o n e n t i a l factor,
ye(x) ,~ b exp
[ ,if(z')
] ,Ix,ldx, f(X')
+
--1
1
= b exp x
--1
f(x') ]
(7.3.31)
2. I < 0 This case is t r e a t e d along similar lines as the preceding one. I n s t e a d of x0 - - 1 we take now x0 - + 1 and we get due to the e x p o n e n t i a l l y small factor from (7.3.16)(7.3.18) ye(--1) = a = C1FI(-1, +I), and therefore
a
C1 =
-t-1
F 1 ( - 1 , + 1 ) = a exp
[/'lxldx, f(x')
-
-1
F u r t h e r we have y~(+l) - b
-
C1
--t- C 2 F 2 ( + 1 , + 1 )
-
C1
1 -t- ~77-:i-~ ~ C2,
l~l)
166
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations which gives +1
C2= f ( 1 ) { b - a e x p [ - / f ( x , )
9
--1
The result is a b o u n d a r y layer t e r m at x = 4-1 and in the intervals - 1 < x < 0 and 0<x< 1-5 we obtain
ye(x)
f(x')
= C1 exp --
=aexp I
+1
~
J"
(7.3.32)
-1
3. I = 0 Inspection of the second t e r m in (7.3.16) gives the possibility of two b o u n d a r y layers, as well in the neighbourhood of x - - 1 as in the neighbourhood of x - +1. We take x0 = - 1 and we substitute the b o u n d a r y values into (7.3.16) to get a simple system for the integration constants C1 and (5'2. (The choice x0 : +1 leads to the same result for y~(x) ). We obtain a = CIFI(-1,-1)
+ C 2 F 2 ( - 1 , - 1 ) = C1 +
1 f(-1)
C2
b = C1F1(+1,-1) + C2F2(+1,-1) and therefore
af(-1)F2(+l,-1) - b C1 = f ( - 1 ) F 2 ( + 1 , - 1 ) -
(7.3.33)
F1(+1,-1)
and
b-aFl(+l,-1) (72 = f ( - 1 ) f ( - 1 ) F 2 ( + l , - 1 )
(7.3.34)
- FI(+I,-1)"
Inserting these results into (7.3.16) gives the approximation of -1 <x<0 and 0 < x _ < + l .
ye(x)
in the intervals
After this formal construction for the cases I > 0, I < 0 and I = 0, which are formally valid outside the interior layer around x - 0, we have still to investigate the possibility of the matching between the outer solution and this interior layer. Therefore we return to the condition (7.3.25) with v(~) given by (7.3.24) with r 2m, m - 0, 1, 2 , . . . . Just as in the case of nonresonance we have to put E2 = 0 and therefore
~2 v(r
= E1 exp ( ' r - x - ) D m ( x ~ ~ ) .
However the parabolic cylinder function Dm (z) with m a nonnegative integer reduces to Din(z) = ( - 1 ) m e [ z2 ~z dm e_ 89 ( ~z) 1 e_ 88 (7.3.35)
(v~)~
3.4
167
The Asymptotic Approximation in the Case of Resonance
where Hm denotes the Hermite polynomial of degree rn - ~, see [148] and [22]. Hence
1
v( ~) - E1 ( V/-~)m Hm ( v/~ ~)
(7.3.36)
and we have for ~ -+ q-c~
ye(v/e~) ~ v(r
~ EI(V~)
m -- E1
It follows from the outer solution
ye(x) - C1Fl(x, xo) - C l x ~/2 exp [ -
x
(7.3.37)
.
i(g(x' ill)dx' f( x'))+ -2--~
ft
t h a t the matching between this solution and the interior layer (7.3.36) is indeed possible with
E1--CI(~) ~/2exp
-
] ( g(x') f (x') xo
)
~ 1 + -2-~ dx' .
(7.3.38)
Moreover the behaviour of ye(x), in the neighbourhood of x = 0, is t h a t of a Hermite polynomial which satisfies Fuchs' theorem: in case ye(x) is not identical zero it is of the form
ye(x) -- ho + hlx + h2x 2 + ' "
,
with (ho, hi) ~ (0, 0). Under the assumption t h a t our formal constructions are justified we have the following theorem THEOREM
4
Let y~ = ye(x) be the solution of the boundary value problem d2ye dye ~-d-~~ + f (x)-j-~ + g(x)y~ = O,
-l<x<+l,
with f and g analytic in [-1, +1], f ( 0 ) = 0
g(o) /'(o)
q-1 --1
we have with 1 = f f ( z ' ) d z '
ye(-1)=a,
and T~(x)dY
ye(+l)-b,
< 0 in [--1, +1]. Whenever
= 0,-1,-2,...
the following approximations of ys(z)"
1. I > O" Be(Z) exhibits a boundary layer in the neighbourhood of x - - 1 . 1
[
y(z') dx'
y~(x) ~ b exp + x
- 1 + 5 1 _ < x _ < - 5 2 , +52_<x_<1,
168
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations where
51 and 62 are arbitrarily small positive numbers independent o f e.
2. I < O" y~(x) exhibits a boundary layer in the neighbourhood of x -
f(~,)
-1
,
-1_x_-62,
+1
+62_x_<1-61.
3. I = 0 : ye(x) exhibits boundary layers in the neighbourhood of as well x = - 1 as x=+l. The approximation outside -62 < x < +62 is given by (7.3.16) and (7.3.33)(7.3.34). In all three cases there exists a matching interior layer around x = 0 and it is given by
v(() with ( = ~x, of degree m.
3.5
7 = - ~ f ( 1O ),,
=
m=
The Turning Point Problem
El(x/2)-mHm(x/~(), ~2 ----
with
7,(0) (o) and H,~ the Hermite polynomial
if(x) > 0
Because this case is not so interesting as the one with discussion to the most simple case
d2ye dye e~+2ax~-aflye=O,
-1 <x<+l,
i f ( x ) < 0 we restrict our
a>O
(7.3.39)
with the b o u n d a r y conditions y ~ ( - 1 ) = a,
y e ( + l ) = b.
(7.3.40)
As in subsection 3.3 we use the substitution
[ OtX2 ]
u = y~(x)exp ~~,
(7.3.41)
which leads again to Weber's equation d2u dt 2
where t =
~/-~x
and n - - l -
1 t2 + (~ + ~ - ~ ) ~ = 0,
(7.3.42)
2"
Similarly as in (7.3.24) we obtain (~x 2
(7.3.43)
3.5 The Turning Point Problem with f'(x) > 0
169
where D~/2 and D-1-~/2 denote the parabolic cylinder functions and where the integration constants C1 and C2 are to be determined by the boundary conditions (7.3.40). There results C1 - exp [~-~c ]Ol { a D ~ / 2 ( i ~ ) - A b D f l / 2 ( - i ~ ) } , and
C2
--exp
[26 ] aD_l_,/2(
2o~) _ bD_l_,/2(-
)
A
with /X : D_I_~/2(
~~
)D~/2(i~~ )
D_I_~/2(+
)D~/2(i~~)
(7.3.44)
To calculate the approximation of ys(x) for e $ 0 we need again the asymptotic expansions of the parabolic cylinder functions, [148, pp.347-348]
D,~(z) - e-Z~z 2 zn(1 +O(z--15 )) , l argz]
<
3/47r
and 1
Dn(z) - e- 88 z'~(1 + 0(---~))
v~ r(-~)
e+'~'rie 88
+ 0(~2) )
(7.3.45)
1 and + ~Tr. 5 for argz between 4-~7r Inserting these asymptotic expressions into (7.3.43) we should distinguish between the cases ~ r and ~ = - 2 m , m = 1 , 2 , .... i) The case ~ r m=l,2,... From (7.3.44) and (7.3.45)we get
{
~--~--] D-l-~/2(C1 (6) -- exp [26.1
)
-'
(a- (-1)~/2b)(1 + 0(6))
and C2(6) = exp [~6] V~/2(i
)
b(1 + 0(6)).
(7.3.46)
Putting this result into (7.3.43) and using again (7.3.45) we obtain after a tedious calculation: ye(x) = a(-x) ~/2(1 + O(e)), - 1 _< x <_ - 5 (7.3.47) and
y~(x) = bx~/2(1 + 0(6)),
+5 _< x _< +1,
(7.3.48)
170
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations where 5 is an arbitrarily small positive n u m b e r independent of e. These approxim a t i o n s are solutions of the reduced equation
dw
2~X-~x
a ~ w = 0;
moreover no b o u n d a r y layers are needed in neighbourhoods of the endpoints and this was already to be expected since the coefficient of d_~ dx in the equation (7.3.39) is negative at x -- --1 and positive at x = +1. To investigate what happens in a n e i g h b o u r h o o d of x = 0 we use for the calculation of D - l - t ~ / 2 ( 0 ) and D~/2(0) the confluent hypergeometric series expansion, see [148, p.347], and there results
v~" 2 ~/4
D~/2(0) = 1~{~ -:_ ~-]-~),
(7.3.49)
D_,_~/2(0) : v/~r(12-89 f~/4)
and by (7.3.45)-(7.3.46)
y~(0) = o ( ~ / 4 ) ,
(7.3.50)
which matches with ( 7 . 3 . 4 7 ) - ( 7 . 3 . 4 8 ) i n a v ~ - n e i g h b o u r h o o d of x = 0. In case /~ > 0 we have uniform convergence at x : 0, but in case fl < 0 the a p p r o x i m a t i o n fails because it becomes u n b o u n d e d for x -+ 0. Finally, we get with aid of (7.3.49) and (7.3.46) for f~ = 0
y~(0) = ~1 (a + b)
(7.3.51)
In this case we have according to (7.3.47) and (7.3.48)
y~(x)
= a(1 + O(~)) for - 1 _< x < 0 and
y~(x) -
b(1 + 0 ( 6 ) ) for 0 < x _< 1
and so there is no uniform convergence at x = 0; we need an interior layer at x = 0 which gives the transition from the value ye ~ a to the value ye ~ b. ii)
The case /3 = - 2 m , m = 1 , 2 , . . . In this case we m a y use the simplification valid for n a nonnegative integer D~(t)
1
d"
~ t2
= ( - 1 ) '~ exp [~t 2] d - ~ ( e - ~ = (v~)
)
-~ e x p [ - 1 t2]H, (_~t
) = e _lt2t~ (1 + O(1~)),
where H , is the H e r m i t e - p o l y n o m i a l of degree n. After s u b s t i t u t i o n into (7.3.43) we get
c~x21 . {
Ye : exp [ - - - ~ j
x2 CI(V/2) I-m exp [-- o~-~E]Hm_I(
+ C 2 D - m ( i v ~ x) }"
x)
(7.3.52)
3.5 The Turning Point Problem with f'(x) > 0
171
Solving as before C1 and C2 from the boundary conditions we obtain
Cl=Cl(r176 C2=C2(r
Hm-1( 2
)
{ D_m(i f ~ ) } - 1 ( 1 + O ( 1 ) )
and therefore with the aid of (7.3.45)
y~(x) = O ( e x p [a(1 - x2)]),
- 1 < x < +1
(7.3.53)
It follows that in the case f l - - 2 m (m = 1, 2 , . . . ) we do not obtain convergence of the solution of our boundary value problem for ~ --+ 0. Summarizing our results we have THEOREM 5
Let y~ - y~(x) be the solution of the boundary value problem d2y~
c~
dye
+ 2aX-d~ x - a~ye = 0, --1 < x < +1, y~(--1) = a, y~(+l) = b, a > 0.
In case 13 ~ - 2 m (rn -- 1, 2,...) the solution ye(x) is uniformly approximated by the solution of the reduced problem: y~(x) = a(-x)~/2(1 + O(e)), y~(x) = b(x)~/2(1 + 0(~)),
-1 <_x <_ - 5 +5 < x < 1,
where 5 is arbitrarily small positive, but independent of e. Further yr = O(~fl/4). In case ~ = - 2 m ( m - 1, 2,...) the solution ye(x) diverges for e $ 0 for all values of x in the interval (-1, +1). Remarks 1. For historical reasons we mention here the first treatment of s singular perturbation problems with a turning point, also called a transition point, by W. Wasow [141]. Other frequently cited papers on the resonance phenomenon are by R.C. Ackerberg and R.E. O'Malley [1] and by B.J. Matkowsky [104]. Further we refer the reader to [111], [112], [127] and [140] where more interesting references are given. Finally we mention the rather recent contributions by Jiang Furu [79], [80]. 2. It follows from the theory above that the sign of ~ (x) at the turning point x - 0 is decisive for the properties of the approximation of the solution ye(x). We suppose in Theorem 4 that ~-, (x) < 0 for all values of x in [-1, +1]; this is not strictly necessary, but this more stringent assumption has only be made to prevent the occurrence of more turning points in [-1, +1]. The case of several turning points, eventually ofhigher order (e.g. f(x) = - x 3 ( x 2 - 1 ) ( x - 2 ) 2, - 2 < x < b, b > 1 and b ~: 2), has been discussed in [104] and the possibility that the turning point may coincide with a boundary point in [1].
172
Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations
3. An interesting problem is provided by the eigenvalue problem
d2y~
dy~
~-j~-~ - ~ - ~ - + ~(~)y = 0,
- 1 < 9 < +1,
y ( + l ) = 0,
There exist denumerably many eigenvalues )~k(6) for each 6 > 0 and Ak(6) ~ k as 6 ~$0, see [56] and [140]. 4. Kevorkian and Cole, [85] or [112], have given an example where f(x) is no longer analytic, viz. f ( x ) = vf-x. 5. The Schrhdinger equation h ~e~yd-~+ (E - r ( x l ) y = 0 with a potential well V(x) gives rise to a boundary value problem with two turning points. Physicists apply the W.K.B. method to obtain a first order approximation of the wave function y; see Remark 7, Ch.7, section 2. Exercises
1. Determine the asymptotic approximation up to 0(r of the solutions of the following boundary value problems and compare the results with the exact solutions a) 6 y " + y ' - 2 x , y(0)-a, y(1)=/3, b) e y " + 2 y ' + y = O , y(0)=a, y(1)=fl. 2. Investigate the solutions of the singular perturbation problems E2y' ' + y = 0 a n d 6 2 y ' ' + y ' + y = 0 ,
0<x<
1, y ( 0 ) = 0 a n d y ( 1 ) = l .
3. Determine the asymptotic approximation up to 0(6) of the solution of the boundary value problem ~y" + ( 2 + 1)y' + 2~y = ~,
y ( - 1 ) = ~,
y ( + l ) = ~.
4. Let ue(x) be the solution of the boundary value problem
{ d2u~ du~ } 6 ~+al(x)-~x +ao(x)ue +bo(x)ue-f(x),
a<x
with u~(a)=a and u~(b)= ,6. The coefficients and the right-hand side satisfy the following conditions bo(x) e C2[a, b], el(x) e C[a, b], ao(x) e C[a, b], f(x) e C2[a, b] and bo(x) < 0 in [a,b]. Prove that for r $ 0 l u ~ ( x ) - w ( x ) - v ( 1 ) ( x -vf a ~)
- v(2) ( bvf - ~x ) = O(v~), uniformly in [a, b];
s(~) ~(1)(~)={~_ ~s(~)}exp[_~v/_b0(a) ],
w(x) -- bo(x)'
v(2)(()=
13- b - ~
exp[-(~-bo(b)],
~= x - a ~= by- e "
Exercises
173
5. Develop a singular p e r t u r b a t i o n theory, similar as in section 2, for the b o u n d a r y value problem
{ d2u~ du~ } due ~ + al(x)-d-~x + a2(x)u~ + bl(x)--~x + bo(x)u~ = f(x),
+
due dx (a) cos~9 + ue(a)sin v9 = due --~-x(b) c o s ~ + ue(b) sin ~ =
a<x
where ~) and ~ are constants with 0 G z9 < 5, 0 G ~ G ~ and where the coefficients satisfy the regularity conditions (7.2.11) with bo(x) < 0 in (a, b). 6. Give an a p p ro x i m a t i o n of the solutions of the b o u n d a r y value problems
~-~-5x2 + 2x
- 1 < x < +1
=0,
with y ( - 1 ) - a,
y ( + l ) -- b.
7. Give an approximation of the solution of the four possible b o u n d a r y value problems
c~-~x2 •
- 1 < x < +1
•
with y(-1)
= a,
y(+l)
= b.
8. Give an a p p ro x i m a t i o n of the solution of the four possible b o u n d a r y value problems
d2y 6~x2 • 2x
4- 2y = 0,
- 1 < x < +1
with y(--1) -- a,
y ( + l ) -- b.
This Page Intentionally Left Blank
Chapter 8 SINGULAR
1.
PERTURBATIONS OF SECOND ORDER ELLIPTIC TYPE. LINEAR THEORY
Introduction
In this chapter we extend the results of the preceding chapter, section 2, to boundary value problems for linear second order partial differential equations of elliptic type. The theory runs largely along the same lines as that for ordinary differential equations. First one constructs a formal approximation satisfying the differential equation and the boundary values up to some positive order of e and consecutively one gives an a priori estimate of the error with the aid of the maximum principle that is also available for partial differential equations of elliptic type. However there are still some complications which arise from the shape of the boundary of the domain of definition. We study the following boundary value problem for the function u~(x, y) defined on a bounded domain ~ C R2:
L~[u~] = eL2[u~] + Ll[u~] = h(x, y),
(x, y) 9 ~,
(8.1.1)
with 02
02
02
0
L2 = a(x, y)--~x2. + 2b(x, y)OxOy" +c(x, Y ) ~ 2 " +d(x, y) Ox
+~(~,y)~o .+f(~,v) .,
(8.1.2)
and
L1 . . . .
0
Oy
g(x, y),
(8.1.3)
while we have along the boundary 0~t of f~ the first boundary condition
~(~, y)Io~ = ~(~, y)Io~-
(8.1.4)
is again a small positive parameter 0 < ~ << 1. To avoid tedious and noninteresting bookkeeping we assume t h a t the coefficients and the right-hand side of the differential
176
Chapter 8. Perturbations of Elliptic Type
equation are continuously differentiable up to any arbitrary order in the closure ~ of f~. The same is assumed for the parameter representation of the boundary 0f~ and for the boundary values along 012 . Weaker conditions will be given later. Moreover we suppose L2 uniformly elliptic in ~, and so
a(x, y)~2 q_ 2b(x, Y)~1~2 + c(x, y)~22 > 5(~ 2 + ~22), V(x, y) 9 ~,
V(~I, ~2) 9 R2,
where 5 is some positive number; it follows that a(x, y) > 0 and c(x, y) > 0 in ~. Further, similarly as in section 7.2, we need in order to guarantee the uniqueness of the solution the condition
g(x, y) - ~f (x, y) ~ 0,
V(x,y) 9
This condition is satisfied for sufficiently small values of e, say 0 < e < e0, whenever we ~ssume g(~,y) > 0 for (~, y) e ~, see [119], p.70. The general form of a first order differential operator is L1
-
0 0 c~(x, Y)-a-- + ~(x, Y)-_a- + ")'(x, y). OX
ay
However, we have chosen L1 = 0y o 9g(x, y) to avoid singularities in the reduced equation (e = 0) giving rise to turning points, see section 7.3. A turning point (x0, Y0) is here characterized by the condition a(xo, Yo) = ,6(xo, Yo) = O. As long as such a point does not exist in ~, it is possible to transform the first order nonsingular operator c~(x, y ) O + fl(x, Y)~u into the form - ~ y by a nonsingular transformation which does not affect the ellipticity of L2. As to the domain l) we consider the following two cases. 1. There are precisely two points A(xl, Yl) and B(x2, y2) on the boundary 0f~ where the characteristics x -- Xl and x = x2 of the operator L1 are tangent to the boundary 0f~. 2. Part of the boundary 0f~ consists of a characteristic line segment x = Xl and/or x = x2; for instance 012 is part of a vertical strip in R 2.
a !
I
I
L X,
Xz
X
Fig. 8.1 The Domain f~
|
I
177
2. The Maximum Principle for Elliptic Operators
These two cases will be treated in this chapter; we make only some remarks for the case that the domain f~ does not satisfy the conditions 1 and 2. In this chapter we present the theory as developed in Ref. [29]. To justify a formal approximation we have to make an estimate of the remainder term; for this we use the maximum principle for elliptic partial differential operators which is treated in the next section 2. In section 3 we give the formal approximation and in section 4 the a priori estimate of the remainder term and the final results for the case that no part of the boundary 0~t is characteristic. The next section 8.5 is devoted to the second case, where parts of the boundary 0~t may be characteristic. Finally we conclude this chapter with some examples of singularly perturbed elliptic equations with turning points. 2.
The Maximum
P r i n c i p l e for E l l i p t i c O p e r a t o r s
The maximum principle for elliptic operators in two variables may be formulated as follows. Let L be the differential operator 02
02
02
0
0
L = a(x, y)-~x2 + 2b(x, y) OxOy + c(x, y)-ff~u2 + d(x, Y)Ox + e(x, y) Oy + f(x, y). := M + f(x, y).,
(8.2.1)
elliptic in a bounded domain 7) C R 2 while the coefficients a, b etc. are continuous within 7) with a(x, y) > 0. If a twice continuously differentiable function (I)(x, y) attains a local maximum in an interior point P of D then M[(I)] ~ 0 in P. If on the contrary (I) satisfies in 7) the differential inequality M[(I)] > 0 then it follows that (I) cannot attain a local maximum in 7); if'moreover (I) is also continuous in 7) then the maximum of (I) can only be attained in some point of the boundary 07) of 7). This is the maximum principle in its most simple form. If the interior local maximum is nonnegative and if f(P) <_ 0 then also L[(I)] _< 0 in P. _
Proof The proof is very simple since we may always apply an orthogonal transformation in 7) such t h a t the value of the coefficient of ~ in P becomes zero. Because we have in the transformed variables a(&, ~) > 0, c(&, ~) > 0 and b~- V -< 0, b~ - ~ - <- 0 it is evident that the above statement of the maximum principle holds, see also [119], pp.61-68, [23], p.321. [:3 As a corollary of this principle we have again the following useful lemma LEMMA 1
If the twice continuously differential functions (I)(x, y) and r bounded domain 7) the relation
IL[~]I <
L[-~],
V(x,y) e 7),
y) satisfy within a (8.2.2)
where L is the differential operator (8.2.1) with f (x, y) < 0 in 7) and if along the
boundary 07) of 7)
I~l __ ~,
(8.2.3)
178
Chapter 8. Perturbations of Elliptic Type
then we have also the relation 7"
lr
(8.2.4)
< 9 for all (x, y) e / 7 ) . 0
Proof:
see the proof of L e m m a 2 in section 2.2 of Chapter 7.
The function
~(x,y)
yields an upper bound for the absolute value of the function
(I)(x, y) in f 7) and so it is called again a barrier function for the function
(I)(x, y).
0
A priori estimates of solutions of elliptic boundary value problems may be obtained by constructing suitable barrier functions satisfying the conditions of the lemma. This will be applied later on when we have to give an a priori estimate of the error of the approximate formal solution. As in the case of ordinary differential equations the condition (8.2.2) may be replaced by the sharper condition, see [119, Ch.2],
IL[r
<
L[-r
V(x,y) e D.
(8.2.2)*
For a useful application of L e m m a 1 we return to our b o u n d a r y value problem (8.1.1)(8.1.4) with g(x, y) > 0. As to the domain f~ we require in this subsection t h a t it is bounded. The regularity conditions are more t h a n sufficient to guarantee the existence and uniqueness of the solution, see for instance [23], p.336. A very simple a priori estimate of ue is given as follows. Since the coefficient e(x, y) is bounded in ~ there exist numbers e0 and a 2 < 1, independent of E such t h a t - 1 + ee(x, y) < - a 2 in fl, valid for all values of e with 0 < r < e0. Further due to the boundedness of the right-hand side h(x, y) and the b o u n d a r y values ~(x, Y)loa there exists also a number m with the property
Ih( x, Y)I < m in f~ Iq~ x, Y)I < m along 0~. It is now easily checked t h a t the function m
9 (~, y) = ~ ( y with
y0)+ m,
min y - Y0 is a barrier function for all solutions u~ with
0 < e < r
Hence
Of~
there exists a number K
depending on a and Y0 such t h a t
lug( x, Y)I < g m
for 0 < e < e0 and (x, y) e ~.
(8.2.5)
A much better a priori estimate is given in the case that the shape of the domain is restricted to the cases 1 and 2 mentioned in the preceding section. It can then be shown
3. The Formal Approximation
179
that there exists a number M independent of ~ such that for sufficiently small values of e one has the a priori estimate
Ju~(x,y) - ~_(x)l < M(y -~/_(x)),
(8.2.6)
where V - ( x ) denotes the set of boundary values along the lower boundary y = ~_(x); see also next section. This result is the two dimensional analog of the one dimensional estimate (7.2.10). The proof is a little bit complicated because the first and second derivatives of ~'_(x), appearing in L [ - ~ ] , are singular in the left and right extreme points of the boundary. We refer the reader to [29], Theorem IV. 3.
The Formal Approximation
In this section we construct a formal approximation of u~ satisfying the boundary value problem (8.1.1)-(8.1.4), where the domain ~t is characterized by the circumstance that there exist precisely two points A(Xl, yl) and B(x2, Y2) on the boundary 0~t where the characteristics x - Xl and x - x2 of the operator L1 are tangent to the boundary 0~t.
~_O,+ !
....
I
r
%~ 9
L
X2
Fig. 8.2 The domain These points divide 0~t into a lower part 0gt_ and an upper part 0~t+ with the equations y = ~'_(x) and y = ~,+(x) respectively. Along 0~t the function u~(x,y) is subject to the conditions
u~(~, y).o~_ = u~(~, ~_(~)) = ~(~, ~_(~)) := v_(~) and
(8.3.1)
~(~, y)to~+ = ~(~,~+(x)) = v(~, ~+(~)).= v+(~).
180
Chapter 8. Perturbations of Elliptic Type
To construct a formal approximation of ue t h a t satisfies the b o u n d a r y value problem up to some positive order of e we start with the reduced equation (6 - 0)
Ow -~ - g(x, y)w - h(x, y), Oy
(x, y) 9 ft.
(8.3.2)
The b o u n d a r y condition is chosen in accordance with (8.2.6). b o u n d a r y condition for w
Therefore we take as
~(~, ~-(.)) = ~_(~),
~ < ~ < ~.
(8.3.3)
The solution of this Cauchy-problem is easily determined and the result reads y
Y
(8.3.4) ,y_ (~1
,7
with
h*(~, ~) = h(~, ~) + g(~, ~)~_(~). It follows t h a t the derivatives of the function w(x, y) with respect to x are singular for x = Xl and x = x2 and the contribution eL2[w] is certainly not uniformly O(e) in all of Ft. Therefore we should exclude arbitrary fixed neighbourhoods V(A) and V(B) of the points A respectively B from our considerations so t h a t eL2[w] is uniformly O(e) in f ~ - V ( A ) - V(B). The function w(x, y) does not satisfy in general the b o u n d a r y condition along the upper b o u n d a r y y = "7+(x) and so we need a correction term. Following Vi~ik and Lyusternik [138] we introduce a local coordinate system in the neighbourhood of the upper b o u n d a r y Off+, see Figure 8.3.
~
B
I
A j
!
|
i
XI
X3,
Fig. 8.3 Coordinates in the neighbourhood of O f +
181
3. The Formal Approximation
Let p be the distance PS from the point P C 9t to the boundary 0~t+ measured along the normal on 0~t+ and O the arc length from A to S measured along 0f~+ (P sufficiently close to 0f~+). The parametric representation of 0~t+ is given by -
~+0),
v = v+O),
with 0 _< 0 _< 00; the point A has the coordinates (x+(0), y+(0)) and the point B has the coordinates (x+(tg0),y+(tg0)). From the data of the boundary 0~t it follows that x+(O) and y+(O) are of class C~176 00]. Finally we consider a lower neighbourhood ~+ (0 < p < P0, 0 < 0 < 00) of the upper boundary 0~t+ that is sufficiently narrow such that the normals from points of O~t+ do not intersect within ~+. Then there exists in (~+ a 1-1 correspondence between the coordinates (x, y) and the coordinates (p, 0). This correspondence is expressed by the relations - ~+(o)+
p~+(,9)
~+(,~)
'
y= y+(O) - p[l + { il+(o) }z] - 89 &+(0)
'
(8.3.5)
where the dots denote differentiations with respect to 0. Substitution of this transformation into the differential equation (8.1.1)-(8.1.3) gives the equation
~02u~ O~u~ 02u~ L~[ur - e a(p, O)-~p 2 + 2/3(p, O) + ~,(p, O) OpO0 002 +r 0) Ou~ ou~ }
+ ~(P' ~)-bY + A(p, ~)u~
_~(p,o)Ou~ -
- ~ - o(p, O)u~ = ~(p, o) ~,(p, o) -Ou~
(8.3.6)
valid for 0 < p _ < p 0 , 0 < 0 < 0 0 . The coefficients c~,/3,~/, etc can be easily calculated" Oxx
, ( p , o) =
o~'
+ 2b(p, 0) ~Op ~ Op
+~(p,~) ~Op
2
'
.................................
O0
(8.3.7)
~'(P' e ) = o v
where ~t(p, O) - a(x, y), b(p, O) = b(x, y), etc. Since the differential operator L2 is elliptic with
a(x, y) > 0 in ~t it follows that also
a(p, 0) > 0 in (~+.
(8.3.8)
182
Chapter 8. Perturbations of Elliptic Type
Furthermore we have it(0, 0 ) = cos(p, Y)lp=0 < 0,
0 < 0 < 00.
(8.3.9)
Because we have assumed that all coefficients of the differential equation and the parameter representation of 0ft+ are of class C ~ it follows that also all coefficients of the differential operator L~ expressed in p and 0 are of class Coo and so we can make the following "expansions": a(p, O) = ~o(0) + a l (O)p + a2(p, O)p ~
~(p, O) = ~o(0) + ~ ( p , O)p "r(;, o) = ~o(;, o) ((p, o) = (o(O) + 6 (p, o)p
n(p,o) = no(p,o) x(p,o) = Xo(p,o) /.t(p, O) = /.tO(O) + ~l('bq)P + #2(P, ~9)P2
.(p. o)
=
~,o(.~)+ -~(p.~9)p
(8.3.10)
O(p, O) = O0(t9 ) "[- 01 (p, O)p, with ao(O)>O
and # 0 ( 0 ) < 0
for o < 0 < 0 0 .
After these preparations we start to investigate the behaviour of u, in the neighbourhood ft+ by stretching the variable p. Putting p = et we may express the differential expression L~[u~] given in (8.3.6) as follows: (8.3.11)
1 {Mo[u~] + eMl[U~] + e2M2[u~]}, Le[u~] -- -~ where
02 0 Mo = ao(O)~-~ - #o(0)~--~
Otl(O)t-~+ 2flo(tg) ~0--~ + {(o(O) -/-/'l(tg)t}~-~ 02
M1 --
- -o(O) ~
02
0
- oo(O)
and 02 02 02 /142 = a2(p, ~)t2~-~2 + 2fit (p, #)t~-D--~ + 70(P, ~) O0 2
+
+ .o(p,
+
o(p,
- tt2(p, o ) t 2 0 - ~l(p, o ) t o , , - o1(p, ~)t.
(8.3.12)
These expressions are valid in ~+ and the expansions (8.3.10) have been chosen such as to provide an expansion of L~ up to O(e). The operators M0 and Mi have
183
3. The Formal Approximation
coefficients which are linear in t and bounded as functions of 0. The operator M2 has coefficients of the form Xi(p, 0)t ~, i = 0, 1, 2, with X~(P,O) bounded in ~+. The boundary layer term vo(x, y; 6) which should correct the function w(x, y), that satisfies in g t - V ( A ) - V(B) the differential equation up to 0(6) and along the lower boundary 0~t_ the boundary condition us (x, 7 - ( x ) ) = ~_ (x), must have the following properties i) v0(x, y; e) is only defined in the neighbourhood ~t+ of 0 ~ + and it satisfies the homogeneous equation
02vo
Ovo
M0[v0] = c ~ o ( 0 ) - ~
#0(0) - ~
= 0
(8.3.13)
ii) it corrects the boundary condition along 0~t+; hence
{~(~. y)+ ~0(~. y; ~)}[o~+ - {~(p. 9)+ ~0(p. 9; ~)}1.=0 = ~+ (~+(o)) or
v0[t=0 = ~+ (x+(0)) - ~(o, O)
(8.3.14)
iii) w(x, y)+ vo(x, y; 6) approaches the function w(x, y) as 6 $ 0 for p 5 independent of 6; this means lim
t ---~oo
vo(t, 0 ) =
0 for 0 < 0 < to0.
5 ~: 0 with
(8.3.15)
The solution of this boundery value problem is given by
vo(x,y; 6) = v0(p, 0; 6) = {~+ (x+(0))
-
~(o, 0)} exp [tr
P]
(8.3.16)
with
~(o) = ,0(~) < 0 for 0 < 0 < 0o. ~0(o) This function has clearly the character of a boundary layer term; it is asymptotically zero for p ~= 0 with p independent of 6. The two-term w(x, y)+vo(x, y; 6) satisfies in ~t+ the differential equation
Ls[w + v0] - 6L2[w] + h(x, y) + Ml[v0] + 6M2[v0]
(8.3.17)
with the boundary condition
(~ + vo)lo~§ - ~§
(8.3.18)
Due to the presence of the term Ml[V0] it follows that Ls[w + vo] - (9(1) uniformly in ~t+ - V(A) - V(B). To obtain a formal approximation satisfying our boundary value problem up to 0(6) uniformly in ~t+ - V ( A ) - V(B) we still need another term. This is easily done by the addition of an extra boundary layer term 6Vl (x, y; 6) = 6Vl (p, O) = 6Vl (6t, 0), that satisfies in ~+ - V(A) - V(B) the differential equation Mo[Vl] + Ml[v0] = 0 (8.3.19)
184
Chapter 8. Perturbations of Elliptic Type with the conditions V 1
(0, bq)
--
0 and lim Vl (t, O) = O. t-+oo
Due to the exponential behaviour of v0 and the properties of the coefficients of the operators M0 and M1 it is clear that v~ (t, 0) is uniformly bounded in f~+ - V(A) V(B); v l - - P l ( O , t ) e ~(o)t with P1 a polynomial of degree two in t. Further vl(t,O) is asymptotically zero for p = et =/= 0, because n(0) < 0. Adding the extra boundary layer term eVl we get
L~[w + v0 + evl] = eL2[w] + h(x, y) + r + eMl[vl] + e2M2[vl] = h(x, y ) + O(e), uniformly in ft+ - V(A) - V ( B )
(8.3.20)
with
(~ + vo + ~,1)1o,+ = ~'+ (~)" To obtain a formal approximation of ue in all of f ~ - V ( A ) - V(B) we have to extend the relation (8.3.20) outside ~+ - V(A) - V(B). This is accomplished by introducing the infinitely differentiable cutoff factor r with the properties
~p0 /
1 5Po
1
for 0 < p <
0
for 5Po 2 ~- P ~- Po,
and by the definition
vi(~,y;e)=,;
e
'
'
i=0,1.
Now we consider the expression
u; (~, y) = ~(~, y) + .~ (~, y; ~) + ~ Applying L~ we get in f l - V ( A ) -
(~, y; ~).
(8.3.21)
V(B)
L~[u*~(x,y)] = h(x,y) + eL2[w] + L~[v~ + ev~] = h(x, y) + eL2[w] + Le[r + eCVl] = h(x,y) + eL2[w] + e-lMo[r + eCvl] + Ml[r
+ eCvl] + ~M2[r
= h(x, y ) + eL2[w] + r + eM2[vo + r
= h(x, y) + r
+ ecru] + eVl] + Ml[VO + eVl]
+ O(e N) + O(e) + O(eN),
(8.3.22)
uniformly in fl - V (A) - V(B). The symbol O(e N) denotes asymptotically zero. This result follows from the fact that the derivatives of the e independent function r are only nonzero for
4. Estimation of the Remainder Term and Final Results
185
2 l po < P < 5Po, but because v0 and vl are both asymptotically zero for
p ~= 0 (p independent of c) we may collect all terms which contain a derivative of r under the symbol o(cg). Summarizing we have
L~[u*~] = h(x, y) + O(e), uniformly in gt - V(A) - V(B),
(8.3.23)
with the conditions u~ It+ : ~o+ (x),
(8.3.24)
where F+ are the parts of 09t+ that bounds the region 9 t - V(A) - V(B), while V(A) and V(B) are arbitrarily small, but e independent neighbourhood of the points A and B. The function u~ is a formal approximation of the solution ue of our boundary value problem (8.1.1)-(8.1.4). To show that u* is really a good approximation we consider the remainder Re(x, y) defined as R~(~, y) = ~ ( ~ , y) - ~; (~, y) = u~(~, y) - ~ ( ~ , y) - ~ (~, y; ~) - ~
(~, y; ~).
(8.3.25)
This remainder term satisfies the conditions
Le[Re(x, y)] = O(e), uniformly in ~ t - V ( A ) - V(B)
(8.3.26)
with the boundary conditions Re[r + = 0.
(8.3.27)
Moreover Re is uniformly bounded along the right and the left boundaries of V(A) respectively V(B) lying in ~t. This latter statement follows from the uniform boundedness of ue (see (8.2.5))and of u;. These conditions are sufficient for the proof that R~ is indeed O(e) uniformly in - V ( A ) - V(B). This will be shown in the next section. 4.
E s t i m a t i o n of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s
First we specify the neighbourhoods V(A) and V(B) of the extremal points A(xl,yl) and B(x2, Y2), see Figure 8.2. V(A) is chosen as the set {(x,y); (xl < x < x l + 5 ) Ngt} and V(B) as the set {(x, y); ( x 2 - 5 < x < x2)A Ft}, with 5 arbitrarily small positive, but independent of e. The remainder Re (8.3.25) is defined in the domain ~ t - V ( A ) - V(B) and it satisfies the boundary value problem:
Le[Re] = O(e), uniformly in ~ t - V ( A ) - V(B), R~(~, ~_(~))
-
o,
R~(~, ~ + ( ~ ) ) = 0,
~
+ ~ < 9 < ~
-
(8.4.1)
~,
~ + ~ < ~ _< ~2 - ~,
(8.4.2)
186
Chapter 8. Perturbations of Elliptic Type
In~(x~ + ~,y)l < M, IR~(x2- ~,y)l < M,
~ - ( X l + 5) < y < ~ + ( x l + 5), " y _ ( x 2 - 5) < y < ~ + ( x 2 - 5),
(8.4.3)
where M is a suitable constant independent of ~. We define the following function ~ ( x )
~(x) = C-c --F-M{XI(X)-t- X2(X)}
(8.4.4)
with C some positive constant, independent of e and to be chosen later on; Xi(x), i - 1, 2, are nonnegative C ~ functions with the properties Xl(x)=l
forx_<xl+5,
Xl(X)-=0forx>-xl+25
X2(x) - 1 for x _ x2 - 5, A graph of the function ~
X2(x) -= 0 for x < x2 - 25.
is given in Figure 8.4.
i i
x,
D'I,C ~
X,§ IS
i I
X~--IS
I i
i I I
I I I
i
j
C~
,
I Xi
i
X;
I i I
I I I
i
i i
i i
i
!
I I
t x~+ ~ 8
jlj-
Fig. 8.4 Graph of ~ ( x )
2
$
J
i
x
I I i
I
I ~'~ X
4. Estimation of the Remainder Term and Final Results
187
It follows t h a t
Le[-~] = -eM{a(x,y)(x~ t + X~) + d(x,y)(xi + X~2)+ f(x,y)(xz + X2)} + Mg(x, Y)(Xl § X2) § cC(-cf(x, y) § g(x, y)). The order relation (8.4.1) implies the existence of a constant K independent of c such t h a t for c sufficiently small, say 0 < e < e0
[L~[R~]I <
V(x, y) E F t - V ( A ) -
Ke,
V(B).
Because all coefficients of L~ and the functions Xi together with their derivatives are uniformly bounded in ft and further g(x, y) > 0 in ft, it is possible to choose C independent of e and sufficiently large such t h a t we get for 0 < e < e0 the inequality L ~ [ - ~ ] > Kc > [L~[R~][. Moreover ~ ( x )
(8.4.5)
is positive along OFt+ and OFt_ and 9 (x) > M for x
=
x 1 §
5 and x -- x2 - 6.
Now it follows from (8.4.1)-(8.4.3) and (8.4.5)-(8.4.6) t h a t Re and so we obtain from lemma 1
~
(8.4.6)
is a barrier function for
IR~(x, Y)I _< ~(x) for Xl + 6 ~_ x ~ x2 - 6 and hence
[R~(x, Y)I-< ~(x)
for
X1
§ 26 _< X _< X2 -- 26
or
IR~(x, y)[ _< Ce
for
X1
§ 26 _~ X < X2 - 26.
(8.4.7)
Therefore R~(x,y) is O(e) uniformly in the part ~* of ~ which is bounded by the verticals x - Xl + 26 abd x - x2 - 26, where 6 is an arbitrarily small positive number independent of e; the same applies to the expression u~(x, y) - w(x, y) - v~ (x, y; e) - ev~ (x, y; e). Finally, because v~ is uniformly bounded in ~* we have also
u~(x, y) = w(x, y) + v 0*(x, y; e) + O(e), uniformly in Ft-*,
(8.4.8)
Hence the solution u~(x, y) of the b o u n d a r y value problem (8.1.1)-(8.1.4) is pointwise approximated uniformly up to O(r in ~t* by the two t e r m w(x, y ) § v~ (x, y; e). So we have obtained the following theorem. THEOREM
1
Let u~ be a solution of the boundary value problem (8.1.1)-(8.1.4) with L2 uniformly elliptic in the domain of definition ft, a(x, y) > 0 in ~ and g(x, y) > 0 in ft. The boundary Oft has only two extreme points A and B where the tangent is vertical.
188
Chapter 8. Perturbations of Elliptic Type
Moreover the coefficients and all other data are of class C ~ the solution u~ is uniformly approximated as
Under these conditions
u~(~, y) = ~(~, y)+ ~; (~, y; ~) + o(~).
(8.4.8)
uniformly valid in a - V(A) - V(B), where V(A) and V ( B ) are arbitrarily small e independent neighbourhoods of the points A and B. w(x, y) is a solution of the reduced equation (e = O) and is given by (8.3.4), whereas v~(x, y; e) is a boundary layer term given by (8.3.16) and (8.3.21): Remarks 1. In case the d a t a of the b o u n d a r y value problem (8.1.1)-(8.1.4) are of class C ~ the approximation (8.4.8) uniformly valid in f ~ - V ( A ) - V ( B ) may be improved by constructing formal approximations of the form rn
~; (~, y ) =
~'~,(~, y ) + i=0
rn+l
~,,,.(~, y; ~),
~
m = 1,2,3,...
i=0
where Ll[w0] = h(x,y),
wo(x,v_(x)) = ~o_(x)
Ll[wil--L2[wi_l],
wi(x,'~_(x))=O,
i=l,2,...m.
The b o u n d a r y layer terms v*(x,y; e) = r e) follow from the expansion of L~ in the b o u n d a r y layer with coordinates p = et and 0. The expansion is of the form L~ = e - l { M o -t- e M 1 q- e 2 M 2 -I-" 9 9-t- em+lMrn+l} and vi(x, y; e), i - O, 1,... m + 1, are determined by the b o u n d a r y value problems V0loa+ = Volt=o=o = ~o+(x+(v~)) - ~0(0, 0),
M0[v0] = 0, lim v0 -- 0, t--~ oo
i
M0[.,] = - ~ M j [ , , _ j ] ,
~ = 1, 2 , . . . ~ ,
j=l
Viloa+ = vilt=o=o = - ~ i ( 0 , 0), and
lim vi - 0
t--~oo
m+l
M0[vm+l] = -
E
Mj[vm+l-j],
j=l
Vm+llon+ = Vrn+llt=p=O=
0,
lim vm+l =
t--~oo
0.
The final result is m
yr~
i--0
i=0
e v~
e) + O(e m+l
(8.4.9)
4. Estimation of the Remainder Term and Final Results
189
uniformly in F t - V ( A ) - V(B) and valid for all m e {0} U N. For details the reader is referred to [29]. 2. It was for reasons of avoiding tedious bookkeeping that we assumed C ~ smoothness of all data of our boundary value problem. However for the validity (8.4.9) it is enough to assume that the coefficients and right-hand side of the differential equation and the boundary values ~+(x) are continuously differentiable up to and including the order (2m + 3) in ~ respectively along OFt. Moreover the parametric representation of OFt. should be continuously differentiable up to and including the order (2m + 6), see [29]. 3. The asymptotic expansion constructed above fails in the neighbourhoods V(A) and V(B) of the extremal points. Several authors [138], [53] studied the behaviour of the solution ue(x, y) in these neighbourhoods. To obtain an approximation uniformly valid in the whole domain ~ is rather complicated and involves matching of the approximation in Ft - V(A) - V(B) with that in V(A) or V(B). A result rather easily to formulate has been given by J. Frankena in [42]. In case the boundary OFt in the neighbourhood of A is represented by lY - Yll = 611 x - xll ~/p and in the neighbourhood of B by lY-Y21 = C21x-x21 ~/q with p and q integers not less than 2, the remainder term R~(~, y) = u~(~, y) - ~ ( ~ , y) -
~ (~, y),
with w and v~ given by (8.3.4), (8.3.16), (8.3.21), may be estimated uniformly in all of Ft. The result reads _
IRel _< C m i n { ( x - xl)l/p(x2 - x) 1/q,
e(x - xl)(1/P)-2(x2 - x)(1/q)-2}, (8.4.10)
_
uniformly in Ft; C is a suitable constant independent of ~. Moreover if s = max(p, q) we have the estimate
Re(x, y) = O(el/28), uniformly in ~.
(8.4.11)
An improvement of the estimate (8.4.10) has been given by J. Grasman in [53]. Jiang Furu has constructed in [81] a formal approximation of the solution ue(x, y) in the neighbourhoods V ( A ) a n d V(B), using multiple scales (p,~,l_ig(p,~))), where g is a function of p and ~ to be determined by some boundary value problem. 4. A generalization in the direction of more general domains Ft is possible. The requirement of only two extremal points of OFt may be relaxed. To fix the idea consider for example a region Ft as sketched in Figure 8.5. Applying the same construction as explained in this section one obtains i) boundary layers along A F and D B which correct the function .w(~)(x,y) determined from the reduced equation and the boundary values along AB. ii) a boundary layer along F C that corrects the function w (2) (x, y) determined from the reduced equation and the boundary values along DC. iii) a "free" boundary layer along D F which smoothes the discontinuity across D F between the functions w (1) and w (2). One might also study for example the interaction of the three boundary layers in the neighbourhood of the point F. For results see lit.[105], [106].
190
Chapter 8. Perturbations of Elliptic Type
B
A I
i
I
i
I l
I
"
1
i
I
I
Fig. 8.5 Domain with more than two extremal points 5. To guarantee the uniqueness of the solution u~ we make the assumption g(x, y) > 0 in ~. Similarly as in the case of ordinary differential equations, see Remark 3 of section 7.2.5, this assumption may be omitted. Due to the continuity of g(x, y) in the bounded domain ~ there exist two numbers 0 and 5 such t h a t
0 + g(~, y) > ~ > 0, Instead of the function
u~(x, y)
v(~, y) e ft.
we consider the function
~ ( ~ , y) = ~-O~u~(~, y). The expression
Ll[ue(x,y)]
becomes
0
Oy (e'~Uue(x' y)) - g(x, y)eaY~e(x, y) = e ay { - ~ y ~ ( x ,
y) - (0 + g(x,
y))~e(x, y)},
and herewith we are in the same situation as before. 6. Formal approximations up to O(r 2) and O(e 3) provide the possibility to approximate also the first and the second derivatives of u~(x). The proof hinges on the application of an interpolation formula and runs along the same lines as given in Remark 6 of section 7.2.5 and the results are similar.
5.1. Introduction 5.
5.1.
191
Domains with Characteristic Boundaries Introduction
It is of interest to study singular perturbation problems associated with the partial differential equation
Ls[us] = eL2[us] + Ll[us] = h(x, y),
(x, y) e a
(8.5.1)
with L2 of order 2 and uniformly elliptic in ft and with L1 of order 1, while the boundary 0f~ of the domain ft contains parts which are characteristics of the operator L1. In this section we treat an example to illustrate the procedure for handling problems of this kind; a more general case has been treated by Eckhaus and de Jager in [29]. Before setting up a scheme for the construction of a formal approximation of u~ we study first a simplified problem in order to get some insight into the behaviour of us in the neighbourhood of a characteristic boundary. Therefore we consider the following boundary value problem
Ls[u~] - e
{ C')2Ue-'~'- -'~y2 O2Ue}
Oy =0,
x>0,
y>0
(8.5.2)
with
us(x,O)=O, d
x>_O,
us(O,y)=qo(y),
y_>O,
qo(O)=O,
(8.5.3)
2
where 99, ~yy, and ~d . . . the need. of this regularity . are uniformly bounded; condition for qo will become clear later on. The solution w of the reduced equation that satisfies w(x, 0) - 0 is simply w -- 0. This solution does not fulfil the boundary condition us(0, y) = qo(y) and so we need again a correction term. We introduce the local coordinate X
= v~
(8.5.4)
and the equation (8.5.2) is transformed into
02us O~2
Ous Oy
02us
=
~
Oy2 "
(8.5.5)
In this preliminary situation we might expect that a first formal approximation of us is given by the solution of the boundary value problem
02v O~2
Ov -- 0 Oy
(8.5.6)
with the conditions v(~,0)=0,
~>0andv(0,
y)-qo(y),
y>0.
(8.5.7)
The solution is easily obtained and it reads (x)
(8.5.s) ~/v~
192
Chapter 8. Perturbations of Elliptic Type
Hence tvl < M ~
2
erf c[
],
where M is the upper bound of the function I~(Y)]" So v is uniformly bounded and analyzing the expression (8.5.8) for large values of f and finite values of y one gets
,(~, X
yo) = o ( ~ ) ,
(8.5.9)
uniformly for x > 5 > 0 and any fixed value y0 >_ 0. 5 may be any positive number, indpendent of e and the symbol O(e N) denotes again asymptotically zero (see also Exercise 8 of Chapter 2). Therefore the function v ( ~ , y) is a boundary layer function and we call it a parabolic boundary layer function since it is a solution of a parabolic differential equation. We put now
~(~, y)= .(~, ~) + R~(~, y), X
where Re satisfies the boundary value problem
02 Re e
OX 2
02 R~ ~ + Oy2 )
ORe 02 v Oy = - ~ -~y2,
x > O,
y >0
with
R~(~, 0 ) = R~(0, y ) = 0,
~ > 0,
y >__0.
(8.5.10)
O2v To show that v ( ~x, y) is at least a formal approximation of ue(x, y) we need sb--~y2 = o(c), uniformly in x > 0, y >_ 0, or eventually in a bounded part of this region, say x 2 + y2 < r 2. However we have from (8.5.8) oo
02v V~{ ~ exp[ Oy2 -(2y)3/2
1~2 ~yy]~'(O) +
/
e-
892
~2
~o"(y- ~-~)at
}
(8.5.11)
~/,/'N and it follows that ~ has a corner sigularity at the origin x = 0, y = 0 whenever ~'(0) ~= 0. The nature of this singularity is clearly revealed if in equation (8.5.11) the origin is approached along any curve ~ = my ~ with m and c~ arbitrary positive constants. The presence of this corner singularity indicates that in attempting a proof for an appropriate estimate of the remainder Re we may expect a difficulty. This difficulty disappears whenever dd-~y(0) = 0 and therefore we change the boundary. condition ue(0, y) = ~(y) a little bit such that the new boundary value ~5(y) has the properties ~(0) = 0 and -~y(0) = 0. Therefore we define ~(y) -- ~ ( y ) - y~o'(O)exp[-- Y l,
(8.5.12)
5.1. Introduction
193
where a is a positive constant to be determined later on. We have indeed ~(0) = 0 and ~'(0) = 0; moreover the change in the boundary condition is small because
~(y)
-
[-~] ~(y) = ~'(0) --exp ~
= O(~),
uniformly for 0 _< y < oo. Instead of the approximation v that satisfies (8.5.6)-(8.5.7) we use now the function that fulfils (8.5.6) and the boundary conditions ~((,0)=0,
~>_Oand~(O,y)=~(y),
y>O.
Similarly as in (8.5.8) we get oo
~((, y) =
e-~
~(y - - ~ ) d t .
(8.5.13)
Assuming ~o"(y) continuous we obtain from substitution of (8.5.12) into (8.15.11)
o~ oy~ =
~
~-
~2 2 ,(o)___~. ~ ) + {~-~
r
dt
~/ v ~ = 0(1) 4- 0 ( ~ - ~ ) , uniformly in any quarter disk x 2 4- y2 < r 2. Putting now
X
ue(x, y) = ~ ( ~ ,
y) +/~e(x, y),
(8.5.14)
(8.5.15)
we have instead of (8.5.10) for the remainder term /re the boundary value problem
r
Ox 2 4. (~y2
(~y = --g'~y2 = O(~) + o ( ~ l - - a ) ,
(8 . 5.16)
uniformly in x 2 + y 2 _ < r 2, with /~e(x, 0 ) - 0 , x _ > 0 and /~(0, y ) = p ( y ) - ~ ( y ) O ( ~ ) , uniformly in y _> 0. The exponent a is still free; we take a = 1 and it follows that ~ ( ~ ) is a formal approximation of ue(x, y). To prove that ~ ( ~ , y ) is indeed a good approximation of u e ( x , y ) we make an estimate _ of / ~ invoking again the maximum principle. A simple barrier function for Re is given by -= v/~C(1 4- y), with C sufficientily large and we get / ~ ( x , y) - O(x/~), uniformly in any bounded domain x 2 + y 2 _ _ r 2, X
u~(x,y)=~(~,y)+O(x/~),
x_>0,
(8.5.17) y_>0. Hence
uniformly i n x 2 + y 2 _ _ r 2,
x_>0,
y_>0.
194
Chapter 8. Perturbations of Elliptic Type
However
(x) v(~, y) -- ~(~, y) --
99 (0)
Y-
e- 89t2 (y -- ~-~) exp
~/v'~-~ = O(v/e), uniformly in x >_ O,
2t2
dt
v~
y _> O,
and so we may finally conclude
X
u,(~, y) = v(-~, y) + o(v~),
(8.5.18)
uniformly in x 2 + y2 <_ r 2, x > O, y >_ O, where v ( ~ , y) is given by (8.5.8). The boundary layer term 0(~,y) is called a regularized boundary layer term. In a first approximation of ue the nonregularized boundary layer term is sufficient; the regularization is only needed to prove that the error is O(V/r uniformly in any bounded region of the domain x _> 0, y _> 0. Summarizing we have the following theorem THEOREM 2 Let uE satisfy the boundary value paroblem (8.5.2)-(8.5.3) with qo, du ' ~dy2 , continuous for y > O. Then the solution ue(x, y) is approximated by
oo X V~~ / 1t 2 x 2 u~(~, y) = v ( ~ , y) ~-~ ~(y - 2 - ~ ) e t + o(v~) z uniformly in any bounded region in the domain x > 0, y > 0. 5.2.
The Singular Perturbation
P r o b l e m in a R e c t a n g l e
By now we have carried out sufficient preparations for the treatment of a singular perturbation problem associated with the differential operator e A - ~ and defined in a rectangle. To be specific we investigate the following boundary value problem 9u~
r
(0 ~
02u~ ~ + Oy 2 ]
Ou, Oy
= O,
0 < X
< el,
0
< y < s
(8.5.19)
with the boundary conditions
Ue(X, 0) : fl(X),
0 < X < el,
Ue(X, e 2 ) : f2(x),
0 < X < el,
U,(0, y) -- gl(Y),
0 < V < 12,
U,(ll, y) = g2(v),
0 < y < e2,
where f l ( o ) -- g~(O),
f~(e~) - g2(0),
f2(0) = g~(e2),
f2(e~) - g2(e2).
(8.5.20)
5.2. The Singular Perturbation Problem in a Rectangle
195
Y D
$ .,,.
Ii
Fig. 8.6 Rectangular domain Moreover we take f~(x), g~(y), i = 1, 2, infinitely differentiable; this condition may of course be weakened, but for the m o m e n t we will not bother about too much bookkeeping. It is clear t h a t the solution of the reduced equation with the b o u n d a r y condition ue(x,O) = f1(x), 0 _< x < s is
W(X, y)
= fl(X),
(8.5.21)
and therefore we need correction terms along the three boundaries AC(x = 0), BD(x = ~1) and CD(y = ~2) and we get b o u n d a r y layers along these three boundaries. The b o u n d a r y layer terms along AC and BD are regularized with the aid of (8.5.12)(8.5.13) and the b o u n d a r y layer t e r m along CD is constructed similarly as in (8.3.16). In this way we a t t e m p t the following approximation of u~(x, y)
ue(x, y)
= fl(x) Jr- Vl ( ~
, y) + 32( e l ~ - x, y) -~- v3 (x, ~2 .. c-- y)
Jr_Re(x, y)
(8.5.22)
with x
x2
Vl(-~,y) --
~I--X y) =
e-~t ~I(Y-- ~t2) dr,
/oo e-
(~2(Y--
.1_.2 )dt (8.5.23)
and
!
(~1 (Y) -- gl (Y) -- f l (0) - gl (O)y e
~2(Y) - g2(Y) - f1(~1) -
r
_
-- --Y--
,
g2(0)Y e
,/;,
,
x
- s~(~)-/~(~)-
~1 ( ~ , e ~ )
~1 -
- ~(
v~
x
~2).
'
(8.5.24)
196
Chapter 8. Perturbations of Elliptic Type
From the estimate (8.2.5) or (8.2.6) it is celar that ue(x, y) is uniformly bounded in the rectangle A B e D and so Re(x, y) is also uniformly bounded in this rectangle. Moreover Re(x, y) satisfies the boundary value problem 2Re
(0
OX 2
-t-
02Re ) Oy2
ORe Oy = Se(X, y),
0 < X < el,
0 < y < g2,
(8.5.25)
with
d2 f l ( x )
~- 02el ( X y) -~- 02V2 ( gl --X , y)
d2r ~2 -- Y] } + ~x2 (x)exp [ -
(8.5.26)
and
R~(~, 0) = - ~ ( ~ ) ~ - ~ ,
R~(~,e2) = 0 gl
Re(O,y) = - - v 2 ( ~ , Y )
-
-
v3(0,
gl
g2
_._It_
,
e y) + gl(O)ye ~ -
g2
--
,
__.E_
After careful inspection of the boundary data along the boundary of a.o. (8.5.20), it appears that we have the uniform estimate
R~(~, y)
(8.5.27)
ABCD, using
(8.5.28)
= O(v~). O(ABCD)
To estimate Re(x, y) in the interior of ABCD we have to investigate the right-hand side Se(x, y) of (8.5.25). We have ef~'(x) = O(e) and by means of (8.5.14) with a = ~1 we have also the estimates ~ ~ou2 ( ~x, y ) = O ( v ~ ) and
e ~ 2 ~ v~ ' Y ) = O(v~), uniformly in ABED, and so it remains to inspect d--~(x)exp[-~2-u] in A B e D " dx 2 e J d2(p _ f ~ , ( x ) _ y I , ( x ) ax 2
where
-
-
1{ 0 2
02
}
~ ~12'01(~1, e2) + ~--~'02(~2, e2) ,
(8.5.29)
X ~1 -- X ~1= ~ andS2= V/~ 9
It follows t h a t d2 ~ exp [ - ~ ]e J = O(~) 1 and Se(x,y) = O(1) in smallneighbourhoods of the cornerpoints C and D. An estimate of Re(x, y), uniformly small in the whole rectangle ABCD is not yet possible and therefore we improve the boundary correction along the upper side CD. We remark that it is not surprising that we are at this stage not yet ready to give a uniformly small estimate for the remainder term; the
197
5.2. The Singular Perturbation Problem in a Rectangle
reason is the overlap of two boundary layers in the cornerpoints C and D. Using the local coordinate 7/ - e2-v the boundary layer term V3 has been obtained as a first approximation of the boundary value problem
02v *
Ov*
C~?~2
OT]
- - +
_~2 02v * OX 2 '
=
O < x < ~,
~2 ,
O < rl < - -
(8.5.30)
with the boundary conditions x il-x v*(x,O) = (p(x) - f : ( x ) - f~(x) - ~ , l ( - - ~ , t 2 ) - ~,~.( V'Z ,t~), =o.
v*(~, ~)-
(8.5.31)
(8.~.32)
-
l~ -y
The approximation ~3(x, ,) - r = (p(x)e- ~ satisfies (8.5.30) with e = 0 and the first boundary condition (8.5.31), while the second boundary condition is replaced by lira v*(x, 77) - 0. Instead of ~3(x, r/) we take now as an approximation of v*(x, rl) ~-~c~
~(~,,) + C~(~,,), where V4(X, 77) is a solution of 02V4 OqV4 -cOrl2 ~ (OrI with the conditions immediately that
02~3 cox 2 -
d2r (x)e -~/ dx 2 '
V4(X, O) -- 0 and
0 < x < ~1,
0 < 71 < -~2----c'
limn__~ V4(X,?7) -- O, 0 _~ X ~ ~1. It follows
d25
Vn(X, ?7) "-- ~X277 e-~"
(8.5.33)
Finally we decompose the function u~(x, y) as
u~(~, y) = fl(~) + ~
(~
' Y) + ~ ( ~1v~-- X ' y)
-[-~3(X '
+ k~(~, y). Now we have obtained for /~e(x, y) the following boudary value problem 02
02 )
0/~e = 5 ( x , y),
(8.5.34)
with
{ '/
02Vl
X
02V2 ~,l--X,y)_.~_E2d4~)
= ~ f~ (~1 + -~y~(-~' Y) + -~y~( v~
-~,Te-"
}
(8.5.35/
198
Chapter 8. Perturbations of Elliptic Type
and boundary conditions similar to (8.5.27). With the aid of (8.5.24) and the definitions (8.5.23) we have
~4~
= O(e-~) and ~ x 4 - O(e-2), uniformly in A B e D
and by means of (8.5.14) with a -
1 we get (8.5.36)
Se(x, y) = O(v~), uniformly in A B E D .
The addition of the boundary values of e2~4(x, y) have changed the boundary values of Re(x, y) only up to O(r 2) and so we have accordingly to (8.5.28)
(8.5.37)
/~e(x, y ) = O(v~), uniformly along O(ABCD). Introducing once again the barrier function @(x) = Cv/e(y + 1),
with C sufficiently large, but independent of e, we obtain by (8.5.34), (8.5.36), (8.5.37) and Lemma 1 the result /~(x, y ) - O(V~), uniformly in A B E D . Hence
.~_e2~4(X, g2 - Y) + O(v~), uniformly in A B E D . C
Finally, because
V4(X,7/)=
d-~-e-ndx, 'i =
(8.5.18), uniformly in A B E D ,
ue(x,y)
: fl(x)-q-
~1
0(})
and
v , - ~, = O(v~),
i=
1,2, see
we have also
(~
' Y ) -]- v 2 (
~'~-
~ ' y) +
,~(~, ~
c-
Y) + O ( v ~ ) '
(8.5.38)
uniformly in the closed rectangle A B E D , with
9 Y) VI(--~,
"{ gl (Y -- ~et2 ' ) -- fl(O)
oo -
-
e-2
-
)
dt,
(8.5.39)
z
"'
,Y)=
and ~3( x,
"g2 -- Y) _--r
g.(Y-
2.t-------~-
exp_ [s -- Y], C
(8.5.40)
(8.5.41)
5.2. The Singular Perturbation Problem in a Rectangle
199
where x
-
12) - v2(
' 12).
(8.5.42)
Summarizing our results and keeping track of the required differentiability of the data we have established the following result THEOREM 3
Let the function u~(x, y) be the solution of the boundary value problem {02u~
in the domain conditions
0 < x < s
02u~} + -O-yy
Ou~ =0 oy
0 < y < ~2, with 0 < e << 1 and with the boundary
ue(x , O)- fl(X),
Ue(X,~2) -- f2(x),
0 _~ X _~ ~1,
us(O, y) -- gl (Y),
ue(~l, Y) -- g2(Y),
0 _~ y _~ ~2.
Under the assumption that the boundary values are continuous in the cornerpoints (fl (0) = g(O), etc.) and that fi and gi (i = 1, 2) have continuous derivatives up to and including the order four, one has for ue the uniform estimate (8.5.38) given above. Remarks 1. The boundary value problem treated in this section has been studied also by L.P. Cook and G.S.S. Ludford [21]. They investigated a semi-infinite strip problem with the aid of Fourier transformation and the results for the rectangle were obtained from the superposition of two infinite strip problems. Further we mention the work by V.A. Nye [110] who studied the boundary value problem with boundary values also dependent on e. 2. The theory of this chapter is useful for the study of the magnetohydrodynamic flow in ducts with a circular or rectangular cross section, see refs [110], [121], [126] and [81].
3. Another interesting approach to linear singular perturbation problems, applicable to as well ordinary as partial differential equations in several variables is due to Y. g a n a i [83], [84]. The singular perturbation problem for u(x) - u(x:, x 2 , . . . , x,~) is written in the form L2[u] + AL:[u] = 0, A > 1, x 9 f~, u(x) = 9~(x), x 9 0~2,
(8.5.43)
where L2 is an elliptic differential operator of the second order, L: a differential operator of the first order and ft a bounded domain. Applying the Lapace transformation y
u~ (x) -- / exp[--Ay]v(x, y)dy 0
(8.5.44)
200
Chapter 8. Perturbations of Elliptic Type one obtains a hyperbolic differential equation 0 L2[v] + -~--Ll[v] = 0,
(8.5.45)
ay
valid in the cylinder f l x ]R1 and with the boundary conditions
v(x, y) = 0, y < 0 and v(x, y) = ~(x)~(y) for y > 0. Investigating v(x, y), in particular its support, and approaching u~ (x) for -4 c~ with the aid of (8.5.44) one obtains an approximation of the solution of the singular perturbation problem. Y. Kanai treats a.o. the elliptic problem
" 0 9+c(~). L2 = Ax and L1 = ~-~bj(x)-~-~. j=l
and the parabolic problem
L2 = / k x and L1 = j=l
bj ~xL] 3
Ot '
(Ax is the Laplace operator). Further he presents an outline for a simple turning point problem in one dimension. For related work we refer the reader also to L. Frank, Singular Perturbations I, North-Holland Publ.Cy, Amsterdam, 1990.
6. 6.1.
Elliptic B o u n d a r y Value P r o b l e m s with Turning Points Introduction
In the preceding chapter we have studied perturbed boundary value problems for ordinary differential equations of the form
d2ye dye c~ + f (x)--~x + g(x)y6 = O,
- 1 < x < +1,
(7.3.1)
where the reduced equation (~ = 0) may become singular due to a turning point, say at x = 0, with f(0) = 0. It appears t h a t the approximation of the solution is highly sensitive for the sign of the derivative f~(x) at x = 0 and for the value of the parameter
~= 2
g(O) f'(O)"
(7.3.3)
P h e n o m e n a similar as those demonstrated in section 7.3 may of course also occur in the case of partial differential equations of elliptic type. In the next subsection we give some examples, however without entering into the details of rather complicated calculations.
201
6.2. Examples of Turning Point Problems 6.2.
E x a m p l e s of T u r n i n g P o i n t P r o b l e m s
6.2.1.
C u r v e s of T u r n i n g P o i n t s
An obvious generalization is provided by the following example
0 2u~ 0 2u~~ Ou~ ~ + 0y~ / + :(~' Y)-0~-y + 9(~, y ) ~ = 0,
(~, y) 9 ~,
(8.6.1)
where ~t is again a bounded domain such as considered in section 8.3. The boundary values are given by
ue(x, y) = ~+(x) along the upper boundary y = ~/+(x) with xl _< x < x2, and
ue(x, y) = T_(x) along the lower boundary y = V_(x) with xl <_ x < x2. Moreover, we assume f(x, y) - 0 along a curve i in fl and f(x, y) =/=0 elsewhere. Further we suppose f,g,~+ and ~_ sufficiently smooth. To discuss the analogue of a simple turning point we make the restriction
f(x, y) = {y - t.(x)}h(x, y),
(8:6.2)
with h(z, y) of constant sign in ~t. The curve y - l(x) is a line consisting of turning points and we finally assume that it passes through the endpoints A and B, see Figure 8.7.
8 A t
I !
! _,
X,
.,.
'~z
Fig. 8.7 Domain with turning points Viewing the results of section 7.3 we may expect that the behaviour of u~(x, y) sensitive for the sign of h(x, y) and for the values of the function ~(x---J). = - g ( x ' l(x)) 2 ~(~, t ( ~ ) )
is
(8.6.3)
202
Chapter 8. Perturbations of Elliptic Type
We expect the following i) Similarly as in Theorem 3 of Chapter 7 we have in the case ~(
'~
h(x, y) < 0 in fi and ~.~x, :/= O, 1, 2 , . . .
(8.6.4)
2
the result that ue(x, y) is approximately zero in any e-independent closed interior of f~ and that ur y) is approximated by boundary layer functions along the upper and lower boundary of 12. ii) Similarly as in Theorem 5 of Chapter 7 we have in the case
h(x, y) > 0 in l~ and fl(x) ~= - 1 , - 2 , . . .
(8.6.5)
2
the results y
"y_(x) <_ y <_ g(x) - 5
(8.6.6)
g(x) + 5 <_ y <_ ~+(x),
(8.6.7)
~-(~) and y
"y+(~) where 5 is an arbitrarily small positive number independent of s. In the neighbourhood of the curve y = ~(x) there is a free boundary layer connecting the approximations (8.6.6)-(8.6.7). This case with g(x, y) - 0 has been worked out in Ref. [76].
6.2.2.
Isolated Turning Points; Nodes
We take as our boundary value problem
( 02u~ 02u~]
L~ [u~ ] = e ~
+ OY 2 ] +
( Ou~
Ou~
x -~-~x + y -~y ) = 0 ,
0 ~ X 2 -~- y2 < 1
with
ue(x, y) - f(cp) for x 2 + y2 _ 1,
(8.6.8)
where ~o denotes the polar angle with 0 _< ~o < 21r. Transforming to polar coordinates we get
L~[ue]-e
02ue ~ + -
10ue
1 02ue
~--~r + - r- -2- 0~0 2
:l=rOue =0,
0_r
0<~<2~r
with
u~(1, ~) = y(~)
(8.~.9)
203
6.2. Examples of Turning Point Problems
and we assume f(99) e C2[0, 27r]. It is clear that the origin is an isolated turning point and we have to distinguish between the + and - sign. i) The attracting node (+sign) Stretching the coordinate r by p =
1-r
(8.6.10)
c
we get in first approximation
op~
op
=o,
~(o,~)=/(~).
It follows from the m a x i m u m principle that us(r, 99) e > 0 and so we have formally in first approximation
(8.6.11)
is uniformly bounded for all
u~(r, 99)..~ f(99), for 0 < r <_ 1.
(8.6.12)
P.P.N. de Groen [57] showed that one has indeed
ue(r, 99) = f(99) q- O ( c r - 2 ) .
(8.6.13)
This result has been obtained rather simply by using again the m a x i m u m principle and an appropriate barrier function. We have L~[r -2] = - 2 r - 2 ( 1 -
2er -2) < - r -2 for r > 2v/e.
(8.6.14)
We choose now the function 9 (r) = s / k r -2
(8.6.15)
as a barrier function for the error us(r, 99)- f(99) in the annulus 2v/~ < r _< 1. Taking ,k > max {4 --
max O
lue(r, 99) - f(99)[,
max
]f"(99) 1}
0 < ~ o < 2_1 r _
o___~,<2~
we get indeed by (8.6.14)
IL~[us(r,99) - f(99)] I = [r~f"(99)l < "ker-2 <- L s [ - ~ ] for 2v/e < r < 1 and
lue( r, 99) - f(99)1 < Acr-2 for r = 1 and r = 2x/~. Hence it follows that (8.6.15) is a barrier function for u6(r, 99)- f(99) follows.
and so (8.6.13)
204
Chapter 8. Perturbations of Elliptic Type
ii) The repelling node (-sign). The solution u~(r,~) is in the disk r < 1 - 5 , with 5 independent of e, approximated by the solution of the reduced equation 21r
if0
w = u~(O, O) = ~
f(cp)d~ = f.
The boundary layer correction is in first approximation given by the solution of the boundry value problem
02v
Ov
cop2 ~- -~p
O,
with v(0, ~) - f ( ~ ) - f and limp_~o~v(p, ~o) = O. This yields for u~(r, ~p) the result
f+
u~(r, ~p) ~ 6.2.3.
(f(~p) -
f-)exp[- l - r ] .
(8.6.16)
A Saddle Turning Point
As a final example of an isolated turning point we treat the following boundary value problem
n~[u~]-e
( ~02Ue C~2Ue) + x - C~Ue O'ae + ~ - ~x - y - ~ - y = 0 ,
-l<x<+l,-l
(8.6.17)
with the boundary conditions
u~(x, + l ) = f+(x),
- 1 < x _< 1,
ue(+l, y) = g•
- 1 <_ y <_ +1
(8.6.18)
where f + ( i l ) = g + ( + l ) and f _ ( + l ) = g+(-1). We assume f+(x) and g• at least twice continuously differentiable. For a first approximation we consider the reduced equation Ow Ow x-0-~x - y - ~ y = 0 .
(8.6.19)
The characteristics are hyperbolas xy - constant along which the solution w(x, y) is constant. To decide which part of the boundary conditions should be satisfied by w ( x , y ) we stretch the x- and the y-coordinate. Stretching the x coordinate by ~+- 1Tx gives in first approximation oe~:
0~+
= 0
6.2. Examples of Turning Point Problems
205
and it follows that boundary layers do not appear along x = +1. The equation (8.6.19) is supplemented with the boundary conditions w(J:l, y ) =
g+(y).
(8.6.20)
Stretching of the y-coordinate by r/+ =
1my
gives in first approximation 1
0r/---~--~+ ~
= 0
(8.6.21)
and so we can correct the solution w(x, y) of the reduced equation by boundary layer functions along the horizontal boundaries y = +1. However, the solution of the reduced equation shows a discontinuity across the y-axis because the boundary values along the vertical boundaries are independent of each other; in particular g+(0) # g_(0) in general. This discontinuity may be removed by the introduction of a free layer. The situation is illustrated in Figure 8.8.
u
• 4-6
-t
Fig. 8.8 A Saddle turning point
y,
206
Chapter 8. Perturbations of Elliptic Type
Starting with the solution w(x, conditions (8.6.20) we obtain
y) of the reduced equation (8.6.19) with the boundary
w(x, y) = g+(xy), x > 0 and w(x, y) - g_(xy), x < 0
(8.6.22)
The discontinuity across the vertical axis x - 0 is removed by stretching the x coordinate by x
~=
v~ u~(x, y)
and by taking as a first approximation of
ffJ(x, y) : g+(xy) +2g-(xy) + 9+(xY) -2 g-(xY) erf ( ~ ) where
(8.6.23)
2/ ~-~~-,-. t
~rf (t) - ~
0
We have outside the free layer ~(x, y) ~ w(x, y) and ~ is together with its derivatives continuous across the y-axis. Further we remark that erf ( ~ ) satisfies the differential equation (8.6.17) and a simple calculation yields L~[~] = O(v/e)
(8.6.24)
Hence zb(x, y) satisfies the differential equation (8.6.17) up to O(vf~) and the boundary conditions along the vertical boundaries up to O(e N) (N arbitrarily large). To obtain a formal approximation of u~ we have still to make a correction by boundary layers along the horizontal boundaries in order to satisfy also there the boundary conditions. These are constructed by solving
0~,~
0~0~
0,--~-~ + ~
= o,
0 <,
< ~
(s.6.25)
with the boundary conditions
v~
O) = f+(x) - ~(x, •
lim v~
77) = O.
~7 ::t: - + c ~
The result reads
v~
r]+) -- v~
1 :t::Y) =
{.f•
o(x,
•
exp [ - 1 T Y]
(8.6.26)
In order to obtain a formal approximation satisfying the differential equation and the boundary conditions up to some positive order of e we have just as in section 8.3 to add extra boundary layers ev~_(x, L ~ ) and ev~ (x, 1+_~). These terms are only needed for
Exercises
207
the proof of the validity of the formal approximation and they may be discarded again later on. As to this proof we refer the reader to [57]. Summarizing the construction one has the result
us(x,y)=~(x,y)+v~(x, 1-Y)+v~
I + Y ) +O(x/~)
(8.6.27)
where ~ is given by (8.6.23) and v~ by (8.6.26). This approximation is valid outside neighbourhoods of the points (0, +1) where the free layer and the boundary layers intersect. Exercises
1. Determine a uniform approximation up to O(e 1/4) of the function satisfies the boundary value problem
( 02us
Ous =o,
09.us )
us(x,y)
that
x 2 + y2 = r 2 < 1,
with us=~(~)
forr=l,
0<_vg_<2r,
~(0)=~(2r),
~=arctanY.
x
(Use the Remark 3 after section 4). 2. Consider the boundary value problem
( 02Ue-Jc-~y O~2Ur 2)
y2 - r2 < 1
with -
e
c
[0,
Give an approximation of ue up to O(e), uniformly valid for - 1 + 6 < x < +1 - 6 with 6 arbitrarily small positive, but independent of e. 3. Develop an approximation method for the boundary value problem
e ( ~02u~ + ~ O2us) +g(x,y)ue=h(x,y) f o r x 2 + y 2 = r us=~o(vg) f o r r = l , with
g,h 9
2<1],
0<~<27r, ~o 9
qo(0)=~0(27r),
2<1
tg-arctan y x
2 < 1.
Give first a formal approximation and consecutively its justification. See exercise 4 of Chapter 7. 4. Develop an approximation method as in Exercise 3, but now for a rectangle ; the boundary values Ue(X , 0) -- f l ( X ) ,
tts(X , ~2) -- f 2 ( X ) , u s ( 0 , Y) = g l ( Y ) , u s ( e l , Y) --- g2(Y)
are infinitely often differentiable and continuous in the corner points. 5. Show that (8.6.16) is an approximation for the repelling node problem.
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Chapter 9 SINGULAR PERTURBATIONS ORDER HYPERBOLIC LINEAR THEORY
1.
OF SECOND TYPE.
Introduction
In this chapter we study the following singular p e r t u r b a t i o n problem
L~[ue(x,t)] = ~L2[ue(x, t)] + Lz[ue(x, t)] = f ( x , t), t > O, -cx~ < x < +cx~,
(9.1.1)
with initial values
ue(x, O) = g(x),
(~U c
- ~ - ( x , O) = h(x),
- c ~ < x < +c~.
(9.1.2)
The differential operators L1 and L2 are given by the expressions 0
0
L1 = a(x, t)-O-xx" +b(x, t ) - ~ . +d(x, t). and
02
L 2 - - Ot 2 9
02 OX 2
9
(9.1.3)
(9.1.4)
The coefficients a, b, c, the right-hand side f and the initial d a t a g and h are for the moment arbitrary sufficiently smooth real valued functions of x and t, respectively x. Furthermore ~ is again a small positive parameter. This initial value problem may be considered as a p e r t u r b a t i o n of the initial value problem Lz[w(x,t)] = f ( x , t ) , - c ~ < x < +c~, t > 0 (9.1.5) with the initial condition
~(~, 0)= g(~),
-~ < 9< +~.
(9.1.6)
As in the preceding chapters we investigate again whether the solution of (9:1.5)-(9.1.6) is a good approximation of (9.1.1)-(9.1.2), and if so we investigate also the order of the approximation w(x, t) in terms of c.
210
Chapter 9. Perturbations of Hyperbolic Type
Problems of this kind have been studied by J.M. Blondel [12], [13], M.G. Dzavadov [28], J. Genet and G. Pupion [52], J.D. Cole [19], D.R. Smith and J.T. Palmer [128], D.R. Smith [129], M.B. Weinstein and D.R. Smith [145], [146], E.M. de Jager [77] and R. Geel [49], [47]. As physical applications we mention so-called overdamped vibration problems as occur in the propagation of electrical signals in a conducting wire of large resistance; in this case the mathematical description leads to the wellknown telegraphist's equation, see [128]. Other overdamped vibration problems are for instance the motion of a vibrating string imbedded in a highly viscous medium and the propagation of radiation through a highly absorbing medium. Singular perturbation problems of hyperbolic type as treated in this and the next chapter are in a certain sense the counterparts of the perturbation problems in Chapter 6, section 6, where we discussed the averaging and the multiple scale technique for perturbed wave equations which are of the type (9.1.1)-(9.1.2) but now with E large. These singular perturbations have only after a long time a cumulative effect on the solution while the singular perturbations of this chapter have as we shall see an immediate influence on the solution. Before we can answer the question whether the solution w(x, t) of the reduced problem (9.1.5)-(9.1.6) may be an approximation of the solution ue(x, t) of the full problem (9.1.1)-(9.1.2) we have to study first the interplay between the characteristics associated with the operator L1 and those associated with the operator L2. This will be done in the next section. In section 3 we give the formal approximation of the solution u~(x,t) and in sections 4 and 5 we show that this formal approximation is indeed a good approximation. To estimate the error term we use the energy integral method that has already been used in Chapter 7, section 1, where we discussed initial value problems for linear ordinary differential equations with a singular perturbation. We follow the theory as has been given in the references [77, 49, 47]. 2. C h a r a c t e r i s t i c a n d S u b c h a r a c t e r i s t i c s
To get some understanding of the interplay of the characteristics of L1 and those of L2 we give in this section the discussion as presented by J.D. Cole [19]. We consider the simple perturbation problem
n~[u~] = e
2u~ Ot 2
(0
02u~ ) Oue b Ou~ Ox 2 + a--o~x + Ot = 0 '
t>0
-cx~<x<+c~,
(9.2.1)
with the initial conditions
u~(x, O) = g(x),
--~-(x, O) = h(x),
- c ~ < x < +c~.
(9.2.2)
The coefficients a and b are constants with b ~= 0 and g and h are arbitrary functions of the class 61 (-oo, +oo) and C(-cx~, +c~) respectively. The solution of the reduced initial value problem o~
~-~ +
bow Ot = o,
w(~, o) = g(~),
(9.2.3)
211
2. Characteristic and Subcharacteristics reads
a ~(~, t) = 9(~ - gt).
(9.2.4)
The subcharacteristics are the lines x = ~t + constant and w(x, t) is constant along these lines. The characteristics of the hyperbolic operator eL2 + L1 are the lines x = i t + constant. We distinguish now two cases i. The subcharacteristics are spacelike, i.e. I~l > 1. In this case we have the situation as indicated in Figure 9.1.
P L'%
\
A
B
C~
Fig. 9.1 Spacelike subcharacteristic Whenever the solution have
w(x, t)
were a good a p p roxi mat i on of
u~(P) ~ w(P)
=
ue(x, t)
then we
w(Q).
However, according to the theory of hyperbolic differential equations the value of u~(x,t) in P is completely determined by the values of g(x) and h(x) with x E AB. Because the value of g(Q) is independent of these values, it is clear t h a t w(P) cannot be in general a good a p p r o x i m at i on of u~(P). Hence it follows t h a t the solution w(x,t) of the reduced problem cannot be in general a good approximation of ue(x, t), whenever the subcharacteristic is spacelike. a ii. The subcharacteristics are timelike, i.e. I~l < 1. In this case we have the situation as indicated in Figure 9.2. Now we cannot state a priori t h a t the solution w(x, t) of the reduced problem is not a good approximation of uc(x, t) for e --+ 0. It will appear in our following discussion t h a t in this case the solution w(x, t) of the reduced problem is indeed a good approxi mat i on of ue(x, t) for r --4 0, whenever we make the additional assumption b > 0. In fact we show t h a t we have for sufficient regularity of g(x) and h(x)
u~(x,t)=w(x,t)+O(e), uniformly in any fixed b o u n d e d domain in the region T h e o r e m 4.
(9.2.5) { - c ~ < x < +c~, t :> 0};
see
212
Chapter 9. Perturbations of Hyperbolic Type
P
A
Fig. 9.2
O,
B
Timelike subcharacteristic
It is of interest to give a physical interpretation of the conditions
I~l a
<
1,
b > O.
(9.2.6)
Therefore we introduce characteristic coordinates =x+t,
ri=x-t
(9.2.7)
and the differential equation for ue becomes
02ue
Oue
Oue
4r 0~0~7 - (a + b) - - ~ - (a - b) ~
= 0.
(9.2.8)
Assuming u~ continuous across the characteristic 7? = rl0 we consider a j u m p in the derivative ~on across this characteristic. P u t t i n g this j u m p
FO_~_~
k = LOw
0.. (~. ~0 + 0) - o.. 0) o~ ~ (~' ~0- .
(9.2.9)
n=no
we get for k the differential equation 4~
dk a~
:
(a - b)k.
and hence k=koexp
a-b
~(,~-~o
)]
,
(9.2.10)
where k0 is the value of the j u m p of ~on in the point (~0, rlo). It follows now t h a t k increases exponentially for t - + +cx>, whenever ( a - b ) > 0, but it decays exponentially for t --+ +cx~, whenever (a - b) < 0. A similar discussion for a j u m p in
213
3. The Formal Approximation
o~ across the characteristic ~ = ~0 gives the result t h a t the value of the j u m p increases exponentially for t -+ +c~, whenever (a + b) < 0, but t h a t it decays exponentially for t -+ +c~, whenever (a + b) > 0. Therefore in the case of exponential decay of j u m p s across characteristics we have the conditions: (a-b) < 0 and ( a + b ) > 0, i.e. b > lal. So we have the result t h a t j u m p s in the derivatives across characteristics decay exponentially for t --~ +cx~, whenever the subcharacteristics are timelike with the additional condition b > 0. In the case of exponential growth of j u m p s across characteristics we have the conditions (a - b) > 0 and (a + b) < 0, i.e. the subcharacteristics are spacelike.
3.
T h e Formal A p p r o x i m a t i o n We consider the initial value problem (9.1.1)-(9.1.4)
Le[ue(x, t)] = cL2[ue(x, t)] + Ll[ue(x, t)] = 6 ~
,
-O~x2
+ a(x, t) ~
+ b(x , t) ~
+ d(x , t)u~ - f ( x t) , ,
(9.3.1)
valid for - o o < x < +c~, t > 0, while the initial conditions are given by C~U~
~ ( ~ , 0) = g(~), We 1. 2. 3. 4.
-~(~,
0) = h(~),
-~
(9.3.2)
< ~ < +~.
make the following assumptions All d a t a are real valued with c(x, t) > O, b(x, t) > O, Vx and Vt > 0. a, b, c, d and f are C~ -c~<x<+c~ and t > 0 . g and h are C~ for - c ~ < x < +c~. There exits a n u m b e r P0 # 0 such t h a t a2
1
_
b2c2 .--
m
p2
> p2,
Vx and Vt > 0
(9.3.3)
5. ~ is a small positive parameter, 0 < r << 1. The assumptions 2 and 3 have been made to avoid some tedius bookkeeping; however, for less s m o o t h functions the analysis can be carried out to a certain extent. As we have shown in the preceding section the a s s u m p t i o n 4 is necessary to obtain an a p p r o x i m a t i o n of u~ in terms of a solution of the reduced problem (~ = 0):
a(x ' t) -~x Ow + b ( x ' t ) ~Ow +d(x't)w-f(x't)'
-c~<x<+c~,
t>O
(9.3.4)
with the initial condition
~ ( ~ , 0 ) = g(~),
-~
< 9 < +~.
(9.3.5)
This initial value problem is a simple linear first order problem and because of b(x, t) ~ 0 it can always be solved uniquely by the m e t h o d s of characteristics; so we may assume t h a t the solution w(x, t) of the reduced problem is known. This function is
214
Chapter 9. Perturbations of Hyperbolic Type
of class C ~ and it satisfies the full differential equation (9.3.1) up to O(e), uniformly in any bounded domain, and it satisfies also the first initial condition of (9.3.2), but in general not the second initial condition. Therefore we introduce a correction term v such t h a t this second condition is satisfied by w + v without disturbing the first condition too much; moreover w + v should also satisfy approximately the diffferential equation at least in bounded domains in t > 0. To this end we stretch the t-coordinate by the transformation t =er. (9.3.6) Substituting this new variable T into L~ and retaining the main contribution of v(x, t) to the differential expression Le[w + v] we must require
02v Ov Ov 2 + b ( x , O ) ~ = O ,
0
(9.3.7)
with the initial condition
o~
o , (~, o ) = h(~),
--cx~
s t (~' o) + - ~
< x < q-oo,
or
Ov Ow 0-;(~,0) = ~{h(~) - -07 (~,0)}
- c ~ < x < +c~.
(9.3.8)
Because (9.3.7) is of second order we need a second boundary condition for v and we take v(x, c~) = lim v(x, ~') = 0. (9.3.9) T--~OO
This condition guarantees t h a t the correction term v(x, t) has only a significant value in a small e neighbourhood of the initial line t = 0. The solution of (9.3.7)-(9.3.9) yields indeed a b o u n d a r y layer function, namely 01tl) t -By(x, O) - h(x) t v(x, 7) = v(x, -~) - c. b(x, 0) exp [ - b(x, 0)~].
(9.3.10)
t) + n~(x, t), u~(x, t) = w(x, t) + v(x, -~
(9.3.11)
P u t t i n g finally
we obtain for the remainder term Re the following differential equation
L~ [Re(x, t)] = - e
(
02w - c2(x, t) Ox 2]
~
+ ec2(x ' t) 02v - a(~, t) Ov - {b(~, t) - b(~, O) } Ov _ d(x, t)v 0~ --
- c~(~ t)
+ ~c~(~, t)
b(x, t) - b(x, O) Ov _ d(x t)v t
T-~T
'
- ~(~, t ) - (9.3.12)
"
4. A Priori Estimates of Solutions of Initial Value Problems
215
From (9.3.10), the regularity of the data and from the fact that b(x,t)-b(x,O) is uniformly t bounded in any bounded domain with t > 0, we may conclude t h a t
LoIRe]- O(e),
(9.3.13)
uniformly in any fixed bounded domain G in the region t > 0. Moreover R~(x, t) satisfies the initial conditions
R~(x, O) = -v(x, O) = O(c)
(9.3.14)
uniformly in any bounded interval of the x-axis, and
OR~ Ot (x, 0 ) = 0 ,
-cr162
(9.3.15)
It follows that the expression w ( x , t ) + v(x, t) satisfies the initial value problem for u~(x, t) up to O(c), uniformly in any bounded domain with t > 0, and so it is a t formal approximation of u~(x, t). To prove that w(x, t ) + v(x, -i) is indeed a good approximation of u~(x, t) we need an estimate of R~(x, t) and this estimate should be small for small values of r This estimate is obtained in the next sections with the aid of the method of energy integrals introduced by K.O. Friedrichs and H. Lewy, [44], [8]. 4.
A P r i o r i E s t i m a t e s o f S o l u t i o n s o f I n i t i a l V a l u e P r o b l e m s for P a r t i a l Differential Equations with a Singular Perturbation of Hyperbolic Type We consider again our initial value problem (9.1.1)-(9..1.4)
e ~-
(z,t) 02u
valid for - c r < x < +cr
Ou
Ou
+ a(x, t) Ox + b(x, t ) - ~ + d(x, t)u - f(x, t)
(9.4.1)
t > O, and with the initial conditions
Ou (x, O) = h(x), u(x, O) = g(x) and -~-
- c r < x < +cr
(9.4.2)
As to the coefficients, the right-hand side and the initial values, we make the same regularity assumption 1-5 as in the preceeding section. The method to obtain a priori estimates of the solution u(x, t) is due to Friedrichs and Lewy [44], [8], but because of the occurrence of the small p a r a m e t e r c we need here a modification of this method as given in [77, 49, 47]. Multiplication of (9.4.1) with 2u gives
O (bu2 -4- 2r -4- -~-~x(au2 - 2ec2uux) = Ot ( - 2 d + bt + ax)u 2 + 2ca 2 - 2ec2u 2 - 2e(c2)~uu~ + 2fu, where we have omitted the arguments of the functions involved. Multiplication of (9.4.1) with 2but and 2au~ gives respectively
0 (~b~ + ~ b c ~ ) + 0 (-2~bc~u~) =
0-~
216
Chapter 9. Perturbations of Hyperbolic Type
and 0 0 (~au 2 -4- ~ac2u 2) Ot ( 2 e a u t u~ ) - -~x -ea~u 2 + 2eatutux -e(ac2)~u~ 2 _ 2aduu~ - 2abutuz
2a2u 2 + 2 a l u m .
Adding these three equations we get o
o
(9.4.3)
oq-~Q1 + ~xQ2 = Q3, with Q1 - bu 2 + 2 e u u t + ebu2t + 2 e a u t u z + ebc2u2z, Q2 = au2 - 2 e c 2 u u x - e a u 2 - 2 e b c 2 u t u z - e a c 2 u x2,
(9.4.3 a) (9.4.3 b)
and after some calculation Q3 = ( - 2 d + a~ + bt)u 2 - 2(but + a u x ) 2 - 2 d u ( b u t + au~) + 2 f ( b u t + a u z ) + 2 f u + E[(2 + bt - a ~ ) u 2 + (2at - 2 ( b c 2 ) ~ ) u t u ~ + ((bc2)t-
< (-2d+a~+bt)u
(ac2)x - 2c2)u~~ - 2 ( c ~ ) ~ ]
2+(du-f)2+u
2+f2
+ c[(2 + bt - a ~ ) u 2 + (2at - 2 ( b c 2 ) ~ ) u t u x + ((bc~)~- ( a ~ ) ~ - 2 ~ ) = ~
- 2(~)~=u~]
< (1 - 2d + 2d 2 + a~ + bt)u 2 + 3f 2 + c[(2 + bt - a ~ ) u 2 + 2 ( a t - ( b c 2 ) ~ ) u t u ~ + ( ( b c 2 ) t - (ac2)x - 2c 2 + (c2)2)ux2 + u2].
(9.4.3 ~)
Hence we have for E < 1"
OQ1
OQ2
0--~ + - ~ x < Q4 + 3f 2
(9.4.4)
where Q1 and Q2 are given by (9.4.3 a) and (9.4.3 b) and Q4 by Q4 -- ~0 ~2 -~- E(")'ll u2 -~- 2~[12~tUx -Jr-~22'tt2),
(9.4.5)
with 3'0 = 2 - 2d + 2d 2 A- ax + bt, 711 = 2 - az + bt , ")'12 =
at -- (bc2)z,
7 ~ = (b~ ~)~ - ( a ~ ) ~ - 2 ~ + (c ~) x "
(9.4.6)
4. A Priori Estimates of Solutions of Initial Value Problems
217
C
A
B Fig. 9.3
X
The domains G and
To estimate u in any compact subset (~ of the half plane t > 0 we enclose G in a trapezelike domain Ft, bounded by the x-axis, a segment parallel to the x-axis with t = T and two characteristics with slopes c-l(x,t) and -c-l(x,t) see Figure 9.3. We integrate the inequality (9.4.4) over ~t and we obtain by Green's t h e o r e m B
-
C
Qlds+ A
D
A
ds
(Qlc+Q2) x/l+c 2 B
VII ~-c 2 C
D
<_/ Q4dtdx + 3 / f2dtdx, ( ds
is positive counter-clockwise) or
/ Q~dx / + C
D
C
ds/ (Q~c+ Q2) x/l + c2 + (Q~c- Q2) v/1§ c2 A
B
D
B
<_/ Qldx + / Q4dtdx + 3 / A
~
f2dtdx.
(9.4.7)
ABCD
This estimate will appear to be very useful because the integrands in the left-hand side are all positive definite in ~t. In fact we have for an arbitrary nonzero continuous function q(x, t) the estimate (see (9.4.3a))
Q1 = bu2 + 2guut ~- sbu2t + 2--cautux-[- -cbc2u2 _> ( b - x/~)u 2 + ~(b- V~)u2t + cbc2u2 + 2~aqutq-Xux _
> ( b - v ~ ) ~ : + ~ ( b - q~ - v ~ ) ~ Because we made the crucial assumptions _ Vt > 0 , there exists a positive function
instance
q2 _ --
b 2c 2 + a 2 2bc 2 9
b > 0 and
q2(x, t)
2
+ ~(b~~ - a~q-~)~.
with
a 2
1- ~
a ~ b-fi <
> p02, Vx and q2
< b, V(x, t) C ~;
for
218
Chapter 9. Perturbations of Hyperbolic Type
Since all functions involved are continuous for t > 0 we have for 6 sufficiently small, say 0 < 6 < 6 0 , Q1 >_ m(u 2 + 6u 2 + 6u2), Vx, t 9 ~t, (9.4.8) where m and 60 are generic positive constants, dependent on the coefficients a,b and c and on the domain l~, but they are independent of 6. 60 and m are generic positive constants which means that they may be lowered, whenever desired, without violating preceding results, e.g. (9.4.8). Further it follows from (9.4.3 a) and (9.4.35)
cQ1 + Q2 = (bc + a)u 2 + 26cu(ut - cux) + 6 ( b c - a)(ut - cux) 2 and
cQ1 - Q2 = ( b c - a)u 2 + 26cu(ut + cu,) + 6(bc + a)(ut + cu,) 2. Using again the relation 1 - g~-/-~2> p2 > 0 we obtain for 6 sufficiently small, 0 < 6 < 60
cQ1 + Q2 >_ m { u 2 + 6(ut T cu,)2}, where m depends again on a,b,c On the other hand it follows from coefficients a, b, c and d that there and on ~t, but independent of 6,
IQ41 5
IQ~I +
V(x, t) 9 ~t,
(9.4.9)
and Ft. (9.4.3~), (9.4.5), (9.4.6) and the regularity of the exists a constant M, depending on these coefficients such that
M ( u 2 + 6u 2 + 6u2),
V(x, t) 9 l~ and 0 < 6 < 60.
(9.4.10)
Substitution of the inequalities (9.4.8)-(9.4.10)into (9.4.7)yields m
/C (u 2 + eu2t + eu~)dx + m /C {u 2 + e(ut - cux) 2} -x/1- - -ds+- ~c D
B A
+ m/{u
2 + e(ut + cux) 2}
D B
A
- M/(U
ds ~/1 + c 2
~2
2 + eU 2 + r
+ K(ft),
(9.4.11)
12
where K(ft) is determined by the initial conditions (9.4.2) and the right-hand side S of the differential equation. We have
K(Q) = M{[lg[I~AB] + r + 3I[fll~ABCD ] with
Jig]lAB
+ ellhll~AB]} (9.4.12)
the L2-norm of the function g with respect to the integration interval
A B and similarly for the other terms. Since the estimates of all integrands are valid in the whole domain ~ the inequality (9.4.11) is also valid for any domain ABC*D* < A B C D with the same values of m, M and K(ft), see Figure 9.4.
4. A Priori Estimates of Solutions of Initial Value Problems
219
• T
A
Fig. 9.4
/
..
C
,,
A'
B"
The domains
B
~'
ABCD, AIBICID! and ABC*D*
Therefore it follows from (9.4.11) -~ (t)
t "y~(,-)
f (u2 +eu2t +eu2)dx-M-M-mf f (u2 +eu2t nt-eu2x)dxdT<-1K(ft)'m o "n (~)
"n (t)
valid for 0 _ < t _ < T and 0 < e < e 0 . Using now finally Gronwall's lemma (Chapter 3, (3.2.23)-(3.2.24)) we obtain the L2 estimate -r~(t)
/ (u2+eu2+eu2)dx<-1K(ft) exp[Mt]m ,~(t) < 1K(ft)m exp[MT] =
C(ft)K(ft),
(9.4.13)
valid for 0 < t _< T, 0 < e < e0, and where C(ft) = 1 exp[MT]. This constant C(ft) will be used as a generic constant, while K(ft) remains fixed and it is defined by (9.4.12); so C(ft) may be raised without violating earlier results. Two other L2 estimates for u may be obtained by applying (9.4.11) to a domain A'B'C'D' (see Figure 9.4) and by using the estimate (9.4.13). We get the inequalities C !
{u + e(ut - cux)2}ds <__C(ft)K(gt)
(9.4.14 a)
{u + e(ut + cu~)2}ds < C(fI)K(~),
(9.4.14 b)
B'
and A !
D'
220
Chapter 9. Perturbations of Hyperbolic Type
valid for all characteristic segments B'C' and D'A' inside ~ and for all c sufficiently small, 0 < e < ~0. Now that we have derived three estimates in L2 norm for a combination of u and its first derivatives it is not difficult to get a pointwise estimate for the function u(x, t) , uniformly valid for ~ sufficiently small in ft. With the aid of the Schwarz inequality we have the following estimates in fl, (see Figure 9.4),
lu2(x,t) - u2(~(t),
u(~,t)u~(~,t)d~ < 2 ] lu(~,t)i" iu~(~,t)id~
t)l-- 2 ] ~l(t) < 2
~(t) u2(~,
t)d~
~(t)
u,(~, t)d~ ~1(t)
}.
Using once again (9.4.13) we get lu2(x, t ) - u2(Ti(t), t I __ e-
89
or
u2(71 (t),
t) < u2(x, t) + C(fl)K(~2)e- 89
After integrating this result with respect to x between the limits have ~2(t) 01u2(')'l(t),
71(t)
and 72(t) we
t) <__ f uZ(x,t)dx + 02C(f~)K(f~)e- 89 71(0
with 01
--
max "/ r ' /'2' t t j - 7 1 t' 'txj j = n B . min ( 7 2 ( t ) - 71(t)) = C D and 82 = 0
0
Once again with (9.4.13) we obtain 01u2(71(t),
t) <
C(f~)g(f~)(1 + 02e- 89
because also
u2(x,t) <_u2(71(t), t) + C(a)g(~)c- 89 we get finally
u2(x,t) < C ( a ) g ( ~ ) { 0 1 1 + (02011+ 1)E- 89}, or after absorption of (020~-1 + 1) in C(D)
u(x, t) <_ v/C(f~)g(f~) e- 88
(9.4.15)
uniformly in ABCD for ~ sufficiently small, say 0 < e < e0. We can derive also pointwise estimates for the derivatives ux(x, t) and ut(x, t); we proceed as follows, see Figure 9.5.
221
4. A Priori Estimates of Solutions of Initial Value Problems
T
d
P (x,'~)
9 ~
A
Fig. 9.5
R
5
Characteristics through P
Let P ( x , t ) be any point in ~ and P Q and P R the characteristics through the point P. We multiply the differential equation (9.4.1) with 2(ut - c u ~ ) and there results after some calculation
2e ( ut - cu~ )
--~ + c-~-~x
--~ - c-~-~x u + 2r ( ut - cu~ ) ( ct u~ + cc~ ux )
(ut -
+ 2 a ( u t - cu~) (2c) -1 ((ut + cu~) -
cu~)) +
b(ut
- cu~) ((ut + cu~) + (ut - cu~))
+ 2 d ( u t - c u x ) u = 2 f (ut - cu~), or
0 c ~o ) (u~-~u~)2 = -~+
b~-a (~,_ ~)~.
b~+ a ( ~ , _ c ~ ) ( ~ + ~ )
C
C
where 5 = a + e ( c t + cox). Introducing positive numbers 0"1 and 0"2 to be determined later on, we may write
~ + ~
(u, - cu~) <
~~
-
c
bc+5 -4
2C
+ ~
(~, - c ~ ) ~
2c
'
1 (ut + c u x ) 2 + - - ( f 1
0.1
0"2
-du) 2 "
Because la[ < bc for t > 0 and because a,b and c are continuous it is possible to define the following constants, positive for e0 sufficiently small
ql
(~)
min bc - a c ~
--
eo
m a x
~2
ct + cc~ c
222
Chapter 9. Perturbations of Hyperbolic Type
and
bc + a
q2(~) = max ~ 12 c
+ s0 max ~
ct + ccx c
These constants depend only on a,b,c and ~, but they are independent of Substituting these constants in the last inequality there results
s.
s -~ + C-~x (ut - cu~) 2 _ { - ql (~) + alq2(~) + a2 }(ut - cu~) 2 1
d_ q2(~) 11 a (ut d- CUx) 2 -4- --a2 (f -- du)2' uniformly in ~ for 0 < s < So. Choosing a l q 2 ( ~ ) = [1 q l ( ~ )
0
O)
~ + c~
and a2 = l q l ( ~ )
(=, - c=~)
2
we obtain the estimate
1
< q,.(~)-(=,
1
+ c~) ~ + -(f
6r 1
_ 2q~ (~t)
-
q~(5)
(7 2
(=, +
c=~): +
-
~=)~
4 (I - d=) ~ ql(~)
uniformly in ~ for 0 < s < s 0 . Consecutively we integrate this inequality along the characteristic P Q and we get with the aid of (9.4.145 )
s(ut - cu~)2(P) < s(ut - cu~)2(Q) +
2q2(~) 1
C(f~)K(f~) ql (12) s
Q (f2 + d2u2)ds.
+ ql(~) P
Using again the regularity of the data of the initial value problem and the estimate (9.4.15) we obtain finally in Ft the uniform estimate
lu,- c~) < v/C(~t)K(f~)s -1, where the constants
ql(~)
and
q2(~)
V(x, t) e ~,
0 < s < s0,
have been absorbed in the generic constant
c(a). Quite similarly we have also
lu~ + c~=l < v/C(12)K(12)e -1,
V(x,t) e ~,
0 < e < e0.
Hence we have finally obtained besides the pointwise estimate (9.4.15) for the solution u(x, t) of the initial value problem (9.4.1)-(9.4.2) also the pointwise estimates for the derivatives ux(x, t) and at(x, t):
ux(x, t) < v/C(f~)K(f~) S - 1
Ut(X , t) < v/C(f~)K(f~) s -1,
(9.4.16)
223
5. The Estimate of the Remainder Term and Final Results
uniformly valid in ~ for r sufficiently small; C(~t) depends on the coefficients a, b, c, d and the domain ~t, while g(~t) is defined by (9,4,12). Summarizing we have proved the next theorem THEOREM
1
The solution of the initial value problem b-~ Ou Ou 02u _ c2 (x, t) 02u } + a(x, t) -~x + b(x, t) -ff[ + d(x, t)u - f (x, t) -cx~<x<+oo,
t>O,
O<e~l,
with u(x, O) = g(x), o~ (~, 0 ) = h(~), - o ~ < 9 < +o0
has under the regularity assumptions 1-5 of section 3 for e sufficiently small the following pointwise estimates, uniformly valid in any compact domain G with t >_ O, u(x,t) < e- 88v/C(f~)K(f~)
(9.4.17)
ux(x, t) < ~-1 v/C(~)K(f~)
(9.4.18)
ut(x,t) < e-1 v/C(~)K(~t).
(9.4.19)
The constants C(f~).and K(~) depend on the coefficients a,b,c,d and the domain f~, while K(~2) depends also on the right-hand side f and on the initial data g and h; the latter is given by the relation (9.4.12). 5.
T h e E s t i m a t e of t h e R e m a i n d e r
Term and Final Results
We return to our formal approximation of the initial value problem (9.1.1)-(9.1.4). The result reads
ue(x,t) = w(x,t) + v(x, t) + Re(x,t),
(9.5.1)
c
where w(x, t) is the solution of the reduced problem
a(~, t) O~ Ow -~x + b(x, t) - ~ + d(x, t)w = f (x, t)
--cx3 < x < cx3,
t > 0,
~ ( ~ , o ) = g(~), t and v(x, -i) is the boundary layer correction given by (9.3.10). The remainder term Re(x, t) satisfies according to the equations (9.3.12)-(9.3.15) the initial value problem
( 0 2 R e _ c2(x, t) O2Re ) ORe Ot 2 Ox 2 + a(x, t) -~x + b(x, t)
+ d(x, t)R~ = O(e),
uniformly in any compact domain in t > 0, with the initial conditions
R~(~, 0) = - ~ ( ~ , 0) = - ~
Ot
b(~,o)
= o(~)
(9.5.2)
224
Chapter 9. Perturbations of Hyperbolic Type
and
0--R~(~,0)=0, Ot
also uniformly valid in any bounded segment of the x-axis. We can now estimate the remainder Re(x,t) and its derivatives by applying the result of T h e o r e m 1 of the preceding section. By (9.4.12) we have g ( g t ) = O(c 2) and hence by (9.4.17)-(9.4.19)
Re(x, t) = O(e 3/4) 0
(9.5.2) 0
-~xxRe(x,t) = O(1), - ~ R e ( x , t ) = O(1),
(9.5.3)
uniformly in any compact domain with t > 0. Using finally (9.5.1),
Ov ,(x, ;t) = 0(~), ~(~,;a,t) = 0(~), -b-;(~, ;)t =0(1) we arrive at the following t h e o r e m THEOREM 2
Let ue(x, t) be the solution of the initial value problem \ c9t2 --
c9x2 ] + a(x, t)
+ b(x, t) -g7 + d(x, t)ue - f ( x , t)
- c o < x < +oc, t > O, with the initial conditions
~ ( ~ , O) = g(~),
OUe (x,O) = h(x), -g-f
-cr162
Suppose that all data are of class C ~ with b(x, t) > O, c(x, t) > O, Vx, Vt > 0 and that the subcharacteristics are timelike in the region t > O. Then we have the following approximations of ue(x, t), uniformly valid in any compact domain G in the half plane t > 0 for r sufficiently small and positive. u~(x, t) = w(x, t) +
0(~3/4),
Ou~ (x, t) = Ow cox -~x (x' t) + O(1)' OUe (x t) = cgw Ot ' --~-~(x,t) + 0 ( 1 ) , where w(x, t) is the solution of the reduced problem
a(x, t) Ow O~ -~x + b(x, t) ~ + d(x, t)w = f ( x , t), w(x,O)=g(x),
--cx3 < x < +cx3.
- o o < x < A-co,
t>O,
5. The Estimate of the Remainder Term and Final Results Remarks 1. The approximation of imation of the form
225
ue(x, t) in G may be improved by constructing an approxm
m
4-
(,~(x, t) - E eiw'(x' t) + E eiv,(x, ~), i--0
(9.5.4)
i--0
with Ll[w0] = f and Ll[W.i] = -L2[wi-1],
i = 1, 2 , . . . m ,
while the boundary layer terms vi are obtained from the expansion of r t--~T. Using the notation
Le - gL2 + L1 - e-1 {M0 + gM1 + . . . + gmMm} + gmMm+l,
(9.5.5) with
(9.5.6)
the functions vi(x, T) satisfy the recursive set of equations
M0[v0] = 0 i
Mo[vi] - - E Mj[vi_j].
(9.5.7)
j--1 The initial conditons for wi(x, t) read
Wo(X , O) -- g(x), Wi+ 1 (X, O) -- --Vi(X,
1, 2 , . . . , m - 1
0 ) , i --- O,
(9.5.8)
and the boundary conditions for vi(x, T) are given by
Ovo Owo OT (x, O) -- e(h(x) - ~ ( x ,
0)),
lira Vo(X, T) = 0
OVioT(X, O) = --C--~OWi(X, 0), ~-~limvi(x, T) = 0, i = 1, 2 , . . . m .
(9.5.9)
In case the data are all C ~ we may expand Cte(x,t) indefinitely. From this we get the following refinement of Theorem 2. THEOREM 3
Let u~(x, t) satisfy the conditions of Theorem 2 then we have the following approximations of u~(x, t), valid for any n C N t3 {0} and uniformly valid in any compact domain G in t >_0 for e sufficiently small positive: Ue(X , t) : i=0 0U e
n--1 ciwi(x, t) + E ~ivi(x' t ) + E
n-1
n--2
t) + i=0
Ou,ot (x, t) = E r i=0
0(~n+3/4 )
i--0
-) + i=0
-~(x,
+
r -~Ovi(x, ~t) + 0 (~,~) i=0
(9.5.10)
Chapter 9. Perturbations of Hyperbolic Type
226
where empty sums are to be taken as zero. The requirement that all data of our perturbation problem are of class C ~ is of course too strong for an asymptotic expansion up to a certain order. The rule is that the number of terms in the asymptotic expansion may be larger whenever the data are more regular. For n = 1 we get with the aid of (9.5.10) the following result. THEOREM 4 The approximation of the solution u~(x, t) of the initial value problem (9.3.1)-(9.3.3) is for n = 1 given by
~,(~, t) = ~0(~, t) + o(~) Ou~ (~, t) = O~o (~ t) + o(~) oz Ou~
~ Owo
' Owo
t
ot (~' t) = --~ (~, t) + (h(~) - --07- (~, o)) ~xp[-b(~, o) ~] + o(~)
(9.5.11)
uniformly valid in any compact domain G in the half plane t >_ O. 2. The theory as presented in this chapter has been generalized by R. Geel [49] to initial boundary value problems defined in strips {(x, t); 0 < x _< s t >__0} and to so-called characteristic boundary value problems; in the latter case the boundary consists completely or partly of characteristic curves associated with the operator L2. 3. The approximations given in the Theorems 2-4 are valid for finite e-independent values of t; an extension of these results for large time scales is given b y W. Eckhaus and M. Garbey in Ref. [32]. 4. Hsiao and Weihnacht have also investigated linear singularly perturbed hyperbolic equations of another very interesting type [68]. These authors considered the following Cauchy problem
e2utt+ut-u**:O,
u(~, o) = y(~),
t>O,
~ OU(x, O) = g(x),
-c~<x<+c~ - c ~ < x < +oo.
(9.5.12) (9.5.13)
The equation (9.5.12) is called "the wave equation of heat conduction" and the perturbation is introduced to overcome the infinite speed of heat propagation. In this case one does not get for r = 0 a degeneration to a first order equation but to a parabolic equation. A formal approximation is constructed by a composite expansion consisting of an outer expansion and boundary layer terms. The proof of its validity is obtained again by the method of energy integrals; it runs along similar lines as given in this chapter, but the estimate of the remainder term requires a somewhat delicate treatment because the trapezoidal domain 12 becomes unbounded for 6 --4 0. Restricting the result of Hsiao and Weihnacht to the first few terms of the asymptotic expansion one has for f and g sufficiently smooth
u(~, t) = Uo(~, t) + ~(u, (~, t) + Vo(., t)) + o ( ~ ) ,
Exercises
227
uniformly in (x,t) for all real x and for 0 _< t _< to with to an arbitrary constant. Uo(x, t) and [/1 (x, t) are solutions of the Cauchy problems
OU
02U
0t
0X 2
= O,
Uo(x, O) = f ( x ) ,
U1 (x, O) = - V o ( x , 0),
(9.5.14)
while V0 satisfies the boundary value problem
O2Vo
Oyo
OT---W + ~
where T -
= O,
Oyo
--ff~T (X, O) = g(x), l i r n V0(x, T) = 0
(9.5.15)
t/r 2.
Exercises
1. Consider the initial value problem
02u Ot 2
02u)
Ox 2
Ou
Ou
+ a ~x q- b --~ + d u - f ( x , t ) , - cx) < x < q- cx~, t > 0
with the initial conditons
~(~, 0)
= g(~),
()U ~ (~, 0) = h(~), - ~
< ~ < +oo;
the coefficients a,b and d are constants. Give minimal conditions on the regularity of the functions that the approximations (9.5.11) hold.
f, g and
2. Investigate the singularly perturbed hyperbolic system
( ) () - ~ q- Ou
e Ovot
1
q- -21(a - b)u q- -~(a q- b)v -- f (x, t), 1
oxOV + 2 1 ( a - b ) u + - ~ ( a + b ) v = f ( x ' t )
-cxD<x<+oo
with u ( x , O ) = uo(x) and v ( x , O ) = v o ( x ) , - c o < x < +c~. The coefficients a and b are constants with a > [b[.
h in order
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Chapter 10 SINGULAR PERTURBATIONS IN NONLINEAR INITIAL VALUE PROBLEMS OF S E C O N D O R D E R 1.
Introduction
In this chapter and the next one we will discuss singular perturbations in nonlinear problems. Due to the large variety of possible nonlinear differential equations and the analytical difficulties concerning the integrability and the uniqueness of solutions we are forced to restrict our exposition to some relatively simple types for which we investigate whether the linear theory as treated in the preceding chapters may be generalized. This means that we discuss in this and the next chapter only second order scalar problems; we start with initial value problems since these are more easily to handle than boundary value problems. In the same way as in the preceding Chapters 7 and 9 we construct a formal approximation of the solution and in order to show that this approximation is indeed a good one we should give an a priori estimate of the remainder term that satisfies a singularly perturbed nonlinear initial value problem with the right-hand side of the differential equation and the initial values being small in some order of ~. Functional analysis provides very useful means for getting a priori estimates of solutions of nonlinear differential equations; so several authors, e.g. Hoppenstead [66], [67], Krein [88], van Harten [64], [62], Besjes [9], Geel [49], [48], Geel and de Jager [46, 75], Genet and Madaune [51]-[102], Hsiao, Weihnacht and Esham [69, 34, 36, 39, 40, 35, 38] applied principles from functional analysis to the study of singular perturbation problems. In section 2 we derive a contraction theorem in a Banach space, due to a.o. van Harten [64], [62], which is very useful to obtain the required estimate of the remainder term. This theorem will also be used in the next chapter where we discuss boundary value problems. Section 10.3 is devoted to initial value problems of the type
r
d2u du 2 + a(X, u) ~ + b(x, u) = O,
u(O) -- a,
du ~x (O) = fl,
(10.1.1)
and section 10.4 to the more general nonlinear initial value problem
d2u du du e-~x2 = f ( u , ~xx) , u(O) = a, ~x(O) = j3
(10.1.2)
230
Chapter 10. Perturbations in Nonlinear Initial Value Problems
The generalization to semi-linear initial value problems with a singular perturbation of hyperbolic type
( 02u ot~
02U ) +a(x, t, u) OU Ou o~ Y i + b(~, t, u ) ~ + c(~, t, ~,) : OU(x, O) = g(x), -~
u(x, O) -= f(x),
o, t > o, - ~
< ~ <
- c o < x < +co
+o0 (10.1.3)
is discussed in section 10.5. Finally we mention other interesting nonlinear singular initial value problems in the remarks concluding this chapter. 2.
A Fixed Point Theorem
In this section we discuss a fixed point theorem that will appear to be very useful in this and the next chapter. Let N be a normed linear space consisting of elements denoted by p and with norm I p l and B a Banach space with elements q and norm [[q[[. Further, let F be a nonlinear map from N to B with F[0] = 0 and we assume that F[p] can be decomposed as F[p] = Lip] + ~[p], (10.2.1) where L is the linearization of F at p - 0. Finally, the operators submitted to the following conditions: 1. L is bijective and its inverse L -1 is continuous, i.e.
L and
[ L-l[q] [ <-- ~-~llqll,
(10.2.2)
where ~ is some positive number independent of q. 2. Denote the ball {p I P E N, [ p [ <_ p} by ftN(p), fi > 0 such that II~[p2]- ~[p~]ll _< re(p) I p~ - p l for 0 _~ p _< p and where lim re(p) = O.
I ,
9 are
then there exists a number
vpl,p2 e aN(p),
(10.2.3)
re(p) is monotonically decreasing for p -+ 0 with
p--~0
Now we have under the conditions (10.2.2)-(10.2.3) the following theorem. THEOREM
1 (Fixed Point Theorem)
If
1
po=sup{p[O~P~fi, p_>o
then there exists for any f E B with [[fl]-< 89
rn(p) ~ ~s
(10.2.4)
an element p E N such that
F[p] = f,
(10.2.5)
[P [ <- 2e-l[[f[[ ~ P0
(10.2.6)
with
231
2. A Fixed Point Theorem
Proof Due to the bijectivity of the map Lip] = q the equation (10.2.5) is equivalent with the equations L[p]=qandq=f-q2oL-l[q]:=T[q], qeB. (10.2.7) We consider the ball ~B(~.p) C B p should satisfy the inequalities
with centre
~
and radius
ip
and moreover
2g-lllfll < p ~ po < ~.
(10.2.8)
For any q G ~B(~.p) we have
I L-l[q] ] -< ~-~llqll-< p -< p0 _< Z, and because q = L o L-I[q] (10.2.9)
~2S(ip) C L[~2N(p)] C L[~2N(~)].
The operator T has the following properties 1. T maps F~s(ip) into ~B(t.p). This is a consequence of (10.2.7), (10.2.8) and (10.2.3); indeed for any q e F~B(~.p) we have the inequalities
[[T[q][[ ~< [Ifll +
1 I1~ o L-l[q][[ <~ -~M-4re(p) I L-X[q] [
1
< -~.p + m(p)p < ~p 2
2. T is strictly contractive on F~B(ip) because one has due to (10.2.3) and (10.2.4) the inequalities lIT[q2] - T[ql]I] = II~ o L - I [ q 2 ] - ~ o L-I[ql]II 1
< re(p) I n-~[q2]- n-~[q~] I < m(p)e-~llq2- qlll < ~llq2- q~[I. It follows that T is a contractive map, mapping the ball ~B(~p) into itself and according to Banach's contraction theorem, see e.g. [109], there exists in ~B(iP) a unique solution q = T[q] of the equation (10.2.7). Hence on account of the equivalence of (10.2.5) and (10.2.7) there exists also a unique solution of (10.2.5) in L-I[~B(~.p)] C ~N(P). Note that the uniqueness of p has not been proved in the whole ball fiN(P) but only in L-l[g~B(t.p)] C F~N(p). Finally we have
I pl = [ L-~[q] [ < t-lllqll < P, and choosing p =
2g-xllfll we get
the estimate (10.2.6).
232
3. 3.1.
Chapter 10. Perturbations in Nonlinear Initial Value Problems
T h e Quasilinear Initial Value P r o b l e m Introduction We consider the following initial value problem with a singular perturbation.
r
d2u
+ o(x u) '
du -~x
+ b(x u) = 0, '
x > 0
(10.3.1)
and the initial conditions are u(0) = c~,
du
~xx(0) = ~.
(10.3.2)
A formal approximation of the solution is obtained by taking first the solution of the reduced equation
a(x, w) -~x dw + b(x, w) = 0,
w(0) = (~,
x > 0.
(10.3.3)
Consecutively a correction term should be added in order to account also for the second initial condition ~d=( 0 ) = ~. To show that this formal approximation is possible and that the remainder term may be proved to be o(e) we need two conditions for the coefficients a(x, u) and b(x, u). i. The functions a and b belong both to the space C1'2(1~,+ x R), i.e. a(x,u) and b(x, u) are for any value of u continuously differentiable with respect to x in [0, cx3) and for any value of x > 0 twice continuously differentiable with respect to u in (-cr ii. a(x, u) > a0 > 0, Vx E I~,+, Vu E R with a0 some positive constant. First of all we remark that these two conditions are sufficient for the existence of a C2-solution w(x) of the reduced problem (10.3.3) in any segment [0, X] with 0 < x < X < X0; X0 is the smallest value of x where w(x) becomes singular; in case w(x) has no singularity the value of X is an arbitrary positive number. The uniform positivity of a(x, u) is needed because it will appear that the construction of the formal approximation breaks down whenever a(x, u) becomes negative in the segment [0, X], see also Chapter 7, formula (7.1.3). 3.2.
T h e Formal A p p r o x i m a t i o n
Let w(x) be the solution of the reduced problem (10.3.3) with the initial condition w(0) = c~. Using our experience with the linear initial value problem in Chapter 7 we write u(x) = w(x) + ~9(~), (10.3.4) where ~ is the stretched variable defined by x = e~. Inserting (10.3.4) into (10.3.1) we get for ~(~) the initial value problem d2v d~ d~ 2 + a ( ~ , w(6~) + 69(~)) d~
d2w dw = -- e~x 2 - a(x, w(x) + eg(x))-~x -- b(x, w(x) + 69(x))
(10.3.5)
233
3.2. The Formal Approximation _ - ~x- ~ while the initial conditions for 9 read valid for 0 < ~ <
~(0) = O,
d9 dw _-7-; ~Yb" aq = ~ - -;:(0).
We expect that the correction term 69(~) is of boundary layer type, i.e. 9 and its derivatives have only a significant value in an ~-neighbourhood of the point x - 0. Therefore we approximate 9(~) by a function v(~) that satisfies the initial value problem
dd~~ 2 + a (0, ~(o)) dv = 0 ,
dv dw ---7 d = ~ - ax----7-(O)
(10.3.6)
0<~
lim v(~) = 0.
(10.3.7)
~--~oo
The right-hand side of (10.3.5) has been neglected because a and b are sufficiently regular and w satisfies the reduced equation (10.3.3). The first initial condition 9(0) = 0 has been neglected since 69(0) has only a small influence on the initial value of u(x). Furthermore we have added the condition lim v ( ~ ) = 0 in order to take care of the ~--~oo boundary layer character of v(~). The solution of (10.3.6)-(10.3.7) is readily determined and it is given by
v(r
=
~'(o)- v(~)~ = a(o,,~)
exp [-- a(O, oOX],
(10.3.8)
C
where we have substituted the initial condition w(0) - a. Due to the requirement a(x, u) >_ a0 > 0 it follows that v(X) has indeed the desired behaviour of a boundary layer. We consider now the function
~(x) -- w(x) + ev(X),
(10.3.9)
e
which is a candidate for the approximation of u. This function satisfies according to (10.3.3) and (10.3.6) the differential equation
d2 ~ d~t d2 w ~ x 2 + a(x, ~t) -~x + b(x, ~) - ~-~x 2 + {a(x, w(x) + cv(X))~ - a(x, w(x)) } dWdx q- {a(r
dv
w ( ~ ) + cv(~)) - a(0, w(0))}~-~ + {b(x, w(x) q-- cv(~)) - b(x, w(x)) },
valid for 0 < x _ < X and 0 < ~ < - / . X With the aid of the assumed regularity of the coefficients a and b (consequently also of w 9 C2[0, X] -see (10.3.3)-) and the behaviour of v as a boundary layer function we obtain by the mean value theorem ~
d2fi
dfi + a(x, ~) ~ + b(x, ~) = O(~)
(10.3.10)
234
Chapter 10. Perturbations in Nonlinear Initial Value Problems
uniformly for 0 < x < X. The initial values of fi(x) follow immediately from ~(o) = ~(o) + ~ ( o ) = ~ + o(~), dft dw dv dw dv d---~(O) = ~x (O) + S-~x(O ) = ~x (O) + -~(0) = t3.
(10.3.11)
Hence the function fi(x) = w(x) + r with w(x) given by the solution of (10.3.3) and v(~) by (10.3.8) satisfies the original initial value problem (10.3.1)-(10.3.2) up to O(e), uniformly for 0 < x < X. Therefore ~(x) is a formal approximation of u(x). To show that this approximation is indeed a good one we introduce the remainder term Re(x), defined by u(x) = fi(x)+/~e(x), (10.3.12) and we make a uniform estimate of /~e(x) for e $ 0 in 0 < x < X. This remainder term satisfies according to (10.3.1), (10.3.2) and (10.3.10), (10.3.11) the initial value problem
~-g-~ + ~ ~ + a ( ~ , a+&) ~ + a~ ] +b(~, a + & ) = o or
d2 Re + a(x, ~t + Re) dRe d~ + {a(x, fi + Re) - a(x, fi)} dx dx 2 d2~t -- a(x, ft) d~t + {b(~, ~ + & ) - b(~, ~1 } = - ~ d - ~ ~ - b(~, ~1 X = O(e), uniformly in 0 < x _ - -
(10.3.13)
6
with the initial conditions d/~
Re(O) = -ev(O),
dx = O.
To estimate /~e(x) with the aid of the fixed point theorem of subsection 10.2, it is useful to homogenize the initial conditions using the simple substitutions
h~(~) = R~(~) + ~,(o),
~(~) = ~(~) - ~,(o).
(10.3.14)
The initial value problem for /~e becomes now d2/~E
c~+a(x,
d/~e d~ fi + / ~ e ) ~ + {a(x, ~ + he) - a(x,~)}~--xx
+ {b(x, ~t + R e ) - b(x, fi)} = O(e), uniformly in [0, X],
(10.3.15)
with the initial condition dh~ h~(o) = - - ~ (o) = 0;
(10.3.16)
3.3. The Estimate of the Remainder Term and Final Results
235
in (10.3.15) we have used again the regularity of the coefficients a and b and that of d~
dx"
It is clear that an estimate of / ~ gives an estimate of / ~ and vice versa and so we consider for the time being / ~ . The function ~(x) is a known function and we have to estimate / ~ from the initial value problem (10.3.15)-(10.3.16). This will be carried out in the next subsection by application of the fixed point theorem of the preceding subsection 10.2.1. 3.3.
T h e E s t i m a t e of t h e R e m a i n d e r
Term and Final Results
The left-hand side of (10.3.15) and the initial conditions (10.3.16) constitute a nonlinear map F from C2[0, X] to C[0, X] with
d2p
dp
d~z
F [ p ] - e ~ x 2 4- a(x, u 4-P)dxx 4- {a(x, ~ 4 - p ) -
a(x,~)}~x
4- {b(x, ~ 4- p) - b(x, ~)}, with
@
p E N := {PIP E C2[0, X], p(0) = ~x(0) = 0}, and F[0] = 0. The linearization of F at p - - 0
(10.3.17)
(10.3.18)
reads
d2p dp { Oa dCt Ob } LiP] = c--~x2 + a(x, ~t) ~ + ~ (x, ~) dxx 4- ~u(X, u) p,
(10.3.19)
and hence
9 ~] "- F[p] - Lip] - { a(x, ~z + p) - a(x, ~t) }
ap
Oa d5 + {a(x, Ct + p) - a(x, ~) - p-~u(x, ~z)} dx Ob + {b(x, ~z + p) - b(x, ~) - p-~u(X, ~z)}. Both operators L and ~I' are well-defined in the normed space N with I P I - max Ip(x)l + v ~ max I dp
t0,xl
t0,xl
~(~)1;
(10.3.20)
the choice of this norm will become clear later on. The Banach space B occurring in the formulation of the fixed point theorem of subsection 10.2 is specified as B := { q l q E C[0, X]} with Ilqll- max Iq(x)l. [0,x]
(10.3.21)
To use the fixed point theorem we should verify the conditions (10.2.2), (10.2.3). Due to the a priori estimate (7.1.27) of Chapter 7 we have from Lip] = q max Ip(x)l + v/~ max I-7-(x)l dp < C max Iq(x)l, [o,z] - [o,x] [o,x] r
(10.3.22)
Chapter 10. Perturbations in Nonlinear Initial Value Problems
236
where C is a constant independent of e. Using the norms (10.3.20) and (10.3.21) it is clear t h a t
I L-l[q] [ _< e-lllqll, Vq ~ B
(10.3.23)
with ~-1 = C and so the condition (10.2.2) is satisfied. Henceforth we use s as a generic constant independent of e which may be raised without violating earlier results. To verify the Lipschitz condition (10.2.3) we investigate
II~(p2) - ~(px)ll
- max I{a(x, ~z + p2) - a(x, (z + [0,x]
@2 pl)}-d-x-x (x)
{dp2 dpl} du :{Oa(x,~4-~)Oa(x,~)}d~ P2
4- {a(x, ~ 4- PI) - a(x, fi)}
dx
dx
4- '~x (x)
~
Pi P2
+
~(x,
~ + a) -
(x, r~) a~ I
Pi
d4 We remark t h a t fi(x) = w(x) + ev(7x ) - evo and so ~-d(x) is uniformly bounded in [0, X] by a number independent of e. Moreover by the assumption a, b E C 1 ' 2 ( ] ~ x R) and the mean value theorem we get
II~(p2)- ~(px)ll _< ~-1 -~ p Ip2 - p l
I,
~pl,p2
e ~"~N(P),
V p E [0, p]
(10.3.24)
,5 may be taken as an arbitrary constant, e.g. /5 = 1 and the generic constant s not depend on e. The function m(p) occurring in (10.2.3)is simply
does
(10.3.25)
r e ( p ) = I. - i - p
v~ and so the condition (10.2.3) is fulfilled. From (10.2.4) we have p0=sup{pI0_o
I,
and therefore l~p 0 =
1
re(p) ___ ~} = ~v~
l~3%/fE.
~2
(10.3.26)
It follows from (10.3.13) t h a t the right-hand side of (10.3.15) belongs to the space B and is uniformly O(e) in [0, X] and therefore smaller t h a n 89 for e sufficiently small. Applying finally the fixed point theorem we obtain the result t h a t there exists a solution ~ R~ of the initial value problem (10.3.15)-(10.3.16) with
I k. I -< 2 - llfll
= o(,),
(10.3.27)
3.3. The Estimate of the Remainder Term and Final Results
237
for ~ sufficiently small. We remark that the size of ftB(t.p) = O(X/~) and by (10.2.2) the size of L-l[FtB(lp)] is also O(v~). Therefore the fixed point / ~ with I / ~ [ O(~) lies certainly in L-I[~B(ip)] and therefore k~ is uniquely determined; see proof of the fixed point theorem. This means that also our original initial value problem (10.3.1)-(10.3.2) has a unique solution for 0 < x < X and ~ sufficiently small. With the aid of the definition (10.3.20) we have with u = ~ + / ~
du d~ max lu(x) - ~(x)l + V~ max I -:-(x) - -:-(x)l = O(~) [0,z] [0,x] g x - or using u(x) -- w(x) + ev(-~) - ev(O) + ke(x)
we get
u(x) = w ( x ) + O(e), uniformly in [0, X] and
du dw d dx(X) - -~X-X(x) + r
x
+ O(~/~), uniformly in [0, X].
(10.3.28)
(10.3.29)
These results give satisfactory estimates for the solution u(x) of the initial value problem (10.3.1)-(10.3.2), and they are quite similar as those for the linear problem, see (7.1.29)-(7.1.30). We formulate the following theorem. THEOREM 2
Consider the initial value problem r
d2u du 2 + a(x, u) + b(x, u) - 0,
0<x
du with the initial conditions u(O) = a, -g~(O) = 3. Suppose a, b C C1'2(1~ • R), and a(x, u) >_ ao > O, Vx E IR+, Vu C N, with ao some positive constant independent of
Further, assume that the solution of the reduced initial value problem a(x, w) dw + b(~, w) = o,
~ ( o ) - ~,
exists in the segment 0 <_ x <_ X . Under these conditions there exists for e sufficiently small a unique solution of the initial value problem and we have the following estimates, uniformly valid in [0, X] ~(~) = ~ ( ~ ) +
du dw d---~(x) = ~ ( x ) where ev(~)
o(~), -c d v(X + ~ ~) + O(v/~),
is a boundary layer term, given by (10.3.8).
Remarks
1. We supposed for convenience sake a(x, u) and b(x, u) to belong to the space C1'2(1~ • 1~). However in our proof we were necessarily restricted to the segment
238
Chapter 10. Perturbations in Nonlinear Initial Value Problems 0 < x _ X < X0, where X0 is the upper bound of x-values where w(x) is still regular. So the semibounded interval ~ may be replaced by [0, X]. Further, it follows that the range of u values is now restricted to an open interval I containing the range Rw of w:
I ~ n ~ - {y c R I3= e [0, x ] with y =
w(x)}.
Hence it is permitted to suppose that a(x,u) and b(x,u) belong to the space C1'2([0, X] X I). The consequence is that a(x, u) and b(x, u) may have singularities in u, provided that these singularities do not belong to Rw. As an example we take the initial value problem
d2u
1 du
Ou
Ed~x2 -~--udxx -~-zt=0,
0 < x < (:x:), u(0) = C 1 > 0,
~x(0) = C 2 .
The solution of the reduced problem reads
C1
w--
0<x
1 + Clx'
and the range of w is C1 1 +C1X
<w~_C1, -
where X
is an arbitrarily large positive constant. The interval I may be chosen as (2(l+C1x),Cx C1 + 1). Application of Theorem 2 gives immediately u(~)
= ~
C1
1 +Clx
+ o(~),
uniformly in any bounded segment. 2. Similarly as in the case of linear equations it is possible to obtain higher approximations of u(x) by putting stronger smoothness requirements on the coefficients a(x, u) and b(x, u). In case a(x, u) and b(x, u) belong to C ~,~176 (]~+ x R) and a(x,u) > a 0 > 0 for 0 < x < o o and - c ~ < u < + o o we have
dO
dO
U(X) ~ ~ 6nwn(X) + ~ 6nTlvn(X--) n=O
(10.3.30)
n=O
and
du
dO n=0
dwn
do
dvn x
--~X (X) -~- ~ 6n+1 "-~X ( C , n=0
(10.3.31)
uniformly in every bounded interval 0 < x < X < X0 where X0 is again the upper bound of x values where wo(x) is still regular. The functions w,.,(x) and v,,(~) are obtained after substitution of (10.3.30) into (10.3.1)-(10.3.2) and collecting equal powers of e. For details the reader is referred to Geel [49], pp.29-37; further the remark above is also applicable in this case.
4.1. Introduction 4.
239
A G e n e r a l N o n l i n e a r Initial V a l u e P r o b l e m
4.1.
Introduction
In this section we generalize the quasilinear initial value problem of the preceding section to the more general nonlinear initial value problem
d2u r
du = f ( u , ~xx),
with the initial conditions
x > 0,
du ~xx(0) =/3,
u(0) = a,
(10.4.1)
(10.4.2)
where f is, apart from some conditions to be stated later on, an arbitrary nonlinear function. We have omitted the x-dependence of the function f, because this is not essential in our discussion and further because it simplifies the notation. It is useful to write the initial value problem (10.4.1)-(10.4.2) in the form
du dt = v(t),
dv E-~ = f(u, v),
(10.4.3)
with u(0) = a,
v(0) =/3.
(10.4.4)
We replaced the variable x by the variable t, since this is more common in the theory of dynamical systems. We submit the function f ( u , v) to the following hypotheses i. f is infinitely often differentiable with respect to u and v. ii. There exists a unique function ~(u) such that
f ( u , ~o(u)) = 0,
(10.4.5)
with the property that the initial value problem du d---t- = ~(u),
u(0) - a,
(10.4.6)
has a unique solution u = uo(t) on some closed bounded segment, say 0 _< t _< T. iii. There exists a positive constant ~ > 0 such that for 0 _< t _< T
of av( O(t),
<
(10.4.7)
where
vo(t) = ~(uo(t)).
(10.4.8)
iv. For the same value of of
Ov (u0 (0),
for all values of )~ between v0(O) and ft.
<
(10.4.9)
240
Chapter 10. Perturbations in Nonlinear Initial Value Problems
The first condition is introduced in order not to be bothered with sufficient regularity as required in the analysis below. The third condition suffices to solve vo(t) from the relation f(uo(t), vo(t)) = 0 and the second condition gives a first step in the outer approximation valid in a segment 0 < 5 <_ t <_ T with 5 independent of e, but arbitrarily small. It will appear that the fourth condition is needed to ensure the appropriate exponential behaviour of the initial (boundary) layer. The condition (10.4.7) corresponds with the condition a(x, u) > a0 > 0 required in the case of the quasilinear initial value problem of the preceding section. 4.2.
The Formal
Approximation
It will appear that in order to prove the validity of the formal approximation we need an expansion up to and including O(e). So we put as an approximation
ft(t) = uo(t) + eul (t) + re(T), ~(t) = vo(t) + ~vl(t) + n(~),
(10.4.10)
where m(T) and n(~-) are the initial layer terms with T = t/e. Using our experience with the quasilinear case we expect
m(,) = ~(~o(-) + ~ml(~) + . . . ) , ~(~) = ~o(:) + ~ ( - )
+..
(10.4.11)
and m ( r ) = n(r). The first terms of the outer expansion follow from the reduced equation (e - O)
duo
f (uo, Vo) -- O,
dt = vo(t),
or from the initial value problem
duo = qO(uo(t)), dt
uo(0)- a
(10.4.12)
and
vo(t) = ~o(uo(t)) "= ~oo(t), valid for 0 _< t _< T. The second terms obtained from the relations
(10.4.13)
eUl (t), evi (t) of the outer expansions are
d
d-~(uo + e U l )
(10.4.14)
= v0 nt- e V l
and
dvo ~2 dvl ~ - ~ + - ~ = f (uo + ~=1, .0 + ~Vl)
Of = ~(~o,
Of ~o)u~ + ~ ( u o ,
~o)~ + o ( ~ ) .
(10.4.15)
241
4.2. The Formal Approximation
We do not calculate ul(t), vl(t) from (10.4.14) and (10.4.15), but from the approximation dvo Of Of ] g--~ -- ~ ~U (U0, V0)U1 -~- ~V (U0, V0)V1j , (10.4.16) making an error of only O(e2). Hence we get
dvo
e2 dvl
-- f (uO nt- CUl, VO nt- ~Vl)
- C - ~ -Jr- W
nt- O(C2),
(10.4.17)
uniformly in [0, T]. Therefore Ul, Vl satisfy the equations dul
dt
= ~l(t)
and
Of
V1 (t) -- (~V (U0, V0))
- l dvo
Of
Of
--~--(t) -- (~--~ (U0, V0) ) (~V (U0, V0))--lul(t).
With the aid of the relation
Of (~o, vo) Ou
Of
d~ auo
+ -a-(~o, ~o) ,.-(~o) = o, uv
we obtain for U1 (t) the differential equation
dul dt
d~
Of
du0(~0)~l + (~(~0,v0))
-1 dvo
dt
(10.4.18)
with the still unknown initial condition ul(0) = -too(0).
(10.4.19)
The initial condition is to be determined from the initial layer analysis below. The solution of this initial value problem gives Ul (t) and consecutively the function vl(t) follows from (10.4.18)
du l
d~
Of
v~(t) = -~-(t)= ~o(~O)~(t)+ (~(~0,~o))
- 1 dvo
-~/-(t).
(10.4.20)
The pair (uo+eul, vo+evl) satisfies the differential equations (10.4.3) up to O(e 2) uniformly in [0, T], but the initial conditions (10.4.4) are not fulfilled. In particular we have uo(O) + eUl (0) = a + eUl (0) and vo(O) + eVl (0) =
d
d uo(O) + e ~ u l (0).
242
Chapter 10. Perturbations in Nonlinear Initial Value Problems
This error is corrected by the introduction of the initial layer t e r m s
~mO(T) and
nO(T) + enl (T) satisfying in their turn the initial conditions d d mo(0) - - e U l (0), no(0) -- ~ - ~-~uo(0) --/~ - vo(0), nl (0) -- -~-~Ul (0) -- -Vl (0). (10.4.21) Further we have
~emo(
) = -d--~rmO(T)-- no(T).
(10.4.22)
Let no(T) + ~n 1(T) satisfy the equation
dno dnl dT + e--~-T = f(Uo + ~Ul + emo, Vo + eVl + no + enl) -- f(uo + eUl, Vo + eVl) :
f ( t t 0 nt. CUl nt. ~ m 0 ,
dvo
v 0 + Ev 1 + n o -~- C n l ) -- E - - ~ - --
e2 dvl --~
-~- O ( ~ 2 ) ,
(10.4.23)
or with the aid of the mean value theorem
dno dnl dT dT f (Uo + eUl + emo, Vo + -CVl+ no + -ca1) -- f (Uo + eUl + ~mo, Vo + eVl + no) + : ( u o + ~u: + ~..o. ~o + ~v~ + no) - f ( u o + ~ : . ~o + ~ : ) of e O--vv(uo + eUl + emo, vo + g'Vl + 'r/,O -t- g'~91'nl)'rtl
=
o] +e~-~u (uo + eUl + e~92mo, vo + eVl + ~92no)mo o.f + O r (u0 + ~Ul + gz92mo, Vo ~- ~Vl + ~2no)no,
where Zgl and ~2 are functions of T with right-hand side with respect to e we get
dno dT
_
of
Ov
0 < Z91(T), Z92(T) < 1. Expanding the
(uo(O), vo(O)+ ,h(~-)no(~-)) 9no(~-), - T > o
(10.4.24)
and
dnl = N(T), dr with N(T)
T> 0
(10.4.25)
the complicated expression
N(T) = ~Of (~o(O), ~o(O)+ ~o(~))~1(~)+ ~Of (uo(O), vo(O)+ 0~(~)~o(~))mo(~) +-
1{Of
(uo(~) + ~Ul(~) + ~o~(~)mo(~), ~o(~) + ~v~(~) + o~(~)~o(~))
Ov (uo(O), vo(O)+ 0,2(r)no(r))
}
no(-).
(10.4.26)
4.2. The Formal Approximation
243
The initial values for no(T) and nl(T) are given by (10.4.21). Solving the initial value problem for no we use now the hypothesis (10.4.9), which gives that the factor of no(T) in the right-hand side of (10.4.24) is negative whenever the value of v0(0) § 02 (~')n0 (T) is between v0(0) and /3. This is certainly the case for ~- = 0 and so nO(T) is monotone till nO(T) vanishes; nO(T) decreases to zero whenever /3 > v0(0) and it increases to zero whenever ~ < v0(0). Hence we have Ino(~)l
=_< I~- ~o(0)1 exp[--~T],
0 _< T < (x~,
(10.4.27)
and so nO(T) has the right assymptotic behaviour as required in an initial boundary layer. The other initial layer function mO(T) follows from (10.4.22) and we have oo
mO(T) -- -- / no('r')dT'.
(10.4.28)
T
Also
too(T)
has the right asymptotic behaviour for ~- --+ r
and moreover
OO
m0(0) = - / ~0(~')~',
(10.4.29)
0
and the still missing initial condition for ul(t), see (10.4.19), is now known. It is in general difficult to get an explicit expression for toO(T) and nO(T) and therefore we have to be content with their qualitative behaviour as represented by (10.4.27) and (10.4.28). Finally we have to investigate the function nl (T) and in particular its asymptotic behaviour. Using again the condition (10.4.9), the boundary layer behaviour of too(T) and no(T) and the regularity of the function f we obtain from (10.4.25)-(10.4.26) that also nl(T) has the right boundary layer character with nl (T) = O(exp[--~(1 -- 5)T]),
(10.4.30)
where 5 is an arbitrarily small positive number independent of r this number is to be introduced because there appears in N(T) after a further expansion with respect to r terms like "rno(T). We consider now the expressions
~(t) -~ r
) § ~Ul(t) § ~mO(T)
and
(10.4.31)
~(t) = ~0(t) + ~v~(t)+ ~0(~) + ~n~(~). On account of their construction, see (10.4.15) and (10.4.23), they satisfy the initial value problem d~ d---t = ~(t) - ~ n l (T) d~ e~-~ = f(~, ~ ) + O(e2), uniformly in [0, T],
244
Chapter 10. Perturbations in Nonlinear Initial Value Problems
with the initial conditions fi(0) = (~,
~(0) = f~.
(10.4.32)
Hence (10.4.31) constitutes a [ormal approximation of the solution (u, v) of our initial value problem (10.4.3)-(10.4.4). To prove that (fi, ~) is indeed a good approximation we put u(t) = fi(t) + R(t), v(t) = ~(t) + S(t), (10.4.33) where (R, S) is a solution of the initial value problem
dR
du
dfi
~t = dt
and
dt = ~(t) - (~(t) -
dS e-~
(dvd~) dt dt
-- e
~,,~(~)) =
s(t) +
~(~)
- f (fi-l- R' ~ + S) - f (~t' ~) -4- O ( e 2)
with R(0) - S(0) = 0. We now make the irrelevant change
R(t) = R(t) + E
nl (--~)dt' = R(t) + E2 0
=
k(t)
+
nl (T)dT 0
(10.4.34)
o(~),
and this gives for the remainders /~(t) and S(t) the initial value problem
dk dt dS
"- S~ (10.4.35)
uniformly for 0 < t < T while the initial conditions are /~(0) ---- S(0) - 0.
(10.4.36)
In the next subsection we derive from this initial value problem an a priori estimate for /~(t) and S(t), which validates the formal approximation (10.4.31). 4.3.
The Estimate of the Remainder
Terms and the Final Result
We use again the fixed point theorem of section 2. Inserting the first equation of (10.4.35) into the second we get d2/~
~-~
d/~
- f(~ + R, ~ + - ~ ) + f(~t, ~) = O(~2),
(10.4.37)
uniformly in 0 _< t _< T, with the initial conditions
k(0) = -~d k (0) -- 0.
(10.4.38)
245
4.3. The Estimate of the Remainder Terms and the Final Result The nonlinear operator F is defined by F[p] = r
d2p
dp
- f(~2 + p, ~ + -~) + f(~2, fi),
with F(0) - 0. As p- and q-spaces we take again
dp
N "= {p E C2[0, T], p(0) - ~-~(0) = 0}, with norm
I pl
-
x I~(t)l, dp tO,T] Ip(t)l + v ~ m[0,a T]
(10.3.20)
m a x
and B "- {q e C[0, T]}, with norm I l q l = max Iq(t)l. [0,T]
(10.3.21)
The operator F is splitted as before in its linear part L and the remainder 9. Hence
F~] = L~] + ~ ] , where
d2p
L[p] = e dt 2
Of Of dp O~z(~z, 9)p - ~v (~Z, ~) -d-[
(10.4.39)
and
dp
Of
Of
dp
(10.4.40)
~ [ P ] - - f ( ~ + P ' ~ + ~-) + f ( ~ ' ~) + 0--~(~' ~)P + ~ - ( ~ ' ~) d--t-"
To apply the fixed point theorem of section 2 we have to check the conditions (10.2.2) and (10.2.3). Due to the a priori estimate (7.1.27) of Chapter 7 we have from L~)] = q
I p(t) I _< Cllqll where C is a generic constant independent of e. Therefore
I L-~[q]l < e-lllqll,
Vq e B,
with ~ = C -1 and condition (10.2.2) is fulfilled. To verify the Lipschitz-condition (10.2.3) we investigate I]~[P2]- ~[pl]ll =
dp2
dpl ~ _ Of ) - f ( f t + p l ' v+ dt / -~u(~z,v)(p2-pl) Of ( dp2 dpl ) dp2 ] _ dpl = m a x f (zt + p2, v -4f (zt + pl v-f) Ov (ft, ~) dt [0, T] dt " ' Of.u+pl( v+ dpl ~(P2 _ P l ) - Of (~2+pl, v + dpl ~ ( dp2 dpl ) Ou ' --~ " -~v - ~ " dt dt dpl Of Of dpl Of
max f(~z+p2 v+ E0,T] ' ~
+{Ouu0f (g + Pl, v + - - ~ ) - Ouu(u' ~) } (P2 - Pl) + { ~vv (72 + Pl ' v + ~ )
(dp2 dt
dpl) dt "
- Ovv (~2, ~) }
246
Chapter 10. Perturbations in Nonlinear Initial Value Problems
Using Lagrange's remainder term it follows with the aid of (10.3.20) that II~#[p2] - ~[p~]ll _%<e-1/9 [P2
-
Pl I ,
(10.4.41)
valid for all P2 and pl in any ball ~N(P) with p E [0, is]; e -1 is a generic constant and t5 may be any positive constant, e.g. /5 = 1. The function re(p) occurring in (10.2.3) is now
re(p) - e -~ ~, E
and condition (10.2.3) is fulfilled. From (10.2.4) we have PO = sup{p 10 < p < 1, m(p) < ~l.} = 2~. 2 p___o and therefore l~po = ~e~3. Since the right-hand side of (10.4.37) is 0(~ 2) we may apply the fixed point theorem and we get finally the result I kl = (10.4.42) It is clear that we needed a formal approximation up to O(~2), otherwise the right-hand side of (10.4.37) would have been O(e) and we could not apply the fixed point theorem. The result is that /~(t) exists uniquely and we have the estimate max dh = o(e2), [0,r] IR(t)l + v/~max [0,r] I-~-I and this gives u(t) : Uo(t ) -4- ~U 1 (t) -~- ~mO(T ) -Jr-O(E2),
uniformly in [0, T]. Moreover we have also dh = O(~3/2) du dt dfi dt+ O(r and therefore
d k - -~-+ dn uniformly in [0, T]. However -~-
du
d t = vo(t) + no(T) + O(~), uniformly in [0, T].
Collecting our results we have the following theorem THEOREM 3
The initial value problem
r
d2u
(10.4.43)
du - f (u, -~-),
O
0(6)=
(10.4.44)
4.3. The Estimate of the Remainder Terms and the Final Result
247
with the initial conditions u(O) = a and -~(0) du -- ~ has under the hypotheses i-iv, stated in subsection 10.4.1, a unique solution in the segment 0 <_ t <_ T. Moreover the following approximations are uniformly valid in [0, T] u(t) -- uo(t) + Cul(t) ~- r
t (~) -~- O(~2),
du duo t ) + O(e), d--~(t) = --d-~(t) + no(~ where the outer expansion terms uo(t), ul(t) and vo(t) are determined by (10.4.12), (10.4.13), (10.4.18) and the initial layer terms e m o ( t ) and n o ( t ) by (10.4.21), (10.4.24) and (10.4.28). Example We illustrate the theory with the famous example of the van der Pol equation , which reads d2u du r + (u 2 - 1) + u = 0. (10.4.45) This equation gives so-called relaxation oscillations which are periodic solutions characterized by a change in time alternately slow and fast. We write (10.4.45) in the equivalent form of a system, viz. du ~v
dt dv U e-dt = ( 1 - u2)v - u = ( 1 - u 2 ) ( v - 1 - u 2 )" Taking initial conditions outside the initial layer
u(0) - a, v(0) = p U0
(10.4.46)
we get as a first approximation valid
v0 = 1 - u~'5
u0 ~: +1,
duo uo dt - 1 - u 2'
u0(0)=a,
(10.4.47)
with
and integration yields t
1
uo(t) - g(u0(t))
3
f
= J uo(t')dt' + a -
1 3 ga .
(10.4.48)
0
We put t
-
(10.4.49)
+ f u(,')d,' : 0
and we get as another equivalent system
dw dt -u, du !u3 e~-~ = u 3
-~- W
(10.4.50)
248
Chapter 10. Perturbations in Nonlinear Initial Value Problems
and by the first equation of (10.4.46) the transformation v -- - ( u -
u 3 + w).
(10.4.51)
The initial conditions u(0) = a, v(0) - ~ change into u(0)=aandw(0)-
~a 3-a+r
which is in accordance to (10.4.49). du We proved that outside the initial layer as well u(t) as ~-(t) are approximated by uo(t) and vo(t) within an error of O(r This means that we have in the variables u and w .1 (10.4.52) ~(t) - ~-u~(t) + ~(t) - ~v(t) = 0(~) and
dw -~t (t) du (t) = ~- ~ ( .t ) .
-u(t) .~(t).
= u 2 - 1 + 0(~)=
-uo(t) .
~- 0(~)
~o(t)
u 2-
(10.4.53)
1 + 0(~).
In Figure 10.1 we have sketched a trajectory in the (u, w) plane
J
7
/
J
%
:
% %
fY
A
/
i
D
/
,'e
/ ~
(
/
,g Fig. 10.1 Relaxation oscillation ~-, -~-) It follows from (10.4.50) that the velocity field (du dw in a point A outside the cubic C (w - 893 - u) is nearly horizontal with -~-d~_ O(~) and it points to the right 1 3 whenever w > 5u -u. Taking A, as in the figure, as the starting point of the trajectory, then the path from A to the cubic is analytically represented by the initial layer. A careful analysis of the velocity field yields that the path is continued by first crossing the cubic C and thereafter it follows C within an approximation of O(c); also the derivative of the path follows
249
4.3. The Estimate of the Remainder Terms and the Final Result
the derivative of C within the approximation of O(r see (10.4.52) and (10.4.53). 1 3 Whenever w < 5u - u and A lies at the other side of the cubic there is no crossing and the path follows again the cubic. This is all in accordance with the theory of this and the preceding section, as long as o ( ( 1 - u 2 ) v - U) "-- 1 - u 2 < - n . So the theory holds as long as the path has not reached the "critical" value u = 1, and the problem arises how to pass the neighbourhood of the "critical" point C (1, - 2). Analyzing the location of the trajectory in this neighbourhood it can be shown that the trajectory contains a point (u*,w*) at a d i s t a n c e o f a t least 0(r 1/3) from ( 1 , - 5 )2 and with W* < - 2 / 3 . d~ becomes O (r ~) and so Inserting these coordinates into (10.4.50) one obtains that -~(u(t), w(t)) gets enough impetus to leave the neighbourhood of (1, - 52) and the path is continued nearly horizontally in the direction of D and the story can be repeated again. The velocity field allows to apply the Poincar~-Bendixson theorem, see e.g. [60, Ch.II] and so there exists a periodic solution. The trajectory of this periodic solution lies in a 1 3 - u for 1 _ < u _ < 2 , w = - 5 2 for neighbourhood of a closed curve defined by w = 5u l U3 - 2 < u < 1, w : 5 -u for - 2 < u < - 1 and w = + 52 for - 1 < - u -< 2. The distance of the periodic trajectory to this curve approaches zero for r $ 0 and it is of order O(r along the branches of the cubic w = 5l U 3 - u. There exists much literature on the van der Pol equation and we refer the reader to Wasow [139, p.301] and O'Malley [112, p.62]. Also should be mentioned the work by Troesch and Urlacher [133], who gave a fine analysis of the behaviour of the trajectories in the neighbourhood of the critical 2 This paper is one of the first of many interesting publications, where points (+1, TS). nonstandard analysis is used in singular perturbation theory [101,133, 134, 17, 4, 136]; see also M. Canalis-Durand, Solution formelle Gevrey d'une ~quation singuli~rement perturb~e; Asymptotic Analysis, 8, (1994). Finally we remark that we discussed in section 3.5 of Chapter 5 the equation d2u
dt----~
+r
This equation becomes with t = r
1 d2u r dT----~
du
2-1)--.
+u-0,
t>0,
0
(10.4.54)
the counterpart of equation (10.4.45), namely
du -~- (U 2 -- 1)~TT + U = 0, T > 0, 0 < r << 1.
(10.4.55)
We have shown in section 3.5 of Chapter 5 that the solution of (10.4.54) is of the form
u(t) = 2(1 - A e - a ) - 89cos(t + ~0) + O(r and the influence of the perturbation r
2
-
1) -~ du is only observable after a long time.
This is in contrast with the solution of (10.4.45), where the perturbation immediately observable in the large horizontal velocity field.
_ d 2 u ed--~
is
250 5. 5.1.
Chapter 10. Perturbations in Nonlinear Initial Value Problems Q u a s i l i n e a r Initial V a l u e P r o b l e m s w i t h a S i n g u l a r P e r t u r b a t i o n of Second O r d e r H y p e r b o l i c T y p e Introduction We discuss the singular initial value problem
EL2[u]+Lliu]=O,
(x,t) e D - { - ~ < x < + c r
t>0},
(10.5.1)
with
02u L2[u] = Or---~ -
c2(x, t) 02u Ox 2
and Ll[u] = a(x, t, u) Ox Ou + b(x, t, u ) -Ou ~ + d(x, t, u).
(10.5.2)
The initial conditions are given by
u ( x , O ) - f(x),
Ou -~(x) = g(x),
- c r < x < +cr
(10.5.3)
To avoid tedious book-keeping we assume for convenience a(x, t, u), b(x, t, u), d(x,t,u) C C ~ ( D x R), c(x,t) c C~176 f(x), g(x) E C ~ ( R ) and in addition we require similarly as in Chapter 9, section 3 i. c(x, t) > 0 in D, c(x, t) uniformly bounded in any strip {(x, t)} with - cr < x < +cx), 0 < t < T * ii. b(x,t,u) > O in D x R , iii. la(x, t, u)l < b(x, t, u)c(x, t) in D x R. The last condition ensures the timelike character of the subcharacteristics, see Ch.9, section 2. As in all preceding sections we start with the construction of a formal approximation consisting of the sum of the solution of the reduced problem with 6 - 0 and a correction accounting for the second initial condition. Consecutively we formulate the initial value problem for the error R~ and we give an a priori estimate using again the fixed point theorem which yields also the unique existence of R~ and by this the unique existence of the solution u of the initial value problem (10.5.1)-(10.5.3). Related initial value problems with more space dimensions have been studied by Genet and Madaune [51], [102]. They discussed a nonlinear differential equation of the type
02u -- As) + a(x, t)Ou e(-O--~ -~ + En bk(x,t) ~Ou + c(x, t)u + F(u) = f(x,t) k--1
with (x, t) E ~ • ~ C R '~. The a priori estimates and approximations of u obtained by these authors are estimates and approximations in the L v norm. 5.2.
The Formal Approximation
A formal approximation of our initial value problem (10.5.1)-(10.5.3) is given by
~),
~o(~, t) = ~o(~, t) + ~ o ( ~ , t
(10.5.4)
251
5.2. The Formal Approximation where w0(x,t) satisfies the reduced problem
Owo Owo a(x,t, wo)-~x + b(x,t, w o ) - ~ + d(x,t, wo) - O, - o c < x < + o c ,
t;>0
To(x, O)- f(x), - c ~ < x < +c~,
(10.5.5)
t -i)
while evo(x, = evo(x, T) is the boundary layer correction. This term is a solution of the boundary value problem
02vo
Ovo
a~---~ + b(~, o, wo(~, o ) ) - ~
Ovo Owo OT (x, 0) = g(x) -- ~ ( x ,
- o,
~- > o
0), lim Vo(X, T) -- O.
(10.5.6)
Despite the regularity of the coefficients a(x,t, To), b(x,t, To) and d(x,t, To) and of the function f(x) the solution of the reduced problem may tend to infinity in finite time or may become multivalued after some time T. Therefore we make the crucial assumption that To(x, t) belongs to C ~ in some region ~0 = {(x,t), - o e < x < +c~, 0 _< t _< T} . The boundary layer correction evo(x, t) follows easily from (10.5.6) and it reads
g(x) - O~oo~~(x,~Jm
evo(x,-~t), = - e b(x, 0, W--o0(-x:{)~ exp [ - b ( x , 0 , wo(x,O))-~tj, . Substitution of ~0(x,t) -
(10.5.7)
Wo(x,t)+ evo(x, t) into eL2[u] + Ll[u] = 0 gives eL2[r0] + Ll[r0] = O(e),
(10.5.8)
uniformly in any bounded domain in ft0 , with
rio(X, O)- f ( x ) + evo(x, 0), O~Zo ot (~' o) = g(~),
-~
-oc < x < +ec
< ~ < +~.
Therefore ro(x,t) satisfies the initial value problem (10.4.1)-(10.4.3) up to O(e) and so it is a formal approximation of u(x, t). However it will appear in the next subsection that this formal approximation is not sharp enough to prove that the error Ro(x, t) is O(e), uniformly in any bounded domain in ~0. We need a higher order approximation, viz.
t) ft l (x, t) -- Wo(X , t) ~- CWl (X, t) -~- ~Vo(X, -~
+ e2vl (x, -~t ) .
(10.5.9)
Substitution into (10.5.1)-(10.5.3) and expansion into powers of e gives after a tedious calculation complicated but harmless initial value problems for Wl and Vl. Fortunately, it is not necessary to calculate them explicitly; we need only to verify their existence and regular behaviour. The second term eWl(X, t) of the outer expansion satisfies the initial value problem OWl
OWl
a(x, t, Wo)-~x + b(x, t, Wo)--~ + dl (x, t, To)W1 = -
( 02w0
Ot2
02W0)
Ox 2 ,
(10.5.10)
252
Chapter 10. Perturbations in Nonlinear Initial Value Problems
with - c r < x < +cr
0 < t < T, and the initial condition Wl(X, O) = - v o ( x , 0).
The function reads
di (x, t w0)
is obtained with the aid of the mean value theorem and it
Oa Owo Ob Owo Od ( x, t, wo ) . d i ( x, t, wo ) = -~u ( X, t, wo ) ---~x + -~u ( X, t, wo ) - - ~ + -~u
t satisfies the boundary value problem The second initial layer correction e2v.i(x, 7) 02711
OqVl
OT----~ + b(x, O, Wo(X, 0))-~T = r
T),
(10.5.11)
T > 0,
with the initial conditions
Ovi OWl OT (x, 0) = -- - - ~ (x, 0) and 7----+oo lim vi(T) = 0. r
T) is an expression regular in x and T and it has the form
r
Ovo
= p0(~, ~0(~, 0))~0(~, ~)+ p~(~, ~0(~, 0))-g2~ (~,,) Ovo
-4- p2(X, lV0(X, 0))Wl (X, 0) ~
0~o (z, ~)
(X, T) -+- p3(X, W0(X, 0)) T-~T
+ p~(~, ~o(~, 0))~0(~, ~) O~o (~, ~); for details see[48],[49]. It follows that Wl is C ~ in - c r 1 6 2 0
u ( x , t ) -- ill(X, t) -- ~2Vl(X, 0 ) -~- R ( x , t ) " = fii(x, t) + R ( x , t ) ,
(10.5.12)
we get for R(x, t) the initial value problem
(
02R OR OR F[R] =e \ ~ - c 2 (x, t) 02R ) + a (x , t, fii + R ) -~x + b(x, t, ttl -4- R). Ot 0~1 + {a(x,t, ~1 nt- R) - a(x,t, ~ l ) } - ~ x -t- {b(x,t, ~tl -t- R) - b(x,t, fii)} 0~0t + {d(x,t, ~ + R) - d(x,t, ~ ) } = O(e2), uniformly in any compact domain f~ C / ) = { (x, t), - c r < x < +cr the initial condition is
R(x, O) = -ff~ OR (z, o) = o,
(10.5.13) 0 _ t _< T}, while
- 0 0 < x < +00.
We show in the next subsection with the aid of the fixed point theorem that R = O(e 7/4) and hence u(x, t) = fil (x, t) + O(e 7/4) and therefore also u(x, t) =
~0(~, t) + o(~)
5.3. The Estimate of the Remainder Term and Final Results
253
5.3. T h e E s t i m a t e o f t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s
To apply our fixed point theorem we should be a little bit subtle in choosing our space N which is defined as follows
N - {PIP e C1(~),
p(x, O) = Op (x, 0)
L2~)] e C ~
= 0}
(10.5.14)
and we state explicitly that the expression L2~] = o_~ . o2p may contain weak or2 _ c 2 (x, t)-5-~2 derivatives. The space B is chosen as the Banach space
B - {q l q C C~
with norm
Ilqll - n~x]q(x,t)l.
(10.5.15)
The left-hand side of (10.5.13) is now a nonlinear map from N to B and it reads
02P F~] =~ -0~
-
c2(x, t) O~p ~ 019 + b(x,t, ~tl + p) OPot ~x2] + a(x,t, ~tl + P)-~x f
0~1 + {a(x,t, ~tl + p ) - a ( x , t , gl)}-~x + {d(x,t, ~tl + p ) - d(x,t,
+{b(x,t,~tl+p)-b(x,t,
ul)}0ul0t
ux)},
(10.5.16)
where gl(X, t) may be considered as a known function; moreover we have F[0] = 0. The linearization of F at p = 0 reads
{ 02p _ c2 (x,t) O2p } Op Op Lip] =c ~ ~ + a(x, t, ~q ) -O-xx+ b(x, t, ~tl) Ot + {Oa -~-uu(X,t,f i l ) ~Ottl
Ob(x,t,~tl)O~tl -0-/-
+ ~
+
Od (x't' -ffuu
}
Ul) P,
(10.5.17)
and therefore 9 ~] = F ~ ] =
{a(x, t,
Lip] ~1 -Jr p ) --
a(x, t,
~1)} "~X
{ b ( x , t, ~t I 7t- p ) - b ( x ,
Oa
+ {a(x,t, ~tl + p ) - a ( x , t ,
U l ) - p - ~ u ( X , t , ul)}
+ {b(x,t,
tt 1
+p)-b(x,t,
?~1)-P-~u(X,t,
+ {d(x,t,
~t I
-~p)-d(x,t,
?~1)-P-ff~u(x,t,
t, ?~1)}0p0t
Off1 Ox O~q
Ob
?~1)} 0t
Od
?~1)}.
(10.5.18)
Theorem 1 of the preceding Chapter 9 gives for the equation Lip] = q with ~Ot 0) = 0 the a priori estimate
p(x, O) -
Ip(x, t)l < e- 88v/C(a)K(a), Op ] < ~- 1v/C(a)K(gt) , ]-~1 Op < c -1 v/C(~)g(gt), ]-~x
(9.4.17)
(9.4.19)
254
Chapter 10. Perturbations in Nonlinear Initial Value Problems
valid for e sufficiently small. We remark that the e-dependence in the argument ?~1 is not relevant for the applicability for this theorem. C(~t) depends on the coefficients of the operator L and these are together with their derivatives uniformly bounded in any compact trapezoidal domain Ft C R 2 (x, t) with 0 <_ t _ T, while K(12) is bounded by 3 m~xlql 2, see Figure 3 of Chapter 9 and (9.4.12). The estimates (9.4.17)-(9.4.19) give rise to the following norm in the space N,
I P I = II~X [p(x,
t) l -4- e 3/4 {
and we get
OP(x,t)[+n~x[cOP(x,t)l} II~X [-~X -'~ "
[ L-~[q] [ <
s
I
(10.5.19)
(10.5.20)
1
with l-1 = C l e - z and C1 some generic constant independent of e. Herewith we have checked condition (10.2.2) of the fixed point theorem. To verify the Lipschitz condition (10.2.3) we investigate
~[p2]- ~[pl][I- m~x {a(x,t, fil + {b(x,
t, 'Ul -'t-/)2) -
+P2)-a(x,t,
0p2
fil + P l ) } Ox
b(x,t, fil + P l ) } - - ~ + {a(x,t,
fil nt- P l ) --
a(x,t,
~1)}
Op2COx Oplox
p2
+ {b(x,t, ~l + p l ) - b ( x , t ,
gl)} \ --~
Ot
+--~x
{-~u
'
-~u (x't' ~l)}dA
Pl P2 P2 -4- ~OU I I { -~ucO b ( x , t , ~t1 --~-A ) _ cObou( x , t , ill)}dA +/{cOd(x,t,-~u ~tl ~t- A ) _ cOd_~u( X , t, ill) }dA. Pl
Px (10.5.21)
From the regularity of the coefficients a, b and d and the definition (10.5.19) of the norm in N it follows by the mean value theorem that P II~[p2]- ~[px]ll < c 2 - ~
I p 2 - pl I ,
~pl,p2 e ['~N(P)
(10.5.22)
for e sufficiently small and with C2 a suitable constant independent of e; f~N(P) is the ball in N with radius p. Hence the function m(p) in condition (10.2.3) equals P re(p) = C2 e3/4~
(10.5.23)
which is monotonically decreasing for p $ 0 with lim re(p) = O. To apply finally the p-~0
fixed point theorem to FIR] = f
it is necessary that
Ilfll--_
1 -~epo,
(10.5.24)
255
5.3. The Estimate of the Remainder Term and Final Results with g = O(E 1/4) and
Po
=
s u p { p l r n ( p ) <-
lg}
=
1 ~ This means that we ~c;~2"
can
p>_o
make an a priori estimate of R whenever [[f[[ < O(~5/4). (10.5.13) is O(~2), uniformly in f~, and so (10.2.6)yields ] R]
The right-hand side of
= O(C7/4).
(10.5.25)
From this result it becomes also clear that a first formal approximation of our initial value problem with t
~(~, t) = ~o(~, t) + ~o(~, ;) - ~o(~, o) + Ro(~, t) is not good enough for getting an estimate of R0(x, t), because we have in this case F [ R o ] = f = O(c) > O(E5/4). From the definition (10.5.12) and the result (10.5.25) we obtain
U(~, t) -- ~0(X, t) -- ~
(~, t) -- ~0(~, ;t) _~2Vl(X , t) 4-~2Vl(X, 0) ] = O(~7/4),
uniformly in any compact domain ~2 with 0 _ t < T. Since w0, Wl, v0, Vl, their x-derivatives and o~0 ot a r e O ( e - 1 ) w e g e t o t , o~1 ot are O(1) and further ~Ovo and o.~ by the definition (10.5.19) of the norm in N
Ou Owo u(x, t) - Wo(X, t) = 0(~), -~x (X, t) - ~ (x, t) -- 0(~) and
Ou Owo Ovo t Ot (x, t) - - - ~ (x, t) - e ~ (x, -)~ - O(c), uniformly in ~.
(10.5.26)
t The terms ~Wl (x, t) and E2Vl(x, ~) do not appear in this final result; they were only needed for the appropriate estimate of the remainder term R(x, t). Summarizing we have THEOREM
4
The solution u = u(x, t) of the initial value problem ( 0 2 U _ c 2 ( x t) O2u) Ou Ou c ~ , ~x 2 4 - a ( x , t , U ) O x 4 - b ( x , t , u ) - ~ 4 - d ( x , t , u ) - - O , o~ (~, o) ~(~,0)- f(~), ~7
-c~<x<4-c<~, t>O,
- g(~),
with the data satisfying the conditions stated in section 5.1, exists and is uniquely determined for e sutticiently small and for (x, t) in any compact domain ~, where the solution of the reduced problem Owo Owo a(x, t, Wo)-~x 4- b(x, t, Wo)--~- 4- d(x, t, wo) = 0
wo(x, O) = f (x)
256
Chapter 10. Perturbations in Nonlinear Initial Value Problems
is regular. The solution u = u(x, t) has the following approximation uniformly valid in ft. u(~, t) - ~o(~, t) = 0(~),
O~ (~, t) - Owo
~
--OT (~, t) = 0(~)
and Ou --or
(~' t)
OWo
-
--~ (~, t)
Ovo - ~ --St-
t
(~' -)~ =
o(~).
evo(x, t) is a boundary layer term and it is given by (10.5.7). Remarks 1. At the end of the previous chapter we mentioned the work by Hsiao and Weihnacht who investigated a Cauchy problem for the equation
"C2Utt -t- Ut -- Uxx = 0.
(9.5.12).
These authors [69] considered also the semilinear perturbation problem
e2utt+ut-u~=F(u),
-oo<x<+oo,
t>0
(10.5.27)
with the initial conditions
u(x, O) = f(x),
eut(x, O) = g(x),
-cx~ < x < +co.
(10.5.28)
The formal approximation is obtained again by an outer expansion and boundary layer terms and its correctness by a contraction mapping applied to the remainder term R. The latter step involves of course also an a priori estimate of the linearized equation for R which is acquired by the method of energy integrals. Restricting the results of Hsiao and Weihnacht to the first few terms of the expandF (z) < 0, z E sion one achieves the following result. Under the assumptions ~-~ R, F(0) = 0, F, f, g sufficiently regular, the solution u = u(x, t), satisfying (10.5.27)-(10.5.28), admits the following approximation t U(X, t) = Uo(X, t) -Jr ~Vl (x, t) -Jr ~Vo(x, -~) Jr- O(~ 2)
(10.5.29)
uniformly valid in the strip So = R x [0, to], where the reduced equation
Uo,t - Uo,~ = F(Uo),
Uo(x, O ) - f ( x )
(10.5.30)
has a bounded classical solution. The second term U1 satisfies the Cauchy problem
Ul,t - Ul,xx = F'(Uo)U1,
UI(X, O) -- - V o ( x , O ) ,
(10.5.31)
while the boundary layer term Vo is the solution of the boundary value problem
v0,~- + Vo,- = ~ -OVo ~ r (x, O) = g(x) and ~lim Vo(x, O) = O,
(10.5.32)
5.3. The Estimate of the Remainder Term and Final Results
257
with T = t / c 2. For details the reader is referred to [69]. A generalization of this result was given by Esham in a Hilbert space setting, [34], [36]. He studied evolution equations of the form (10.5.33)
c2utt + ut - A[u] - F(u), where A is a positive definite self-adjoint operator.
2. Other nonlinear hyperbolic-parabolic singular perturbation problems are reported in [39, 40]. Esham and Weihnacht consider in [39] perturbation problems of the form e2utt+ut-g[u]u~=f(x,t), t>O, O<x
0
The initial conditions are
u(x,O) = f(x),
r
= g(x),
O< x < 1
and the boundary conditions u(O, t) = u(1, t) = O,
t > O.
An application concerns the oscillations of an extensible elastic string. The same authors consider in [40] the case where g[u] is replaced by some function h(u~). The results are similar as those obtained above in (10.5.29). Other variations on this theme are presented in [35, 38]. 3. The modelling of nonlinear waves leads to interesting singular perturbation problems. Consider the following initial value problem for a solution of the well-known transport equation 0--t + ~
o(1u 2) = 0 ,
-c~<x<+c~,
u(x, O) = Uo(X)
t>0
(10.5.35) (10.5.36)
with -~-oo
f
uo(x)dx < oo.
To fix ideas suppose uo(x) > 0 with only one maximum. Because the wave velocity equals u(x, t) the top of the wave will travel faster than the b o t t o m and so the
258
Chapter 10. Perturbations in Nonlinear Initial Value Problems solution is in general multivalued which is not acceptable in physics. (see Figure 10.2) This multivaluedness m a y be removed by a d m i t t i n g solutions which m a y be discontinuous across some curve x = s(t) in the (x, t) plane, a so-called shock wave.
!
Fig. 10.2
Shock Wave at t = to
Let us now make the explicit a s s u m p t i o n t h a t the t o t a l "mass" is a conserved quantity. This m e a n s +~
d f u(x,t)dx dt
O,
or
s(t)-0
-dtd f
oo
u(x t)dx + -~d
-oo
I
~(t)-0
u(x, t)dx -
~(t)+0
oo
-~Ou(x, t)dx +
I
-oo
I -~~ (x, t)dx
8(0+0
+ { u ( ( s ( t ) - o),t) - u ( ( ~ ( t ) + o ) , t ) } ~
ds
= o.
S u b s t i t u t i o n of (10.5.35) gives ~1[u2] --[u] d8 and the velocity of the shock wave becomes
ds
l[u] 2 d-t- = 2 [u] '
(10.5.37)
where [f] denotes the j u m p across the shock f((s(t) + 0), t) - f((s(t) - 0), t). On the other h a n d u m a y be considered as a generalized function U; taking U _= 0 for t < 0 and accounting for discontinuities we have the following relations between the distributional and the "ordinary" derivatives of U and u respectively
OU Ot and
B
Ou Ot
ds
[u]~(~ - ~(t)) ~ + uo(~)~(t)
1 2 OsU _ 0~lU2 + Ox - O x
1 -2
[u~]~(~- ~(t))
5.3. The Estimate of the Remainder Term and Final Results
259
and by (10.5.37) we get for the transport equation (10.5.35) together with the initial condition (10.5.36) the equation
Ot + - ~ x
U 2) = u 0 ( x ) 5 ( t ) ,
U=Ofort
or because U _-- u except for a set of measure zero 7t-oo
~(x, t)
(~, t) + ~
(~, t) ~0~
(10.5.38)
,
t>O
--oo
valid for any smooth test fucntion ~ with compact support in the region t >_ 0. u = u(x, t) is the discontinuous solution of (10.5.35)-(10.5.36) where the shock cuts off equal areas from the area below and above the curve u = u(x, t), see Figure 10.2. Also this discontinuous solution is not quite satisfactory and therefore one introduces a small dissipation such that u becomes continuous, however, with a steep gradient across the curve x = s(t). Oleinik [115] considered the singular p e r t u r b a t i o n problem
0u. o--t- + -~x
u~
= e Ox 2 ,
- c o < x < +co,
t > O,
e > O,
with the initial condition
u~(~, 0) = ~0(~).
(10.5.39)
First we remark that this initial value problems is uniquely solvable for all E > 0 and its solution is single valued for e sufficiently small. Therefore the introduction of a generalized solution is not necessary; see [147], pp.96-97, where the Cole-Hopf transformation is treated. Oleinik showed t h a t u~(x, t) converges for c -+ 0 in distributional sense to a certain function u(x, t) in any interval [a, b] of any straightline t - to > 0 and moreover t h a t this function satisfies the relation (10.5.38). E s h a m [37] took instead of the transport equation the Burgers equation
Ou
Ou
o-~ + u ~
= Z~,
Z>0,
and he studied the effect of a p e r t u r b a t i o n with a small term ~utt. In particular he investigated the initial boundary value problem
e2 02u Ou 02u Ou -~+--~-fl-~x2+U-~-~x=f(x,t), u(0, t ) = u ( 1 , t ) = 0 ,
u(x,O) = g(x),
-1 <x
0
0
~
Ou
- h(x),
O < x < l.
The limit with ~ --+ 0 gives a transition from a hyperbolic to a parabolic differential equation. Finally we mention the work by Lax and Levermore [93]; they added to the
260
Chapter 10. Perturbations in Nonlinear Initial Value Problems
equation (10.5.35) a small d i s p e r s i o n t e r m and they considered the singular p e r t u r b a t i o n problem
072
072
0372
0--~ + u -~x - e O x 3 '
- co < x < + co ,
t > O,
with the initial condition ~(~, 0)=
~0(~),
-~
< 9 < +~.
This differential equation is the well-known Korteweg-de Vries equation; it describes long-waves in relatively shallow canals.
Exercises 1. Give a first order approximation of the solution of the system du dt
= v,
dv e--rr = -t-u n - v ,
n=2,3,...,
t > O,
with ~(0) = ~.
~(0) = ~ > 0,
Specify the t-interval where the approximation is valid. 2. Investigate the initial value problem
d2u ~-gi~ +
(du) 2n+1 -gi
-e ~=0,
with
n=0,1,2,...,
du ~(o) = ~,
7i(o)
=
t>0,
~.
3. Determine a first order approximation of the solutions, including their first order derivatives, satisfying the initial value problems i. e { 02~ - c2(x , t) ~02~ } + - ~ + so~ inu--O, -co<x<+oo, t>O -~ < 9 < +~. u ( x , 0) -- lr sin x, ~o~ (~, 0 ) = g ( ~ ) ii. e{ -O-i o2'` V - c2 ( x , t ) ~ } + -o-~~- sin u = O, - o o < x < +co, t > O, Ou 0) = g ( x ) , -cx~ < x < +co. where g ( x ) is an u ( x , O ) = sinx, -bT(X, arbitrary C~176 Sketch the solution w ( x , t) of the reduced initial value problem for the values t = 0 , t = t l > 0 and t = t 2 > > t l (seeRef. [83]).
Chapter 11 SINGULAR PERTURBATIONS IN NONLINEAR BOUNDARY VALUE PROBLEMS OF SECOND ORDER 1.
Introduction
In this chapter we turn our attention to singular perturbations in nonlinear boundary value problems of second order. As in the preceding chapters 7 and 8 we first construct a formal solution consisting of a combination of an approximation of the solution of the unperturbed problem with ~ = 0 and a correction due to the loss of a boundary condition. As to the proof of the justification of this procedure we investigate again the error in the formal approximation and show that this error becomes small when the perturbation parameter goes to zero. We treat nonlinear ordinary differential equations and also partial differential equations of elliptic type, the latter less extensive than the former because of complex and tedious calculations. In section 2 we discuss the boundary value problem
d2u du ~-d--~ + a(~ ' ~) ~ + b(~ ' u) - o,
o<x
u(O)--a,
u(1)--/3,
(11.1.1)
with a(x,w)=fiO for 0 < _ x _ < l and all values of wEIR. We get a boundary layer at x = 0 whenever a ( x , w ) > a o > O and one a t x - 1 whenever a(x, w) < a0 < 0. The justification of the construction is established by using again the fixed point theorem of section 10.2; it will appear that the verification of the condition 10.2.2 requires some extra effort. In the next section 3 we drop the condition a(x, w) -71=0 and this may give rise to the occurrence of interior layers (shocks). The theory is illustrated with the well-known Lagerstrom-Cole example
d2u du ~-ff-~x2 + U--~x
u = O,
u(0)=a<0,
u(1)=13>0
for the cases / 3 > a § and f l < a + l . Another interesting example is the boundary value problem
d2u
du
e~x2 + U ~ xx = 0 ,
u(-1)=a,
u(+l) -/3
262
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
for several values of a and ft. This example shows the "capriciousness" of singular perturbations, a phrase from W. Wasow [144]. Section 4 is devoted to autonomous conservative equations of the type ~2 d2 u
+ f (u) = O,
u(O) - a,
u(1) = ~,
and we use phase-plane representation and some geometric intuition. In section 5 we present the theory of Haber and Levinson [59] concerning the more general boundary value problem
d~ (d~) ~-~2 = / u , ~ ,
u(0) = a and u(1) = ~,
where .f(u, v) is submitted to the condition
0/( dg ) Ov g ( ~ ) ' ~ ( ~ )
>0,
0___x<~and ~
h(x),-~x(X)
<0,
5 : < x _ ~ 1,
where g and h are solutions of the reduced equation ( r with g ( 0 ) = a , h ( 1 ) and g(&) = h(&). The theory is rather technical and therefore we do not give the proof of the asymptotic approximation of the solution, but the result will appear to be very plausible by the experience already obtained in the preceding sections. Finally, we treat in the last two sections the elliptic boundary value problem
02u 02u~
-c ~
Ou + -~y2 ] + a(x, y)-~y + b(x, y; u) -- 0,
(11.1.2)
where (x,y) belongs to a bounded domain ff and u(x,y) satisfies a boundary condition along the boundary Off. The analysis is facilitated by the assumption ~Ob(x, y, u) < 0 for all (x,y) e f and all u E ]~; the reason is that this requirement is sufficient for the application of a generalization of the maximum principle or the fixed point theorem necessary for the approximation of the remainder term. The general problem with the equation
( 02u 02u)
c \ cOx2 + ~
Ou
Ou
+ a(x, y, u ) ~ W b(x, y, u)-~y + c(x, y, u) -- O,
(~, y) e ~
(11.1.3)
involves many complications as may be inferred already from the theory of section 2 where the boundary value problem (11.1.1) has been discussed. Moreover the behaviour of the solution in the neighbourhood of the extremal points of Off where the subcharacteristics are tangent to 0fl is another source of very unattractive calculations, see Chapter 8, section 3; the interested reader is referred to [53, 62, 63, 78].
263
2.1. The Formal Approximation 2. 2.1.
B o u n d a r y V a l u e P r o b l e m s for Q u a s i l i n e a r O r d i n a r y D i f f e r e n t i a l E q u a tions The Formal Approximation
As mentioned in the introduction we start with singularly perturbed boundary value problems for ordinary quasilinear differential equations. So we discuss the following boundary value problem
d2u du 6-~x 2 Jr- a(x, u) ~x -q- b(x, u) = O,
0 < x < 1,
(11.2.1)
with the boundary conditions u(O) = c~,
u(1) =/3.
(11.2.2)
The parameter e is small positive and further we make the following assumptions i. a ( x , u ) _ a0 > 0 for all values of x E [0, 1] and for all values of u E IR. This condition may be replaced by a(x, u) < a0 < 0 and the following development may be carried out in a similar way. This requirement is introduced to prevent a singularity in the reduced equation and the occurrence of interior layers (see section 7.9,.6) ii. the solution of the reduced boundary value problem
a(x, w) -~x dw + b(x' w) = O'
O < x < l,
w(1) -- ~,
(11.2.3)
is uniquely determined and differentiable in the segment [0, 1]. iii. To avoid tedious bookkeeping we assume for convenience that the coefficients a(x, u) and b(x, u) are infinitely often differentiable with respect to x and u. It follows from the assumption above that the solution w(x) of (11.2.3) is also a C ~ function in [0, 1]. The choice of the boundary value w(1) - /3 is in accordance with the condition a(x, u) > a0 > 0, see (7.2.12)-(7.2.14). The formal approximation is constructed again with the aid of a composite expansion N
N+I
Z
Z
k=0
k=0
c
+
(11.2.4)
To obtain an estimate of RN(X;---c) which is uniformly small in [0, 1] it will appear that it is sufficient to take N = 1. The function To(X) is defined by the unique solution of the reduced problem (11.2.3) and the function Wl(X) satisfies the equation
e --d-~x2 +
d x 2 + . . . + a (x , w o + e W l + . . . ) + b(z, w0 + 6Wl + ' " )
( -~xdwo+ e --d--xx + " " ) = 0,
264
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
or in a first approximation
dwo dwl {a(x, Wo + ewl) - a(x, w0)}--~x + ea(x, wo) dx d2wo .-~{b(x, w 0 --~ ~ W l )
-
b(x, Wo) } - - c
dx 2
or with the aid of the mean value theorem
dwl{Oa dwoOb } a(x, wo)--~ x + ~u(X, w o ) ~ + ~u(X, wo) wl =
d2wo dx 2 ,
(11.2.5)
valid for 0 < x < 1, while the boundary condition is w1(1) = 0 . We remark that" this boundary value problem for Wl and generally all the boundary value problems for the terms wk of the so-called outer expansion are linear, so that no complications in the construction of the outer expansion arise. To meet the boundary condition w(O) - a we should introduce the boundary layer terms ~kvk(~c) with ~ - - ~ . Putting
u(~) ~ ~ k ~ k ( ~ ) + ~k~k(~)"= W + Y and assuming that the outer expansion W is known we get for the boundary layer part V the differential equation
d2W l d2V ( dW l dV ) ~--s + - - ~ + a(x, W + Y) ~ + -~7 ( + b(x, W + Y) = O, or
1 2V1
{
dW + b(x, W + V) - ~d~ 2 -~- a(x, W + v) dV + a(x, W + V)-~x
} = -~
,
or
dzV dV dW d~ 2 -+-a(x, W + V) ~ + 6{a(x, W + V) - a(x, W ) } - ~ x + r ( d2W dW "1 =- r + a(x, W) ~ + b(x, W) ~ := d(x,r In case we take N = 1 we have d(x, r - O(r equation as follows ~r d2vk _~_ + a{e~, ~r (wk(r
W + V) - b(x, W)} (11.2.6)
uniformly in [0, 1]. We write this
vk(~)}~r kdvk d~
+ ~[a{~, r ~ ( ~ (~) + ,~(~))} - a{~, r ~ ~ ( ~ ) } ] r ~ - dwk ~ (~r + ~[b{~, ~ ( ~ ( ~ ) + ~(~))} - b{~r ~ ~ ( ~ ) } ]
= d(~, ~).
265
2.1. The Formal Approximation
Using the regularity of the functions a(z, u) and b(x, u) and expanding this expression into powers of e we obtain after straightforward calculations the following differential equations for the boundary layer terms v0, vl, v2"
d2vo
dvo
d~C-Z--- 4. a (0, wo(O)+ vo(~))-~__ = O,
d2Vld~ + ----~
dVl 4a(O, wo(O) + vo(~r --~
"(11.2.7)
-~uOa(0,
w0(0) + v0(~)) -d-~Vl dvo (~r =
dvo
el(~, V o ) T ( ~ ) 4. fl(vo) -- fl(O)"-- gl(~, Vo) a;
(11.2.8)
and d2v2 d~---7 4. a(0, w0(0) + v0(~)) ~
dvl
~- e2(~, VO, V l ) - ~ ( ~ ) ,
dvo
Oa (0, wo(O) + vo(~C))-d-~ v2(~) + Ouu dvo
+ e3(~, VO, V l ) ' - ~
4. f2(~, VO, Vl) -- f2(~, O, O)
J
(11.2.9)
:= g2(~, vo, ~l)
where el, e2, e3, fl and f2 are smooth differentiable functions. The equations (11.2.7)(11.2.9) are valid for 0 < ~ < c~ and we have as boundary conditions Vo(O ) -- OL -- Wo(O),
Vl(O) -- --Wl(O),
v2(O ) -- 0
(11.2.10)
and lim Vo(~)- lim v~(~)- lim v2(~)= O.
(11.2.11)
The equation (11.2.7) is a nonlinear equation while the other two are linear. The solution of (11.2.7) with the conditions (11.2.10)-(11.2.11) is readily obtained with the aid of the inverse of the function v 0 - v0(~); putting ~ = ~ ( v 0 ) w i t h ~ ( v 0 ( 0 ) ) - ~ ( a - w 0 ( 0 ) ) 0 and ~(0) - c~ we get for ~ the equation
---+
a(O, ~ o ( 0 ) + ~o) = O,
where the prime denotes differentiation with respect to v0. Hence already anticipating limcx~ ~d~ = 0, we obtain ~--+ vo
1
~'(~o)
_- - / a(O, wo(O) + r/)dr/:=
Q(vo),
0
and therefore
Vo
~ - v(~o) - +
/
a-wo(O)
de Q(~)"
(11.2.12)
266
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
Because a(0, w 0 ( 0 ) + y) > a0 > 0 we have ~'(v0) negative or positive according to v0 being respectively positive or negative and it follows that v0(~) decreases or increases whenever v0(~c) is respectively positive or negative. Moreover
~0
d~ 2 ~ - a
(0, w0(0)) dvo
for v0 small and so we have negative exponential behaviour of v0(~) for ~r large which proves the boundary layer character of the function v0(~r Using the smoothness of the right-hand side of (11.2.8) and the boundary layer character of vo it is not difficult to solve consecutively the boundary value problems (11.2.8), (11.2.9), (11.2.10), (11.2.11). The equation (11.2.8) is equivalent with the equation
d2Vl~-d~----I- ~d { a (0, wo(O) +
vo(~)) 9Vl (~) } = gl(~, Vo)
(11.2.13)
or
dvl d----( +
a(0, w0(0)
+
v0(~))
9V l - -
/
gl (r/, v0(r/))dr/,
(11.2.14)
oo
where we have already taken care of the condition lim Vl (~) = 0. Due to the exponential behaviour of v0 and the form of gl(r/, v0(r/)) as given in (11.2.8) the integral in the right-hand side of (11.2.14) converges. A solution of the homogeneous equation corresponding with (11.2.14) is according to (11.2.7) given by d,o d~ and so we get with the boundary condition Vl(0) = -Wl (0) and by variation of constants after a straightforward calculation the result
v~(() -
-
[
W 1 (0) ~,
d~ (0)
dvo
+ \ d~ (7"1) { f gl 07', vo(rl/))drl/}drl --~-(~r
(11.2.15)
o
From the boundary layer behaviour of v0 we get immediately from (11.2.15) that also vl(~) has the right boundary layer behaviour. The boundary value problem (11.2.9) with (11.2.10)-(11.2.11) may be solved in a similar way and the result reads
v~(~) =-
-~(~) o
) { / g~(~', v0(~'), v~(~'))a~'}e~
-~-(~1.
(11.2.16)
~?
Summarizing all these results we have constructed the following attempt for an asymptotic approximation of the solution u(x) of our boundary value problem (11.2.1)(11.2.2)
G
:= ~(~1 + R(~, ~1.
(11.2.17)
267
2.2. The Estimate of the Remainder Term and Final Results
From (11.2.6) and the boundary conditions for w~ and v~ we get that ~(x) satisfies the boundary value problem d2u
e--~x2 + a(x, ft)
dfi
+ b(x ~) = O(e 2)
uniformly in [0, 1], with ~(0) = c~, ~(1) = ~ + o ( e N ) , (N arbitrarily large and O(e N) means asymptotically zero). Therefore the function g(x) is a formal approximation of the order 2 for the solution u(x) of the boundary value problem (11.2.1)-(11.2.2). Moreover, because
r with
d2ttd2R 2+r
(dftdR) 2 + a(x, fi + R ) ~x + ~
g ( 0 ) + R(0, e) - c~ and
f i ( 1 ) + R(1, e) = ~,
+ b(x, g + R ) - O, we have for the remainder term
R(x, e) the boundary value problem d2R . e~+a(x,
dR d~ ft + R ) ~ + {a(x, ft + R) - a(x, ft) }--~x + {b(x, ft + R) - b(x, ~) } -
O(e2), uniformly in [0, 1], with R(0) - 0 and R(1) - o(eN). To estimate this remainder term it is useful to have homogeneous boundary conditions. Therefore we change R(x) by an amount o(eN), which has of course no influence on the estimate of R(x). We introduce
R(x) - R ( x ) - R(1). r
and ~(x) = g ( x ) + R(1)r
where r C C~[0, 1] with e.g. r - 1 for 3/4 _< x _< 1 and r 1 0 _< x _< 5" Due to the regularity of the coefficients a and b we get finally d2R
d/~
- 0 for
d~
c--~x2 + a(x, ~t + R) -~x + {a(x, ~ + [~) - a(x, u)} d---x + {b(x, ~ + it) - b(x, ~)} = O(e2), uniformaly in [0, 1],
(11.2.18)
with the boundary conditions R(0) = R(1) = 0. It is now our task to derive from this boundary value problem an estimate of the /~(x). This will be carried out in the next section with the aid of the fixed point treated in the preceding chapter. Then it will also become clear why we had a formal approximation up to O(~ 2) and why a formal approximation up to not sufficient to produce an appropriate estimate of the remainder term. 2.2.
T h e E s t i m a t e of t h e R e m a i n d e r
(11.2.19) function theorem to make O(e) is
Term and Final Results
The boundary value problem for the remainder term /~(x) is given by the equations (11.2.18)-(11.2.19).
268
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
In order to apply the fixed point theorem we introduce the nonlinear map F ~ ] from the normed space N into the Banach space B. The space N is defined as the linear space N := { P I P e C2[0, 1], p(O) = p(1) = O} (11.2.20) with norm
I pl
d2p ~(~)1 + e 2 ~O,al:~I~1,
- max Ip(x)l + ~ max I dp to,~ tO, l~
(11.2.21)
while the Banach space B is given as B := { q l q e C[0, 1]}
(11.2.22)
Ilqll = max Iq(x)l. [0,1]
(11.2.23)
with norm
The map F[p] is defined by
d2p dp d~ F[p] - e--~x2 + a(x, fL + P)-~x + [{a(x, fL + p) -- a(x, u)}~xx + {b(~, ~ + p) - b(~, ~)}],
(11.2.24)
where fi is the given formal approximation of the solution of the boundary value problem (11.2.1)-(11.2.2). F ( 0 ) = 0 and the linearization of F(p) in p = 0 reads
d2p dp { Oa dfz Ob } Lip] - e --~-~x2 + a ( x , u) -~x + ~ ( x , ~ ) -~x + ~ ( x , ~ ) p
(11.2.25)
and hence
dp {a(x ~ p) a(x, ~) Oa (x, fL) } d~ q2[p] = F[p] - LiP] - { a(x, fi + p) - a(x, fi) } ~x + ' + - P--~u dx + {b(~, ~ + p) - b(~, ~1 - p ~Ob (x, fi)}
(11.2.26)
To apply the fixed point theorem of Chapter 10 we should verify the conditions (10.2.2) and (10.2.3).
The condition (10.2.2) This condition concerns the bijectivity of the operator L and the continuity of L - I , i.e. ] L-l[q] ] < l-lllqll,
Vq E B,
(11.2.27)
where I is some number independent of q. Assuming bijectivity one might at first sight try to prove this relation with the aid of the maximum principle and the barrier function technique as has been used in Chapter
269
2.2. The Estimate of the Remainder Term and Final Results
7, subsections 7.2.3 and 7.2.5. However we meet here a rather unpleasant situation since for the application of the maximum principle we should require
Oa
dfi
--a~(~' ~) " ~
Ob
+ ~u (~' ~) < ~ < o
(11.2.28)
Because ~da = O ( 1 ) in the neighbourhood of x = 0 and because we did not fix the sign of oa (x, ~) in this neighbourhood we are not sure that we can fulfil this condition. The trick to introduce i5 = pe -~ does not help either since in that case t5 satisfies the equation d2i~
L[~] =e-~x2 + {a(x, ~t) +
2sO} di6
--~x+
[ Oa d~t Ob ] ~02 + ~a(x, ~t) + { -~u (X, {t) -~x + Ou (x, Ct)} ~ = e -ox 9q(x). Due to the regularity of a(x,~), b(x,~), a(x, fi) > a0 > 0 and the boundary layer behaviour of ~ there exist positive constants A,B, C and K, independent of e, such that
0 < a(x,~z) <_A,
Oa uu'[--ff-(x'u)l< B,
Ob
d~t
K
'au-[-z-(x'~)[ < C and [~x[ < - - .
Therefore z9 must satisfy evq2q-vqA+
A-C= C
(
V~0q-~--~
---~-c A - - - - 4 - C < 6 < 0 E
'
but this is not possible without restrictions on A, B and K. The remedy to overcome the difficulty in (11.2.28) has been given by A. van Harten [62]. He proved the following lemmas. LEMMA 1
Let M be the differential operator d2 d M := c~(x) d---~x2+ ~(X)~xx + 7(x), with ~, # ~ d ~ ~o~ti~uou~ ~nd ~(~) > ~0 > 0, W e [0, 1]. If t h ~ X E C2[0, 1] and positive constants io, 51, 52 such that
O < 51< X(x) < 52,
M[X] < -io < O, V x E [ 0 , 1 ]
~i~t~ ~ f,~ctio~
(11.2.29)
then the solution of the boundary value problem M[p]-q,
p(O)-p(1)=O,
where q E C[0, 1] satisfies the estimate max [0,1]
Ip(x)l
< --
52eO1 max [0,1]
Iq(x)l.
(11.2.30)
270
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
Proof The function r = /l:/[r] :=
x-lp
satisfies the equation M[rx] = q or
d2r { dx }dr a ( x ) x ( x ) ~ + 2a(x) (x) + ~(x)x(x) + M[x]r(x) = q(x).
Since M(X) is negative we may apply the barrier function technique for estimating r. We take as barrier functions the constant ~o 1 max Iq(x)l and we get immediately [0,1]
J~r[~01
max [0,1]
Iql] <
-max
.
Iql < I/1~/[r(x)] I < + max Iq[ < -~7/[- l o 1 max [q[]" .
.
.
Hence we obtain by the fundamental lemma of subsection 7.2.2
[r(x)l
< l o 1 max [q(x)[, [o, ll
-
Vx e [0, 1],
or
]p(x)l < ~olX(x) max [q(x)l < ~o~52 max [0,1] [0,11
Iq(x)l,
Vx E [0, 1].
We remark t h a t this estimate guarantees the uniqueness of the solution of M[p] - q; (see also e.g. [89]). LEMMA 2
Let L be the differential operator as given in (11.2.25) then there exists a function X(X) satisfying the conditions of LEMMA 1, i.e. )(.(X) E C2[0, 1] and there exist
positive constants 51 and 52 independent of r such that L[X]=-I+O(r
O<x<
1
(11.2.31)
with 0 < 51 < X < 52-
(11.2.32)
Proof The function X is constructed as the first order approximation of the solution of the b o u n d a r y value problem
L[X] = where the constants k0 and X has the familiar form
- 1 with x(O) = ko and X(1) = k l , kl
are chosen such t h a t (11.2.32) holds. The function
with (I)0 the outer solution and ~0, ~1 the boundary layer terms. For details of the proof the reader is referred to [62].
271
2.2. The Estimate of the Remainder Term and Final Results
COROLLARY Combining both lemmas we obtain from Lip] = q, p(O) = p(1) = 0 the a priori estimate max [o, 1]
Ip(x)l <- go~52 max Iq(x)l, [0,1]
(11.2.33)
1 with 62 and go independent of c; take for instance go = ~-
To proof finally (11.2.27) we need also a priori estimates of dd~(x) and d-~(X);dx2 these estimates can easily be made by using the differential equation L~v] = q and the interpolation theorem that has been used in section 7.2.5, Re ma r k 6. We write L ~ v ] = q as
d2p dp L[p] = e~-~x2 + A(X)~xx + #(x)p = q, with
Oa d~z Ob A(x) = a(x, ~) and #(x) = ~u(X, u) dxx + Ou (x, ~). From Lip] = q we have
max d2P[0,1] I~x21-< ~-1{ /~071~Iq' + ~071~[AI" ~0,al~ I~xl dp
+
I~o,31~'#] I~o,31~IP] }
9
(11.2.34)
From the interpolation theorem follows the existence of positive constants c and d such that
m~x dp (x ) l < c r~o?~] I--~x ~P2 (~) I+ d c - l r~o?l~ Ip(x)l, [0,1] I-d--xx
(11.2.35)
where c may be any positive number and d depends on]y on the interval of x-values, see section 7.2.5, Remark 6. Taking c - ~r 1 max IA(x)l} -1 we get with d independent of e. [0,1] e-1 max [0,1]
I,~(~)1max dp d2p [0,1] I~(~)l < ~1 ~07~ I~(~)1 + 2d~-~{ ~07~ 1~(~)1}2 ~o,31~ Ip(~)l
and substitution into (11.2.34) gives
[0,1] ~X21 -~ 2e- ~O,alfflq(x)[+ 2e Using now (11.2.33), max
[0,1]
2de { ~O,al~[A(x)l + ~O,al~:[#(x)[ ~o,al~[p(x)[.
I~(~)1 - o(1),
max
[0,1]
I,~(~)1 - o(1) we obtain
finally
the
results max I d2p
[0,1] ~X2(x) I ~ C2c-2~o?1~ Iq(x)l,
(11.2.36)
and with the aid of (11.2.35)
m~x I~x(X) dp I _~ C1~-1 [o, 1]
~o71~ Iq(x)l,
(11.2.37)
272
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
where C1 and C2 are constants independent of e. Hence (11.2.33), (11.2.36) and (11.2.37) yields the required estimate
[L-l[q]
[ = ]p[
= m a x [ p ( x ) l + e m a x I dp Eo, j
<
Eo, j
d2p
r §
e-lllqll, Vq e B
(11.2.27)
with 1-1 independent of e. We proceed now to the second condition (10.2.3) of the fixed point theorem. T h e c o n d i t i o n (10.2.3) To check this condition we investigate
I]~(P2)- ~(pl)[[ ~ max [0,1]
dp2 { a ( x , ~t + p2) -- a ( x , ~t + pl)}--~x [
-[- max {a(x, fi + P l ) - a ( x , [0,1]
fi)}
-~- max {a(x, ~t + P 2 ) - a ( x , [0,1]
~t + P l ) -
Oa
dx
dx
[
Oa d~t ( P 2 - Pl)~u(X, U-~- Pl)}~XX I
Oa
d~t
-~- max (P2 --Pl){ ~UU(X, '~) -- ~U (X, U "[" Pl) } dxx [ [0,1] Ob
max {b(x, ft - I - p 2 ) - b ( x , ft - [ - P l ) - (P2-PI)}o--~( x, 'u +Pl)[ [o, 1] -t-
Ob (x, ~) - Ob
max
[o,~]
We consider two elements pl and p2 in the ball ~N(P) with radius p _ 1 and we get on account of the regularity of the coefficients a(x, u) and b(x, u), the formal approximation fi (11.2.17) and the definition of the norm (11.2.21) the result
IIIII(p2)- ~I/(pl)II--~ c c - l P
I P 2 - Pl I ,
'v'p e aN(p), p ~ 1,
(11.2.38)
where C is a constant independent of e, and it follows that condition (10.2.3) holds. Now we are finally in the position to apply the fixed point theorem to the boundary value problem (11.2.18)-(11.2.19) for the remainder R. The function re(p), see (10.2.3), is the monotonic function C e - l p and the number P0 (10.2.4) becomes p0=sup{p[0_P_<
1, c e - l p <
~.} = 2C"
Because the right-hand side of (11.2.18) is O(e2), uniformly in [0, 1] we can satisfy the condition I l f l - - O(~2) < ~ p o -- 4 C '
3. Transition Layers
273
for e sufficiently small; so we get by (10.2.6) the result that /~ exists with
IRI
(11.2.39)
~ 2g-~ Ifll = o(~2) 9
Moreover the solution /~ is also unique since /~ lies clearly in the domain L - l [ f t B ( g P o ) ] - L-1 [gts( ge]J st2~] (see proof of fixed point theorem) From this we obtain also the result that the original boundary value problem (11.2.1)(11.2.2) has a unique solution for e sufficiently small. Summarizing we have the following theorem.
THEOREM 1 Be given the boundary value problem d2u
du
~a-fi~ + ~(~' u) ~ + b(~, ~) = o,
0< x < 1
(11.2.40)
with the boundary conditions
u(O) -- a,
u(1) -- ft.
(11.2.41)
The coefficients a and b are of class C~([0, 1], ]R) with a(x, u) >__ao > O, Vx E [0, 1], Vu C 1R and with ao some positive constant. Under the assumption that the solution of the reduced boundary value problem
a(x, w) --dTx dw +
b(x, w) - O, - 0 < x < l,
w(1)=/~
exists uniquely in [0, 1] we have the result that also the solution of (11.2.40)-(11.2.41) exists uniquely for e sutticiently small. Moreover the following approximations hold, uniformly in [0, 1], Ue(X) -- W O ( X ) +
and
du~
CWl(X)-~- VO(x) + gVl (X) "~-0(C2 )
d~o ~
d~ (~) = ~
+
~)+0(~) ~~
"
(11.2.42)
(11.2.43)
The outer solution terms Wo and 6W 1 a r e given by (11.2.3) and (11.2.5) and the boundary layer terms vo and CV1 b y (11.2.12) and (11.2.15).
3.
Transition Layers We consider again the boundary value problem r
d2u du 2 + a(x, u) -~x + b(x, u) = 0,
with the boundary conditions
0 < x < 1,
(11.3.1)
274
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
u(0) - a,
u(1) =/3.
(11.3.2)
In contrast with the preceding section we drop the conditions (i) and (ii) of subsection 11.2.1. This means t h a t a(x, u) may become zero for some value of u and x with 0 < x < 1 and there is the risk t h a t the reduced equation
a(x, w) dw + b(x, w) = O,
(11.3.3)
becomes singular; take e.g. a(x, u) = u and aft < O. In this case we are in a situation similar as t h a t of a turning point, discussed in section 7.3, with the difference t h a t this point is not a priori given by some value of x, but its location depends in general on the differential equation and the b o u n d a r y values. Instead of the conditions i) and ii) we suppose now t h a t there exist functions UL (x) and uR(x) which satisfy the requirements duL a ( x , UL ) ~ nt- b(x, UL ) = O,
O < x < &,
duR a(x, u R ) - ~ + b(x, uR) = O,
_< x < 1,
uL(O) =c~ and
(11.3.4)
uR(1) -- fl,
(11.3.5)
while a(x, UL) < O,
0 < X < ~ and a(x, uR) > 0,
~ < x < 1.
(11.3.6)
Assuming t h a t our b o u n d a r y value problem has a solution we may infer from T heorem 2 of section 10.3 t h a t UL(X) and uR(x) are good approximations of u(x) up to O(e) in the intervals 0 < x < ~ and ~ < x < _ 1 respectively. du (x) in x - 0 and x - 1 are We remark t h a t according to this t h e o r e m the values of ~-~x of no relevance in an O(E)-approximation of u(x). In the supposition t h a t UL(X) and uR(x) are known we have now to construct an interior layer connecting u ,.~ UL(X) and u ~ uR(x); it will appear t h a t this layer has the appearance of a shock wave whenever UL(5:) =/=uR(~). This transition is analyzed by stretching the x-variable around the still unknown value of ~ by x-~ -- ~ ,
(11.3.7)
and we represent the transition layer by a function v(~) that should satisfy in approximation the relation U(X) ,~ UL(X ) nt- V(~) -- UL(X) + V
E
with lim v(~) = 0 and lim v(~) = uR(~ + 0) - UL(X, -~- 0). ~-~-~ ~+oo
(11.3.9)
The latter limits implies t h a t the solution UL(X) of (11.3.3) can be continued into an interval 0 < x < ~ + (f, where 6 is arbitrarily small positive but independent of e.
275
3. Transition Layers Substitution of (11.3.8)into (11.3.1) gives
r
+ -d2v ~ + ca(g: + ~-~, UL "+"V)~ dui 1 dv ) \ dx +-e-~ +r - d=~ ~ a (~ + ~ , ~ ( ~ + ~ ) + ~(~))-~dv + 6{a(& + r
UL(X + c~) + v({)) -- a(5; + r
UL(X + r
duL (~c+ e{)
+ ~{b(~ + ~, u~(~ + ~) + ~(~)) - ~(~ + ~, u~(~ + ~)} = 0. Assuming sufficient regularity of the functions a and b we get in first approximation the following boundary value problem for v(~)"
d2v dv d{-----5 + a(~, UL(5:) + v)--~ = O,
(11.3.10)
with lim v({) = 0 and lim v({) = ~-+-c~ ~+m Integration of (11.3.10) with
UR(~C)-- UL(g:).
(11.3.11)
lim dv {-+-m 7~(~) = 0 yields
dv f a(~, UL('X) + v(~')) ~-~(~')d~' dv d~c + = O, --(20
or
uL(~)+v d---~+ ,,L (i:)
a(~:, rl)drl = O, with ( ~lim v(() - UR(~:) -- UL(~:). +~
(11.3.12)
uR(~:) > UL(~:) the function v(() is monotonically increasing from zero to uR(~) - UL(~) when
In case
w
f
a(~, ~)d~ < O,
for all values of w between UL('Yc) and uR(&); quite similarly in case uR(~c) < ui(~,) the function v(~c) is monotonically decreasing from zero to uR(x) UL(~C) when -
a(bc, rl)dr] > O, ,,L(~) for all values of w between
un(~c) and UR(~C).
276
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
So it follows that we obtain a monotone transition whenever w
-
f
(11.3.13)
< 0,
UL(~,) for all values of w between UL(YS) and uR(&). Finally, because ~d" approaches zero for ~ -+ +c~ we obtain from (11.3.12) the following equation for the location of the transition layer
/
(11.3.14)
a(&, rl)drl = 0.
We illustrate this discussion with a few simple but interesting examples. Examples 1. A well-known example has been provided by Lagerstrom and Cole; it is given by the boundary value problem
d2u du e-d~x2 + U-~x
u = O,
O < x < 1,
u(O) = c~, u(1) =/3.
(11.3.15)
The reduced initial value problems are
duL
UL--dx
UL -- O,
duR
un---dx- - - u R = 0 ,
UL(O ) = 5,
O < X < ~ and
UR(1)--~,
&<x
from which it follows
u i ( x ) = x + 5, O _ x < 2 ,
anduR(x)=x+~--l,
& < x <_ 1.
(11.3.16)
The condition (11.3.6) puts a restriction to the choice of the values of a and fl; it is necessary to take a < 0 and ~ > 0. (11.3.17) There is no problem to continue U L ( x ) in the interval 0 < x < & + 6, 6 > 0 and 6 independent of E; therefore the equation for the transition layer is determined by (11.3.12), which becomes in our example ~+ad-v
d---~+
or
dv
d~
~drl = 0,
1 2 { (~ + ~ + ~)~ - (~ + ~) ~}'
3. Transition Layers
277
with lim v ( ~ ) = ( ~ + / 3 - 1 ) - ( ~ : + a ) = / 3 - a - 1 . The transition point x = :~ is determined by the condition (11.3.14), which reads &+fl-1
/
1 r/dr/= ~{ (2 4- fl - 1) 2 - (2 + a) 2 } = O,
and therefore ~+~-
1- +(&+a).
In case of the + sign we get a = f l - 1 and the solution of the boundary value problem (11.3.15) is simply u = x 4 - a , 0 < x < 1 and there is no shock. The other possibility is 1 ~- ~(1-a-~), (11.3.18) and this puts another restriction on the boundary values of a 0<~<1.
and
/3, because
tt
-!
4,'1
~
,"
t
--'t
X
Fig. 11.1 fl > ce 4- 1 Interior Shock
Shock layer region in a -
fl plane
It follows that a and fl must satisfy the inequalities - 1 < a + fl < +1.
(11.3.19)
278
Chapter 11. Perturbations in Nonlinear Boundary Value Problems Moreover UL(Z) and u R ( z ) should be negative respectively positive in 0<x<& and & < x < 1; this gives ~ + c ~ < 0 and & + 1 3 - 1 > 0 and with the aid of (11.3.18) we get besides the conditions (11.3.17) and (11.3.19) the condition /3 > a + 1.
(11.3.20)
It follows t h a t the shock layer gives a positive j u m p from v = 0 to v = r SO
~+(~-t-v
I rldrl= i, +o,
and
(l+ot--,6)-I-v 89 O
f rldrl
which results in 7dv> 0 . Summarizing we come to the conclusion t h a t the conditions (11.3.17), (11.3.19) and (11.3.20) are necessary and sufficient to obtain a shock solution. The situation is illustrated in Figure 11.1. A n a t u r a l question is to ask what happens when the shock condition (11.3.20) does not hold, i.e. < c~ + 1. (11.3.21) The solutions UL(X) = ~ + x and ult(:r) = 3 - 1 + x are sketched in Figure 11.2.
/ /
.....
->
a I
~--71
l
I
/ 4 /
Fig. 11.2
Interior weak shocks, fl < a + 1
In this case there does not exist a point $ with
a(x, uL) = UL(X) = a + x < O, O < x < ~c and
~(~, ~R)
= uR(~) = ~-
1 + 9 > 0, ~ < 9 < 1;
therefore the discussion of the case with ~ > c~ + 1 breaks down. However the reduced equation has also the trivial solution u ( ~ ) = ~ 0 ( ~ ) -= 0,
3. Transition Layers
279
and so it is quite natural to expect that in the case approximation for u ( x ) is valid
/3 < c~ + 1 the following
u ( x ) = x + ol for 0 < x _< - a u ( x ) -- O f o r - c ~ _ < x _ < 1 - / 3 u ( x ) -- x + ~ -
l for l - /3 _< x _< l.
(11.3.22)
This approximation is called an angular approximation. The example with all possible values for c~ and /3 has been treated extensively in [24], where also the proof of the correctness of the approximation is given by application of the maximum principle; this principle may be generalized for quasilinear differential equations, see [119]. Another approach to analyze quasilinear boundary value problems is by using some geometric intuition applicable to autonomous equations and therefore also to our example. The independent variable will be denoted by t, which is more common in a discussion such as follows. The boundary value problem is of the form u(O)=c~
e-d-~ + d F ( u ) + g ( u ) = 0
u(1)=/3>O
(11.3.23)
and we introduce the so-called Lidnard transformation du
z = e--~ + F ( u ) .
(11.3.24)
The differential equation becomes the "fast-slow" system du
e-d-[ = z - F ( u ) dz d--[ = - g ( u ) .
(11.3.25)
In case of the Lagerstrom-Cole example we have du
~--J-i = zdz d-t - u,
1
-~u 2, (11.3.26)
and it is easy to sketch the trajectories of this dynamical system in the (u, z)-plane; see Figure 11.3. d~t = O ( ~ ) As long as z - S u1 2 is significantly different from zero -&and dz u = e z _ ~ 2 -O(e). Therefore the time used to move along finite stretches of d-~
the horizontal parts of the trajectories is to be neglected in a first approximation.
280
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
D \\\
t
9 t/
\ A~8
""
>
ts
Fig. 11.3
Trajectories in the (u, z)-plane
Further it is clear t h a t the trajectories with neighbourhood of the curve
z > 0 are most of the time in the
1U2 , Z = F(U) = -~ and along this curve we have
dZ ~-V-dt
udU dr'
which is nothing else as the reduced equation with ~ - 0. The first case /~ > c~ 4- 1 corresponds with the curve A B C D ; the part A B with UL(t), the part C D with uR(t) and the part B C with the shock layer. The second case ~ < c~ 4- 1 is represented by the situation t h a t the trajectory passes t h r o u g h an e-neighbourhood of the singular point (u, z) - (0, 0). Since dz -- u -- O(~) the solution u - u(t) will stay some time in the d~dt ---- O(1) and ~neighbourhood of u - - 0 , see Figure 11.3. The Li~nard transformation is very useful to obtain quickly a good image of the behaviour of the solution. For m a n y other interesting examples the reader is referred to O'Malley [112], Lutz-Goze e.a. [101, 133, 134, 17, 4, 136]. The latter authors use n o n s t a n d a r d analysis and they have obtained m a n y interesting results, as well old as new. A n o t h e r example is taken from an interesting and stimulating paper "The capriciousness of singular perturbations" by W. Wasow [144]. The b o u n d a r y value problem reads
d2u
~-~ +
du
u~
= 0,
-1 < ~ < +1,
u(-1)
= ~,
~(1) = ~.
(11.3.27)
3. Transition Layers
281
In case a < 0 and ~ > 0 the reduced equation has solutions
UL(X) = O~, --l <_x < l,
UR(X) ----fl,
--l < x <_ +l
all point of the interval ( - 1 , + 1 ) have the property UL(X) < 0 and UR(X) > 0 and therefore we cannot apply the theory of this subsection. However, and
by Theorem 2 of section 10.3 both solutions might be considered as an approximation of u(x). To decide which of the solutions UL(X) or UR(X) approximates u(x) depends on the possibility to construct an appropriate boundary layer in the neighbourhood of x = +1 in case we take u(x) ~ c~ < 0 or in the neighbourhood of x = - I in case we take u ( x ) . ~ > O . So let us assume that U L ( X ) - - a < O approximates u(x) for - - l _ < x < 1 and further that the boundary value u(1) = ~ is, for the time being, arbitrary; it may be positive, zero or negative. Stretching the x-coordinate in the neighbourhood of x = 1 by lmx C
gives for the boundary layer term v(~c) the boundary value problem d 2 (c~ + v) d~ 2
-
d(c~ + v) d~
( c ~ + v ) ~
= 0,
0 <~<
c~,
with v(0)=~-c~and
lim v(~c ) = 0 .
Integration from ~ = +oo with -~(c~) -- 0 to some value of ~c gives
__dv_- _lv2(~ ) + av(~) d~
and integrating once again with v(0) = / 3 -
v(~)
= v
( 1 - x) e
=
~+,~2ae'~ ~-~
(11.3.28)
2
e~e
=
a yields the result 2aexp [aL~] ~+~ - e x p [c~/-~] ~-~
.
(
11.3.29
)
To obtain in the case ~ > a a monotonically increasing boundary layer function v(~c) and in the case ~ < a a monotonically decreasing one, it follows from the right-hand side of (11.3.28) that one should require ~ - a < - 2 a . Hence we get for a+/3<0, a<0 (11.3.30) the approximation
u(x) ~ c~ + 2c~ ~+~ ~--c~
exp [c~L~] "
(11.3.31)
282
Chapter 11. Perturbations in Nonlinear Boundary Value Problems In the same way it can be shown t h a t for a+/3>O,
fl>O
(11.3.32)
the solution of the b o u n d a r y value problem (11.3.27) is approximated by exp[- fll~] u(x) ,~/3 + 213 (~-~a+Z_ exp [ -/3---F]l+x "
(11.3.33)
The case a + fl = 0, a < 0 gives the approximation
u(x) ,~ a,
O < x <_ 1,
-l_<x
and there is a transition layer around x - 0. An easy calculation gives 2c~
u(x)~a-
t , ' l + e x p [ ~j)ra~"
-1<x<+1.
(11.3.34)
As to the choice of the b o u n d a r y values a and /3 the only case left is the case a > 0 and ~ < 0. By T h e o r e m 2 of section 10.3 we cannot use the reduced equation with u ( - 1) = a or u ( + l ) = fl as starting point for the approximation of u(x). The only escape is to use the trivial solution u ~- 0 of the reduced equation and we have u(x) ~ O for - l < x < + l , and in general two b o u n d a r y layers in the neighbourhoods of x = - 1
and x = 1.
We leave it to the reader to show t h a t the approximation is given by a u(~) ~
~_ i + . + 1 2
~
+
1
-- p l--x" 2
(11.3.35)
It is interesting to note t h a t one meets here an example with b o u n d a r y layers which are not of exponential type. The dependence of the solution of our b o u n d a r y value problem on the values of a and is indicated in Figure 11.4. It is of interest to investigate also the transition from one region to a neighbouring region in the a,/3 plane. Therefore one should make a and /3 dependent on ~, e.g. u ( - 1 ) = a - ~ and u ( + l ) = -c~ + r a < 0. For this the reader is informed in a paper by A. Boh6, using n o n s t a n d a r d analysis [15]; the b o u n d a r y layer shifts as an initial layer from the right to the left.
4. Autonomous Conservative Equations
283
l
!
! . . . . . . . . . . . . . . .
t
~ o t
Fig. 11.4 c z - f~ dependence of the solution
4.
A u t o n o m o u s Conservative Equations
All equations in the preceding sections contained a nonlinear damping term and therefore we discuss in this section also conservative systems. In particular we consider boundary value problems of the type
~2d2u
+f(u)=O,
O
(11.4.1)
with u(O) = c~ and u(1) = j3. Integrating this equation along a trajectory gives
~2(du) 2 + v ( ~ )
= H,
(11.4.2)
284
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
where H
V(u)
is the conserved energy and
the potential, defined by
dV du = f(u).
(11.4.3)
Introducing phase plane variables (u , -~-) d~ we get from (11.4.1)
du ~-dt -- v,
dv s--~ -- -f(u),
(11.4.4)
v : + v f 2 ( H - V(u)) 89
(11.4.5)
with To solve the b o u n d a r y value problem we should choose the energy H such t h a t the trajectory in the (u, v)-plane passes through two points with u = a and u = ~ in a time interval precisely equal to one; this requires the relation i 1 =
,6
/ / dd tt d=u 0
,8 --du
=
~
e
/du
(11.4.6)
~(u) .
~
From this we see at once t h a t any solution should have the property t h a t v(u) becomes O(~) in the neighbourhood of one or more points of the trajectory; this means t h a t V(u) must approach the constant H in these neighbourhoods. This gives already an indication for the choice of H; taking a m a x i m u m of Y(u) we have .f(u) = 0 and it follows from (11.4.4) t h a t the t r a j e c t o r y passes an O(~) neighbourhood of a rest point and so (11.4.6) can be satisfied. As an example let us study the p e n d u l u m d2 u
e-d-~- + s i n ( r u ) = O '
O
1,
u(O)=a,
u(1)=~,
(11.4.7)
with - 1 < a < ~ < + 1 . It follows t h a t
V(u)
1
= --
cosru,
(11.4.8)
71"
with m a x V(u) = 71 for u = 2 n + l , The system (11.4.4) becomes
n integer.
du
dv
e-~- = v, with
e - ~ = - s i n ( ru),
(
1
v = +V/2 H + - cos r u
)2
(11.4.9)
1
.
(11.4.10)
71"
1 H = 1 and The phase portrait is sketched in Figure 11.5 for three values of H, H > 3, H < 1. 71"
4. Autonomous Conservative Equations
285
V
I \
Fig. 11.5 Phase portrait of the pendulum The points S1 and $2 are saddle points and O is a centre. The p e n d u l u m makes 1 full swings when H > _7r1 and it is confined in its movement when H < 3" A "construction" of possible solutions of our boundary value problem runs as follows. Consider two values of the energy, viz. HI = ~1 _ O(r p > 2 and H2 = 0 < 6 < _2 with 6 independent of ~. These two values give according to (11.4.10) 71" two trajectories T1 and T2 which connect points A, A' and B, B' where u = a and u - fl respectively. There is an infinite number of possible trajectories, e.g. ABB', ABB'A'AB,... and A'AB, A'ABB',.... Let us take from these possibilities an arbitrary travel scheme, then the time used to reach B or B' from A or A' along the trajectory Ti approaches infinity for c --+ 0 and t h a t along the trajectory T2 approaches zero for c -+ 0; this follows from an estimate of the quantity
T=r
j Vdu f v ~ ( H + ~ c o1 s --~=r o~
u) 89
o~
Because the travelling time is a continuous function of H there exists certainly a value of H, between HI and /-/2 such t h a t this time is precisely equal to one, ( H = ~1 0-( 8 2 ) ) . Hence any travel scheme ABB', ABB'A'AB,... or A'AB, A'ABB',... is possible and there exist an infinite number of solutions of our boundary value problem. Every travel scheme contains the points P a n d / o r Q, each counted with a certain multiplicity, where v = 0. To pass the neighbourhoods of these points takes time intervals of order O(1). The parts of the trajectory in the neighbourhoods of the rest points S1 and $2 give in the (u, t) diagram horizontal line segments with u = - 1 and u = +1 respectively. These line segments are connected with each other by steep transition layers which correspond with the trajectory parts PABQ and QB'A'P, see Figure 11.6.
286
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
"tt §
l
=
Fig. 11.6 Solution A ~A B
Solution A B B I A ~ A B
O'Malley gives in [112] a whole gallery of several interesting phase portraits and the corresponding (t, u) diagrams. An interesting one is given by the b o u n d a r y value problem ~2 d2u (11.4.11) -d-~ = 1 - u 2 , u(O) = c~, u(1) = / 3 , with - l < c ~ < f l < 2 . The H a mi l t o n equations are
du ~--~ -~ V~
d~)
~-~- = 1 - u 2,
(11.4.12)
and conservation of energy gives
(
v = =t=x/2 H -
1
~1 U3+ u
.
(11.4.13)
The phase portrait is sketched in Figure 11.7. The separatrix is given by (11.4.13) with H - 5" 2 One can show in a way similar as in the preceding example t h a t there exists for H = 52 _ O(e2) an infinite number of "travel schemes" which connect the points A or A ~ with B or B ' in a time interval precisely equal to 1.
4. Autonomous Conservative Equations
287
vI
Fig. 11.7 Trajectories of (11.4.11)
tt 2.
2
--1
Fig. 11.8 Spike solutions
ABBtAIAB
A'ABBIA'ABBtAIAB
The time used to travel along any part of the trajectory outside the neighbourhood of P is always O(~) while the time to cross this neighbourhood is of order O(1). The path
288
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
A B B I A p corresponds with a sharp spike in the (u, t) diagram, where u jumps during an ~-time interval from a to +2 and from +2 to c~; this is also in accordance with 1 (11.4.11) and (11.4.12) which gives a curvature of O(~) for u > - 1 +5, 5 independent of ~. A sketch of two possible solutions is given in Figure 11.8. For more literature on the interesting subject of this section we refer the reader to R.E. O'Malley [112, 113], and to R. Lutz and M. Goze [101], where is also treated the nonautonomous case
._C2d2u -~
+ f (t, u) = O,
u(O)=c~,
u(1)--fl.
The latter authors use nonstandard analysis and they show in their book the strength of this m e t h o d for many problems where a small or large p a r a m e t e r is involved. It is however outside the scope of this text to give an introduction to nonstandard analysis. Lutz and Goze present in [101] a rather general t r e a t m e n t of the b o u n d a r y value problem (11.4.1) where V(u) may have any number of local minima and maxima, and they give also a formula for the location of the jumps. 5.
A More General Case
In this section we discuss a generalization of the boundary value problem (11.3.1)(11.3.2). Instead of the semilinear equation (11.3.1) we consider now the equation
C~x2-- f
u, dxx '
O<x
(11.5.1)
with (11.5.2)
u(0) = c~ and u(1) = ~. We assume again t h a t the reduced equation f
w, ~
= O,
(11.5.3)
has a left solution WL(X) = g(x) for 0 < x _ ~ with g(0) = c~ and a right solution WR(X) = h(x) for ~ < x < 1 with h(1) = fl; moreover g(~) = h(2) = ~ and r ~-~(x) = #2. This means t h a t the reduced equation has a so-called = angular solution. We assume #1 < ~2 which is of course no restriction for what follows. The most fundamental paper on this b o u n d a r y value problem is by Haber and Levinson, who discussed a generalization of the b o u n d a r y value problem (11.5.1)-(11.5.2), [59]. They proved the following theorems for the b o u n d a r y value problem
e-~x2 = f
x,u,-~x,e
,
0<x
with
0<e<
u(0) = a and u(1) = ft.
289
5. A More General Case
THEOREM A Suppose f(x,u,v,e), ~
and ~
Of( a~
are of class C and
dg
~'g(~)'~(~)'~
)
>0,
Of( dh ) Ov x,h(x), ~ ( x ) , 0 <0,
0<~<~, ~_<x_< l,
(11.5.5)
and f(2, ~, v, o) > o,
~ < v < ~2.
(11.5.6)
Then for sumciently small e > 0 there exists a solution u(x) = O(x, e) of the boundary value problem (11.5.4) such that for ~ $ 0 O(x, e) -+ wi(x) = g(x), uniformly in 0 _< x _< ~: and O(x,e) -+ wR(x) = h(x), uniformly in 2 < x < 1,
while
dg
dO (x, e) --4 (x), uniformly in 0 < x < 2 - 5 dx dxx -
and
dO dh dx (x,r ~ --d-~x(x), uniformly in ~ + 5 < x < 1,
with 5 arbitrarily small positive, but independent of e. THEOREM B The solution O(x, e) is unique in the sense that given an ca > 0 there exists 51 > 0 such that for sumciently small e > 0 there is no solution fig(x,r of (11.5.4) other than O(x, e) which satisfies I~,(~,~)- g(~)l < ,~, ~ - ~ < and
Ifig(X, E) --h(x)[ < ~1,
x-~<w.
Another significant result has been given by O'Malley [113]. He determined a formal asymptotic expansion of the solution O(x, e) and demonstrated its validity. Finally we mention the paper [24] by D5rr, Parter and Shampine; they discuss the case where f is quadratic in ~du and the validity of an asymptotic approximation is shown by using a generalization of the maximum principle. We do not give here the proofs of the Theorems A and B, because they are rather technical; however we make the assertions plausible by means of a stability argument that follows from our Theorem 3 in Chapter 10, concerning nonlinear initial value problems. This theorem states that the solution u = u(x) of the initial value problem
d2u (du) u, ~
e~x2 =f
,
O<x
u(O)=a,
~xx(O)=9'
290
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
is approximated uniformly by the solution of the reduced problem
( dw)
f w, -~x --0'
w(O)--o~,
as long as w exists and f(w, v) satisfies the inequalities -~(w, -~-~) < - ~ and dw ~0 ) 0 v , A) <_ - ~ , ~/A with A between -~-(0) and 7; ~ may be any positive number. In particular one has the results
du dx
u(x) - w(x) 4- 0(~), x > 0 and
dw +O(E), dx
= --
z >_ ~ > o,
uniformly in ~ and (f independent of ~. Trying to solve the boundary value problem (11.5.1)-(11.5.2) we consider two value problems
d2u
(du)
initial
du
~ x 2 -- f
U,~xx ,
x<~,
u(~)~,
~XX(;~)----)~I
~x 2 --f
u, ~xx '
x>~,
u(~) ~ ,
~xx(X)--)~2.
i
and
Because of the two conditions (11.5.5) Theorem 3 of Chapter 10 gives the result that these initial value problems have solutions which approximate
wi(x) = g(x)
for x <__& and
wR(x) = h(x) for
x >_ ~.
One may now expect that one can choose an initial point (x, w) in the neighbourhood of (~, ~) and further )~i and ~2 as functions of ~ such that the resulting total solution passes precisely through the points (0, c~) and (1, ~). The condition (11.5.6) is also plausible because it guarantees that the solution u is concave in the neighbourhood of (~, ~) as it should be. As an illustration we take the following example from [144]. The boundary value problem reads
d2u
(du) 2
e~-~x2 - 1 -
dxx
,
u(0)=c~,
u ( 1 ) = ~ w i t h l f ~ - c ~ I < 1.
(11.5.7)
As solutions of the reduced equation we have two possibilities i) WL(X)=g(x)=a--x, w R ( x ) = h ( x ) = x + ~ - l , with 2 = ~-#+i 2 ii) w L ( x ) - - g ( x ) = c ~ + x , w R ( x ) = h ( x ) = - x + ~ + l , with ~ = ~-~+i 2
"
The function f (u ~--~) d~ is 1 - - ( d~ ) 2 and we obtain ,
Of( d g ) ~ x x= Ov g(x), ~x(X)
-2
Of( h(x), ~xx(X) dh)
(x) = 4-2 and ~v
dh
= -2~xx(X) = ~=2.
291
6.2 The Nonlinear Generalization of the Maximum Principle
It follows now from the condition (11.5.5) that the second possibility as an approach for the solution of the boundary value problem (11.5.7) should be ruled out. This is in accordance with the result
u(x)-clogcosh
1
(x-
1
(c~-fl+l))
+
1
(c~+fl-1)+O(s),
(11.5.8)
which may be verified by solving (11.5.7) with the aid of elementary quadratures. 6. 6.1.
B o u n d a r y V a l u e P r o b l e m s for Quasilinear Partial D i f f e r e n t i a l E q u a t i o n s of E l l i p t i c T y p e
Introduction Complementary to the preceding sections on singular perturbations of quasilinear
ordinary differential equations we present in this section some quasilinear singular perturbation problems for partial differential equations of elliptic type. The method to obtain useful approximations of solutions of boundary value problems is similar as that for ordinary differential equations; one constructs again a composite expansion consisting of the outer solution (c = 0) and corrective boundary layer terms accounting for lost boundary conditions. However, complications may be expected of computational and, with regard to the proof of the validity of t h e procedure, also of fundamental nature. We mention the case that the characteristics of the reduced equation may be tangent to the boundary of the domain of definition and further the circumstance that the solution may depend in a "capricious way" on the boundary values as has been shown in the examples of the sections 3 and 4. Nevertheless we discuss in this section some boundary value problems of semilinear type, viz.
( 02u
02u )
~ OX2 + ~
Ou -~- a-~y + f ( x , y; u) -- 0
(11.6.1)
where (x, y) belongs to a bounded domain ft, while u(x, y) satisfies the boundary condition u(x, y) = ~(x, y) for (x, y) 9 0ft. In the subsections 6.3 and 6.4 we take a -- 0 and a - - 1 respectively, while 0I (x, y; u) < 0 for all (x,y) 9 ft and all f ( z , y; u) is submitted to the requirement b-~ u 9 ]R. This condition facilitates the calculation of the estimate of the remainder term by using a generalization of the maximum principle for nonlinear elliptic equations. This generalization is presented in the next subsection 6.2 6.2
T h e N o n l i n e a r G e n e r a l i z a t i o n of the M a x i m u m Principle
Let F(x, y, u, p, q, r, s, t) be a continuously differentiable function of all its variables. Ou q - ~yy, Ou r - ~O2, u s = ~~ u and t - ~02u we may With u - u ( x , y ) and p = b-~, formulate the boundary value problem.
(
OU OU O2U O2U O2U)
n[u] = F x, y, u, Ox' Oy' Ox 2' OxOy' Oy 2 with u(x, y) = ~(x, y) e 07).
= f ( x , y), (x,y) 9 7 : ) ,
(11.6.2)
292
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
The differential expression in the left-hand side is called elliptic at a point (x, y) e 1) with respect to the function u = u(x, y) iff
OF ~2
OF ~ OF z +~-;~ +-bY >0, V((, r/) 9 ]R2, (~, 77) # (0, 0).
(11.6.3)
The differential expression is elliptic in 1) with respect to u = u(x, y) iff (11.6.3) holds in each point (x, y) C l). To formulate a maximum principle for the nonlinear elliptic differential equation of the type (11.6.2) we consider a function w(x, y) that satisfies the inequality
(
OW OW 02W 02W 02W)
F x , y , w , Ox' Oy' Ox 2' OzOy' Oy 2
< f(x,y)
and we get
F(x, y, u, u~, uy, u ~ , uxy, uyy) - F(x, y, w, wx, wy, wx~, w~v, wyy) > O. Introducing the difference v(x, y) = u(x, y ) - w(x, y) and using the mean value theorem we obtain for v(x, y) the linear differential inequality
no[v]-
~
o-~-~-~x2+ ~
N 0~ +
ooxoy +
~
o-~y2 +
oN + ~u oV>~
(11.6.4)
where the coefficients are evaluated at the arguments uo -- w + O ( u - w), uox wx + O ( u x - wx), etc. with O= O(x,y) and 0 < O(x,y)< 1. If we now make the crucial assumption that the functions u(x, y) and w(x, y) exist, that L0 is elliptic in l) with respect to u0 and finally that (-b-4~)0~_< 0, V(x, y) C ~9, then it follows from (11.6.4) that v(x,y) cannot have a positive maximum in an interior point of :D. From this simple statement one obtains the following lemma which is a generalization of the maximum principle for linear elliptic equations, see Chapter 8, section 2. LEMMA 1
Let the twice continuously differentiable functions ~(x, y), ~ l ( x , y) and ~2(x, y) satisfy within a domain ~) the inequalities L[~I] < L [ r
< L[@2], V(x,y) E :D,
(11.6.5)
where L is the differential operator defined in (11.6.2), elliptic in l) with respect to the functions ~ + 0 1 ( g 2 1 - ~ ) and ~ + 0 2 ( ~ 2 - ~ ) and OF < 0 for all ( x , y ) E T ~ and for all functions u(x, y) C C2(:D); 01,02 are defined as 0 above. If moreover
9 ~(~, y) < ~(~, y) < ~,(~, y), then this relation holds also for all points (x, fl) C f).
v(~, y) e or,
(11.6.6)
293
6.3 Elliptic Equations without First Derivatives
The proof is simple; it is similar as the proof of the maximum principle for ordinary differential equations, as given in Lemma 2 of Chapter 7, section 2.2. Finally, we remark that the inequality (11.6.5) may be replaced by the sharper condition L[~I] _< L[O] < L[ql2],V(x,y) e 7),
(11.6.5")
see [119], Chapter 2, section 16 and Chapter 1, section 1. If we want to apply this lemma to estimate the error in an asymptotic approximation of the solution of a singularly perturbed nonlinear boundary value problem we are hampered by the condition
OF _< 0, Ou
V(x,y) e T~,
Vu e C2(D).
(11.6.6)
This is rather serious as may already become clear from the following semilinear example
02u ~ Ou u) = O. aij(xl,x2, u, u~,, u~.2) OxiOxj + bi(Xl,X2, U)~xi + d(xl,x2, i,j=l
i=l
To be sure that OF < 0 one needs the restrictions that aij and bi do not depend on u and that Od <_ 0. The latter condition may be relieved a little bit; in case the coefficients {bi} do not depend on u one can use the trick of Remark 5 in Chapter 8, section 4 and the requirement ~-~_ Od < 0 may be replaced by the condition
Od
3~1, ~)2, 5 E IR such that b i l l -F b2~2 + Ouu <- - 5 < 0,
(11.6.7)
with t91,~2 and 5 independent of ~. The lemma holds of course also for ordinary differential equations; despite its limitations DSrr, Parter and Shampine showed its practical utility in their study of singular perturbations in nonlinear ordinary differential equations [24].
6.3 Elliptic Equations without First Derivatives We study the boundary value problem
-5-~~+ oy~}
+ f(~,y,~)
(x,y) Cgt,
= 0,
0<e<
(11.6.8)
with the boundary condition
u(x, y) = ~o(x, y),
(x, y) e O~t.
(11.6.9)
We take Ft bounded and the data infinitely often differentiable with respect to their arguments. In particular, the parameter representation (x(a), y(a)) of OFt is smooth in a. In view of the application of the maximum principle we suppose
Of(x,y,u)< Ou
5<0,
V(x y) e ~
VuE]R.
(11.6.10)
294
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
Finally, we assume t h a t (11.6.8)-(11.6.9) has a solution and the same for the reduced equation f(x,y,w) = 0 . (11.6.11) It follows from (1.6.10) and the m e a n value t h e o r e m t h a t this solution w is unique and the implicit function theorem gives w - w(x, y) with w E C ~ ( f t ) . In order to construct with the aid of w = w(x, y) an approximation ~ = fi(x, y) of the solution of (11.6.8)-(11.6.9) we have to introduce a b o u n d a r y layer all along Oft, which takes care of the b o u n d a r y condition (11.6.9). As in C h a p t e r 8, formula (8.3.5) we introduce local coordinates (p, a) in an interior neighbourhood V of Oft; p denotes the distance PS from a point P E V to the b o u n d a r y Oft and a the arc length from a given point So E Oft to the point S E Oft. Since Oft is smooth, the neighbourhood V can be defined as 0_p_
(x, y) and (p, a).
then we have in V
x = x(a) + p - ~
1+ ~(a)
y=y(a)-p
~-~
1+
' ,
(11.6.13)
where the dots denote differentiation with respect to a and where one should pay attention to the right sign of the square root. Expressing our dependent variables in (p, a) we retain the notation of our function symbols, e.g. f(x, y, u) with (x, y) e V is written as f(p, a, u). The differential equation (11.6.8) becomes in (p, a) coordinates
{ 02u
1 02u
-$Bp~ + -~ ~
(02P +
02p) Ou
( 020
-5~~ + -$~y~ N +
+ oy~]~
+y(p,~,,u)-O, (11.6.14)
where J
is the Jacobian Ox
Stretching the coordinate p by p = V~ T we obtain for the b o u n d a r y layer function in first approximation the b o u n d a r y value problem
v(% a)
02v OT~ +
--
(11.6.15)
f (0, ~, ~ ( ~ ) + . ) = 0
with v(O, a) = @(a) - t~(a) and
lim V(T, a) = 7"--+OO
O,
(11.6.16)
6.3 Elliptic Equations without First Derivatives
295
where ~ ( a ) = w(x(a), y(a)) and q3(a)= p(x(a), y(a)). With the aid of f(O,a, Cv(a)) = 0 and the mean value theorem it follows from (11.6.15)that
o~~ = Or 2
Of (o, ~, ~(~) + O~)v Ou
with 0 < 0 < 1 . It is in general not well possible to give an explicit formula for the boundary layer function v, but it is possible to establish its proper boundary layer behaviour. Because ~Ou < - 6 < 0 it follows that V(T, a) as function of T is concave for v > 0 and convex for v < 0. As long as v(0, a) = ~ ( a ) - ~ ( a ) is not zero, it is impossible that the function v(~-, a) can change its sign for T > 0, because we have lim V(T, a) = 0. It follows T---+ oO
that V(T, a) decreases monotonously for V(T, a) > 0, and increases monotonously for ~) < o v(T,a) 0 we have due to v ( % a ) > 0 and ~(~, ov
02v Ov Ov < 6v~ O~"2 Or Or It follows that
< 5v 2 o~ or from which we obtain
(0~T)2 > Sv 2 for T > O.
T
T
OV
- - OT 0 - , ~) > a ~(~, ~). Integrating once again we get
~(~,~) = o(~ - ~ ) = o ( ~ - ~ ) ,
(11.6.17)
which yields the exponential behaviour of the boundary layer term. Of course, the same reason holds for the case v(0, a) < 0. We consider now the function P
~(~, y) - ~(~, y) + r where r
~),
(11.6.18)
is a suitable C~176 off function, e.g. r
0~p<
1
~P0, and r
_= 0,
3
~p0_~P_~P0,
with p0 as defined by (11.6.12). The function ~(x, y) is a formal approximation of the solution of the boundary value problem (11.6.8)-(11.6.9) and it satisfies by construction the boundary value problem e ~ + ~
+f(x,y,~)--O(v~),
uniformly i n f , ,
(11.6.19)
296
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
with
~(~, y ) [ o ~
= 0.
Putting
u(x, Y) -- u(~, Y) + R(x, Y) we get for the remainder term R(x, y) the boundary value problem
~ \ o ~ + -g-~y~ + ~ -g-fi~ + -g-fiy~ + I (~,v, ~ + R ) - O, (~,v ) e a with
R(~, V) l o~ = o. Using (11.6.19) we obtain
02R 02R) r ~ + ~ +f(x,y,~+R)-f(x,y,g)=O(v/e)uniformlyina,
(11.6.20)
R(~, V)[o~ = 0. It is now not difficult to give by means of L e m m a 1 an a priori estimate for the error R(x, y). Barrier functions are provided by taking ~1,2 = + K v / ~ with K sufficiently large. For ~1 - + K x / ~ we have
r
+ f ( x , y , fz + g~l) - f ( x ' y,~) : -~u Of (x 'Y' ~ +O1Kx/~)Kv~ < -e~Kx/~,
and similarly for ~2 = - K x / ~
eASe + f(x, y, fi + #2) - f(x, y, ~) > +5 KV~. Hence R - - O ( v / ~ )
uniformly in ~ and we have the following theorem.
THEOREM 2
Let u(x, y) be the solution of the boundary value problem
h with u(x, y) - ~o(x, y) along Of~ and suppose that the reduced equation
f(~, v, ~ ) = 0 has a solution in f~. Under the assumptions that all data are infinitely often differentiable and that
, o f v,~) < -~ < o, o~(~
v(~,v) e ~ ,
WER
297
6.3 Elliptic Equations without First Derivatives
with 5 independent of e, then the following approximation for the solution holds u(x,y) - w(x,y) + r
a) + O(x/~), uniformly in ~.
(11.6.21)
The function w(x, y) is the unique solution of the reduced equation f (x, y, w) = 0 and -e- a) is the boundary layer term with exponential behaviour. The function v/~ v(T,a) is given by the boundary value problem (11.6.15)-(11.6.16) and r is an appropriate C~ off function.
r
Remark
As to the proof of the validity of the approximation (11.6.18) it is also possible to use the fixed point theorem of Chapter 10, section 2. We consider again the boundary value problem (11.6.20) for the remainder term R(x, y)
02R
F(R) "= ~ ~
02R)
+ ~
+ f ( x , y , ~ + R) - f ( x , y , ~ ) - g(x,y) = O(v/~),
(11.6.22)
uniformly in Ft, with R(x,y)log t = O. The linearized differential operator L reads Lip]- r
02p
~
02p~
+ ~]
Of
+ ~-~u(x, y, ~)p
and therefore
~o] = f(x, y, gz + p ) - f(x, y , ~z) - ~u Of (x, Y, u)P : ~-U-gU2 02f (x, Y, u + Op)p2. The space N is chosen as
N = { p i p ( x , y) C C2(Ft), p(x, Y) logt = 0}, with norm
I P(x,Y) [ = m a x Ip(x,y)l, and the space B as
B - {qlq(x,y) e C(h)} with norm [[q(x, y)[[ =
max_ [q(x, Y)I.
Because O--4(x, of y, 5) < - 5 < 0 we may apply the maximum principle to the linear boundary value problem Lip] -- q, Ploa - 0; choosing a suitable barrier function one easily shows that the condition (10.2.2) of the fixed point theorem is fulfilled, i.e.
]L-~[g]l ~ ~-lllgll,
Vg ~ B,
298
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
where g-1 is independent of r The Lipschitz condition (10.2.3) becomes
IIr
r
02 f
= m~x~ ~
02 f
= max~t ~
02.1'
(~, y, ~ + e~V~)V~ - ~
(~, y, ~ + 0~p~)Vl~
(x, y, ~t + 02P2)(P22 _ p2) +
< C 1 m a x {([Pl[ + [P2[)IP2 - Pll} +
f~
02 f (x ft + Olpi) } p 2 ~u2t.x ' y, (t + 02P2) - ~u 2 , Y,
{02f,
C2max {]pl[ 2- IP2 - Pll} < Cp IP2 - il91 I F~
,
where C1, C2 and C are constants independent of e and this inequality is valid for all pl, p2 in a ball ~N(P) with [ p] < 1. Applying finally the fixed point theorem to the boundary value problem (11.6.22) we obtain the result that the remainder term uniquely exists and moreover max IR(x, Y)I = O(v/~ ). From the unique existence of R(x, y) we get also the unique existence of the solution u(x, y) of our boundary value problem, provided the reduced equation has a solution. Therefore we may omit in Theorem 2 the assumption concerning the unique existence of the full boundary value problem (11.6.8)-(11.6.9); however the condition that f ( x , y, w) = 0 has a solution in ~ should be retained. We illustrate the theory with two examples. Examples 1. Consider the boundary value problem
~Au - u - g2(x,y)u 3 = O,
(x,y) e ~,
u]ogt = ~0]o~,
(11.6.23)
where g(x, y) is a positive Coo-function and ~t a bounded domain with 0~t of class Coo. By our theorem we get
u(x,y) = r
a) + O(v/~), uniformly in
where v(7,a) is the boundary layer function determined by (11.6.15)and (11.6.16) and r a suitable C~176 off function. Constructing an infinite asymptotic series for u(x, y) one obtains that u(x, y) is asymptotically equal to zero uniformly in any closed e-independent interior domain of ~. 2. Another boundary value problem is from the often cited paper by Berger and Fraenkel [6]; it reads as follows
~
+ ~ - g~(~, y ) ~ = 0,
(~, y) e ~,
~]o~ = 0,
(11.6.24)
299
6.3 Elliptic Equations without First Derivatives
with the same assumptions as in the example above. Berger and Fraenkel showed that there exists besides the trivial solution also a unique 1 positive solution u(x, y) that tends to g-(E~,y) as e $ 0 outside a narrow neighbourhood of width O(x/~) concentrated along 0Ft. Just as before a formal solution is easily constructed, viz.
s where r ~Ou
v(T,a)
y) = g(x, y-------~+ r
, a) "= ~
g(~,y)
is the boundary layer function with
a C~ off function. The condition o~ y, ~) < - 6 < 0 and we have
+ ~(p, a; c),
(11.6.25)
v ( O , a ) - - { g ( x , y ) } - l ] o ~ and
y, u) < - 6 < 0 implies
ol Ou (x,y,~) - 1 - 3{1 + g(x,y)~(p,a; ~)}2. The function ~(p,a; ~) is monotone in
~
and it increases from
- { g ( x , y ) } -1 to
zero and so we have ~(p, a; ~) _< 0. It follows that ~ (x, y, ~) is certainly not negative in a neighbourhood of the boundary 0Ft and we cannot use Theorem 2. Nevertheless the formal solution (11.6.25) can be justified and this is possible by a modification of the proof of the fixed point theorem of section 10.2. The remainder term R(x,y) = u ( x , y ) - ~t(x,y) satisfies according to (11.6.25) the boundary value problem
02R ~)2R~ FIR] = e -~x2 + -~y2 ] + ( 1 - 3g2~t2)R- 3g2(tR2 - g 2 R 3 = h(x, y) = O(x/~), (11.6.26) uniformly in Ft, with RIo a = O. The linearization of F is n[p]=e
02p
~ + ~
02p )
+(1-3g2~2)p,
and therefore
V~[p] = _3g2~tp 2 _ g2p3. The coefficient (1 - 3g2g 2) is not negative in all of ~t and so we cannot apply the maximum principle to obtain an a priori estimate of the solution of
L[p] - q E C(~t) in the space N = {PIP E C2(~t), Ploct - 0}. As remedy one uses instead of the spaces C ( ~ ) and N Sobolev space H~(Ft) respectively, see next chapter.
the spaces L2(Ft) and the
300
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
Berger and Fraenkel show by using the known expression of fi(x, y) that
cOp) 2 s
cOp 2
~2
p2 dxdy] s
:= C~llpll~, vp e H,~(r~). This means that L is a positive operator (strongly coercive) in H0~(f~) and it follows from the Lax-Milgram theorem that L is invertible and one has the estimate
Ilpll~ -~ IlqllL_____~,
(11.6.27)
(see next chapter). Therefore the constant ~ in (10.2.2) equals e. To apply the fixed point theorem we need also condition (10.2.3) and the requirement that the right-hand side of (11.6.26) satisfies the estimate
1 IlhllL2 _~ ~ p o -
O(~p0),
with p0 -+ 0 for e --~ 0. However h(x,y) -- O(vf~) and so in order to use our fixed point theorem we need an expansion of the formal solution into more than one term. Taking a sufficient number of terms such that Ilhlli2 = O(~ 2) we get according to (10.2.6)
IIRII~ _< 2~-~llhllL~ = O(~), and so
u(x, y) -+ ~(x, y)
for e $ 0 in the norm of H~(n), which means
f{ ( ~a (,., _ ,~))= + ( ~a (._ ~))~ + (~- ~1~} e~dy= o(~1.
(11.6.28)
~2
As to some definitions we refer the reader to the next chapter and for details of the calculations to the paper by Berger and Fraenkel [6].
6.4 Elliptic Equations with First Derivatives We should like to discuss in this final section the two dimensional generalization of the boundary value problem treated in section 2. To avoid nonessential complications we take the perturbing operator EL2 as eA and the domain f~ is a circular disc in R2. The quasilinear boundary value problem analogous to that of section 2 then reads
( 02~
02~'~
\ o~ + ~
O~
Ou
] + a(~, V; ~) ~ + b(~, V; u) ~ + c(~, y, ~) = O,
(11.6.29)
valid for 0 < x 2 + y2 < 1, while the boundary condition is
u(x, Y)lan = (p(x, Y)lon,
x 2 + y2 -- 1.
(11.6.30)
6.4 Elliptic Equations with First Derivatives
301
Despite these simplifications there are still many complications which make this boundary value problem very unattractive. We have already experienced in section 2 the difficulties to overcome in the proof of the validity of the formal procedure. Besides these we have now the extra complication that the solution of the reduced problem becomes singular in the points A and B where the subcharacteristics are tangent to the boundary 0ft. This complication was already met in Chapter 8 where the linear case has been treated; an acceptable solution was found by presenting an approximation of the solution valid outside arbitrarily small neighbourhoods of the points A and B. The same will be done in this section but not without making a further serious simplification. The boundary value problem we shall now discuss is given by
~
+ 0y ~ ] - N + f(~' y; ~1 = 0,
0 < x 2 + y2 < 1
(11.6.31)
with the boundary condition
u(x, y) = ~(x, y) for x 2 + y2 = 1.
(11.6.32)
The essential simplification lies in the fact that the coefficients of the first derivatives in (11.6.29) are independent of x, y and u. (11.6.31)-(11.6.32) become in polar coordinates
~02u e\~+rN
l Ou
1 02u~ +~-~002]-sin
OOu Or
cos 0 0 u r 00
~-f(r, O, u) = O
(11.6.33)
valid for 0 _< r < 1, 0 < 0 < 27r, with the boundary condition u(1, 0) = ~o(cos 0, sin 0 ) : = ~o(0), 0 _< 0 < 2~r.
(11.6.34)
We denote ~(0) by ~+(0) for 0 < 0 <_ 7r and 7r < 0 _< 27r respectively. The assumptions to be made are 1. f ( x , y, u) and ~0(0) are smooth functions of their respective arguments; moreover -~ is bounded and negative, i.e. there exist constants 5 and ~ such that
Of
< ~uu(r, 0, u) < 5 < 0 for 0 < r _ 1, 0 < 0 < 2r, - c ~ < u < +c~.
(11.6.35)
2. The reduced problem 0~t) = f ( x , y, w), 0y
0 < x 2 + y2 < 1
(11.6.36)
with w(x,y)= ~_(0) for (x2+y 2) = 1, y < 0, has a solution w = wo(x,y) which is of class C ~ in 0 _ x 2 + y2 _~ 1 with the exception of the points (x, y) = (• 0), where the subcharacteristics x - +1 are tangent to the boundary. The condition
0/
-u(-' ' ~0 ' ~
< ~ < 0
302
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
may be relieved by putting u - e+~ and by taking 0 sufficiently large. The construction of the formal approximation runs along the same lines as in Chapter 8. The outer solution is given by (11.6.37)
w(~, v) = ~0(~, v) + ~ , ( ~ , v) + with
OWo
Oy
(11.6.38)
= f ( x , y , wo), Wo(COSO,sinO) = ~o_(0), ~r < 0 < 2zr
OWl Oy
Of (x, y, w0)w 1 ~-- AW0, W 1 (COS 0, sin 0) = 0, r < 0 < 2~ Ou
(11.6.39)
The solutions wi(x, y) are singular in the points (:t=1, 0) and so we consider only points (x,y) with {(x,y) 1 0 < x 2 + y 2 < 1, - l + 5 < x < 1 - 5 } where 5 is an arbitrarily small positive number, independent of e. To satisfy the boundary condition along the upper part of the boundary with 0 < 0 < ~ we need a correction term which is represented by boundary layer functions
v(~, v) = ~o(t, o) + ~v~(t, o) + . . .
(11.6.40)
with x - r c o s 0 , y-rsin0 and t = 1-,- = e The differential equation in local coordinates (p, 0) is
02u Op2
1
Ou
1
02u)
( 1 - p ) Op t ( l - p ) 2 002
valid for 0 < 0 < 7 r Substitution of
and 0 < p <
Ou +sinOoo
cosO O u + f ( p , O , u ) ( 1 - p ) Oz9
0
(11.6.41)
1.
u(x,y) =Wo((1 - p) cos0, (1 - p) sin0) + eWl((1 - p) cos0, (1 - p) sin0) + - . . + vo(t, zg) + evl(t, 0 ) + ...
(11.6.42)
gives for Vo and Vl the boundary value problems
02vo Ot---ff-
073 0
+sinO~=O,
0<0<7r,
O
vo(O, O) = ~o+(0) - wo(cos 0, sin 0) lim vo(t, 0) = 0
(11.6.43)
and
02Vl Ova Ovo ~Ovo Ot 2 + sin 0 0--~ -- Ot + cos o - ~ - f (0, 0, wo(cos 0, sin 0) + vo) + f(O,O, wo(cosO, sinO)) : = h ( t , O ) , 0 < 0 < Ir, O < t < cx~, Vl(O,O)=Oand
lim vl(t,O) = 0 .
t--~ oo
(11.6.44)
303
6.4 Elliptic Equations with First Derivatives For 0 < c ~ < ~ ) < l r - c ~
and 6 = l - c o s c ~
we get the results
vo(t,~9) = {qa+(0)- w0(cos~9, sin0)} e x p [ - t sin~9]
(11.6.45)
and t
vl(t, 8) = J e-rsin~{H(T) + A}dT,
(11.6.46)
0
where
H(T) -
]
h(T',~)eT'sin~dT ' and A = - s i n ~
o
J
H(T)e-rsin~dT
o
The correction term v0 has exponential boundary layer behaviour and the same is true for Vl because ~ is bounded and so Ih(t, zg)l < g ( 1 + t ) e -tsin~, where g is a constant. After introducing again the C ~ - c u t off function r with r - 1 for 0 < p < 51 and r for ~2 _
~(~, y) = ~0(~, y) + r
a) + ~v~ (t, a)}.
(11.6.47)
This function satisfies by construction the relation ~'~uniformly in the region 0 < the boundary conditions
X2
0~ -z- + f ( x , Y, ~) - 0 ( ~ )
oy
(11.6.48)
@ y2 < 1 with - 1 + 5 < x < 1 - 6. Further 5 satisfies
~2(x,y)=qo(x,y)forx 2+y2-1with
-1+5_<x_<1-6.
(11.6.49)
As to the values of ~(x, y) along the line segments x -- • - 6) within the disk we only know that these are bounded by some constant independent of e. The remainder R(x, y) defined as R(x, y) - u(x, y) - ~z(x, y) (11.6.50) satisfies the boundary value problem
eAR
OR -~y + f(x, y, u ) - f(x, y, ~t) --
eAR
OR of -~yy + -~u (X, y, ~z+ O(x, y)R)R = O(e),
uniformly in the region 0 < x 2 + y 2 < 1, - 1 + 6 < x < 1 - 6 , conditions are R-0forx2+y 2-1, -1+6<x<1-5 and IRI_<M along the segment x = + ( 1 - 6 ) .
(11.6.51) while the boundary
304
Chapter 11. Perturbations in Nonlinear Boundary Value Problems
The latter assertion follows from the fact that not only ~(x, y) but also the solution u(x, y) is uniformly bounded in the disk x 2 § y2 < 1. This is easily proved with the aid of the generalized maximum principle of section 6.2 and the barrier functions ~ l ( x , y ) - =kKl(1 + y) =t=K2, with the constants K1 and K2 sufficiently large. Choosing finally the barrier function (8.4.4)
~2(x,y) - Ce + M{Xl(X) + X2(x)}, with C sufficiently large we obtain the result: R(x, y) - O(e), uniformly in the region {(x, y ) [ x 2 + y~ _< 1, - 1 + 26 < x <_ 1 - 25}. Summarizing we have obtained THEOREM
3
Let u(x, y) be the solution of the boundary value problem r
~U
O<_x 2 + y 2 < 1
oy
with the condition u(x, y) = 9~(x, y) for x 2 + y2 = 1. If f and ~ are smooth functions of their respective arguments and -~(x, y, u) is bounded with
of <: o u ( X ' y ' u ) < 6 < ~ f o r 0 < x 2 + y 2 < 1, - e e < u < + c ~ , and if the reduced boundary value problem has a smooth solution wo(x, y) then the function u(x, y) is approximated as
~(~, y)
= ~0(~, y) + r
- p 0) + o ( ~ ) s
uniformly in any region {(x,y) [ 0 _< x 2 + y 2 < 1 , - 1 + 6 < x _< 1 - 6 } , where the boundary layer function vo is given by (11.6.43) and where 6 is an arbitrarily small positive number, independent of e. Exercises 1. Give a first order approximation for a solution of the boundary value problem
d2U=u2
~-~
(du) 2 -
~
,
with
~(0) = ~ > 0,
u(1) = fl > 0,
t3e>a>flora
2. Give a first order approximation for a solution of the boundary value problem
d2u
(du) 2
Exercises
305
with u(0)-a>0,
u(1)=/3>0and0<x/~
1
< ~,
0< ~
<
1
Check the restrictions on the boundary values. 3. Give a generalization of Theorem 1 of this chapter by considering the following boundary value problem in an annular domain ft:
02U
10u 1 02u \ Ou + -," ~ + -~-d-~ ) - a(~, ~, ~) ~ + b(~, ~, ~) - 0
0 < rl < r < r2, 0 ~ ~) < 2~r, with the boundary conditions
~(~, o) - ~ ( ~ ) ,
~(~, ~) = ~ ( ~ ) .
The assumptions regarding the coefficients a, b and the boundary values T1 (0) and ~2 (~) are similar as stated in Theorem 1.
This Page Intentionally Left Blank
Chapter 12 PERTURBATIONS 1.
OF H I G H E R O R D E R
Introduction
In this final chapter we direct our attention to more general scalar perturbation problems. Up till now our perturbations were all of second order. A much more general situation is presented by the operator r
+ L1,
(12.1.1)
where L2 is a differential operator of arbitrary order m and L1 a differential operator of order k with 0 <_ k < m. To treat this kind of perturbation problems a more detailed specification is required. Whenever L2 may be of any order the same applies also to L1. As long as we deal with linear ordinary differential equations, boundary value problems are in general still tractable, but whenever we have to do with partial differential equations there is a plethora of problems. For instance L2 and L1 may be independently of each other elliptic, hyperbolic or parabolic and so there are already nine different cases of degeneration for r -=- 0 and in six cases we get a change of type of the operator. Apart from linearity or nonlinearity each problem requires its own theory, in particular if one wants to investigate the justification of a formal approximation of the solution. It is obvious that a general theory is not well possible and existing theories are far from complete. It seems sensible that the direction of research in the multitude of perturbation problems is to be determined by significant problems from mathematical physics, e.g. the equation of Navier-Stokes for the stream function, (in two dimensions). One of the most advanced texts presenting many boundary value problems with different degenerations is by J.L. Lions [100]; this author applies variational methods which are often used in modern textbooks on partial differential equations. Another important contribution is by M.I. Vi~ik and L.A. Lyusternik [138] treating boundary value problems for ordinary and partial differential equations, in particular the degeneration of an elliptic operator of higher order to one of lower order and the degeneration of an elliptic operator to one of first order. In connection with this paper we mention also the work by Besjes [9, 10, 11], who among other things improved the results by Vi~ik and Lyusternik by providing estimates in the maximum norm instead of the L2-norm. Further we refer the reader to the papers by D. Huet with L2 elliptic and L1 of order zero, [70, 71, 72] and by D. Huet [73] and W.M. Greenlee [55] with L2 and L1 both elliptic; these papers
308
Chapter 12. Perturbations of Higher Order
have been written in more or less the same spirit as in the book by Lions. Finally it is proper to mention also the well-known book by W. Wasow [139], where boundary value problems for ordinary differential equations have been treated. A striking difference with perturbations of the second order lies in the circumstance that with second order problems a simple inspection of the differential equation gives a decisive answer to the question where to locate the boundary layer. This is no longer true for perturbations of higher order and it may even happen that a boundary layer construction induces severe complications, see [142]. It is outside the scope of this textbook to give an extensive treatment of perturbations of higher order and therefore we restrict our considerations to the case of ordinary differential equations and to the case where both L1 and L2 are elliptic. In section 2 we deal extensively with boundary value problems for singularly perturbed ordinary differential equations. We investigate the location of the boundary layers and we give only a formal approximation. In section 3 we treat partial differential equations of elliptic type with an elliptic degeneration and we use the variational method to obtain an approximation of the solution of boundary value problems in arbitrary bounded domains in ] ~ . Using some theory of elliptic partial differential equations, it will appear that the construction of an approximation of the solution of the boundary value problem and the proof of its validity is very simple if we assume that the operators L1 and L2 are uniformly strongly elliptic and besides this that L1 is positive. Therefore we give first a short survey of the theory of elliptic equations stating the most important definitions and theorems; for proofs the reader is referred to well-known textbooks, e . g . A . Friedman [43] or S. Agmon [2]. One of the reasons to discuss the elliptic-elliptic degeneration is that it has a direct interesting application in the theory of thin shells. The boundary value for the displacement of a plate Ft, under lateral pressure and clamped along its edge OFt, reads r
(~u - A u = p(x, y), (x, y) C Ft with u[o a = -ff~n[Oa = O.
(12.1.2)
This boundary value problem degenerates for 6 = 0 to the membrame problem with
- - A w -= p(x, y), (x, y) e Ft and W[o ~ - O.
2.
Perturbations
2.1
Introduction
(12.1.3)
o f H i g h e r O r d e r in Ordinary Differential Equations
We consider the following scalar boundary value problem with a perturbation of higher orer Em-kLm[y] + L k [ y ] - f ( x ) , 0 < x < 1, (12.2.1) where
dm "~ Lm - dx---~ + Eai(x) j--1
dm-i dxm_----~
(12.2.2)
2.1 Introduction
309
and
dk k dk_ j Lk = bo(x)~xk + E b j ( x ) dxk_ j ,
(12.2.3)
j=l
with bo(x) :/: 0 in [0, 1] and with the coefficients aj, bj and the right-hand side f infinitely often differentiable and finally 0 _< k < m; as boundary conditions we take Bi[y](0) = ai,
i-
1,2,...r < m
(12.2.4)
and Bi[y](1)=j3i,
i=r+l,r+2,...,r+s=m,
(12.2.5)
where
d~,~ .x, dj B~ = dx~------7 + E 7~J d-xxJ' j=o -- 1
(12.2.6)
with 0_
Our main task is to investigate the influence of the perturbation r on the solution of the reduced problem with ~ -- 0. But here we meet immediately the question how to define the reduced problem. In case k -- 1 and m - 2 one has the choice out of only two boundary conditons and a simple inspection of the unperturbed operator LI tells you which boundary conditon should be taken; in case bo(x) > 0 one fulfils the boundary condition at x - 1 and in case bo(x) < 0 the condition at x - 0. The choice of the proper boundary condition in the reduced problem is determined by the first term in the expansion of the operator r + L1 after the coordinate x has been stretched as x - r or x = 1 - r and it is the stability of the resulting boundary layer solution that fixes the proper choice of the boundary condition in the reduced problem. The same considerations apply also in the case m > 2, but it will appear t h a t we get boundary layer equations at both end points of the interval [0, 1]. They read in first approximation
dm V dkV dm U dkU d~ m -[- b0(0) d---~ - = 0 and ( - 1 ) m~dr/m + (--1)kb0(1) ~ -- 0.
(12.2.7)
The number of stable solutions determine the number of boundary layer corrections at each end point and hence also the number of boundary conditions left over for the reduced boundary value problem which, however, may not always be solvable.
2.2 T h e Formal A p p r o x i m a t i o n We return to our boundary value problem (12.2.1)-(12.2.6) and we shall give besides the construction of a formal approximation of the solution also the conditions under which this approximation is possible. The outer expansion
W(x; r
Er rt:O
(12.2.8)
310
Chapter 12. Perturbations of Higher Order
satisfies the recursive system
Lk[wo](X) = f (x), Lk[w,](x) = O,
n - l, 2, . . . (m - k - 1 ) ,
Lk[w,~](x) = - L , ~ [ w n - m + k ] ( x ) ,
n > m-
k,
(12.2.9)
valid for 0 < x < 1. Because we cannot state a priori the boundary conditions for W ( x ; e) we leave these open for the moment and remark only that each wn is determined by (12.2.9) up to any linear combination of k independent solutions of the homogeneous equation Lk[w] = O. Since W ( x ; e) cannot satisfy all m boundary conditions (10.2.4)-(10.2.5)we need boundary layer corrections at x = 0 or/and at x = 1. Stretching the x coordinate as x = e~ we get for the boundary layer function V(~; e) the equation
dm V dkV ,,~ din_iV k dk_ Jv d~----~ + b o ( e ~ ) - ~ + E e J a j ( e ~ ) d~m_ j + E e J b j ( e ~ ) d~k_ j = 0 , j=l j=l valid for ~ > 0. Expanding V(~; e) as oo
(12.2.10) n=O
we get the recursive system
dmvo
dkvo
d~ m + b 0 ( 0 ) ~
= 0,
d~vn dkvn d( m + b0(0) d---~ - - r=((), where
r,(()
n > 1,
is a linear combination of preceding terms
(12.2.11) ve
and their derivatives
(l < n) with coefficients being polynomials in (. The characteristic equation of the hemogeneous equation reads Ak (Am-k + b0(O)) = O,
(12.2.12)
and we denote its roots with negative real part by /zj, j = 1, 2 , . . . q. It follows that the functions v~(~) with the right decaying boundary layer character are given by q
vo(~) = ECjoet'J~,
(12.2.13)
j=l
and
q
v,(~) = E ( c j , . , + ~p~)(~))e t'j~, j=l
(12.2.14)
2.1 Introduction
311
where the coefficients Cjo and cjn are to be obtained from boundary conditions still unknown and where Fin _(1)(~) are polynomials in ~ completely determined by rn(~). Similarly stretching the x-coordinate at the other end of the interval [0, 1] by x = 1 - er/ we get for the boundary layer function U(r/; e) the equation (_l)m
dmU
dkU
d-----~ + (-1)kb0(1 - er/)-~@ +
m
k
j=l
j--1
dm-Ju Eejbj(1 1)k-J dk-jU = 0 , E ~jaj(1 - er/)(-1)m-/&Tin--------7 + - er/) ( d@-J valid for r / > 0. Expanding U(~}; e) as oo
u(,;
~
(12.2.15)
= n=0
we get the recursive system (--1)m-k dmu0
d ku0
(__l)m_kdmun
dkun
d~Tm -~-b0(1) d - ~ --0,
dr/----~ + b0(1)
d@
= s~(r/)
n _> 1,
where sn(r/) is again a linear combination of the preceding terms derivatives with l < n and with coefficients being polynomials in 7}. The characteristic equation reads now + b0(1)) = 0,
and we denote its roots with negative real part by similarly as in (12.2.13)-(12.2.14)
(12.2.16) ul
and their
(12.2.17)
uj, j = 1, 2 , . . . p . Then we have
p
Uo(rl) = Edjoe~'Jn
(12.2.18)
j--1
and
p
un(rl) - E (djn + rlp~.2)(rl))e ~'~'1
(12.2.19)
j--1
_(2) with the stillunknown constants dj0 and din and where the polynomials pin (~I) are completely determined by s,~O?). Because bo(x) ~ 0 in [0, 1] it is clear that the number p equals the number of roots with positive real part of the equation (Am-k + b0(0)) = 0 and so p + q = m - k, whenever this equation has no purely imaginary roots; we assume that this is
312
Chapter 12. Perturbations of Higher Order
the case and the other case with p + q -- m - k - 2 is not considered here, see Refs. [142], [61]. A formal approximation oo
y ~ Eenyn(x)
(12.2.20)
n--0
consists of the outer expansion W ( x ; e), (12.2.8), and the boundary layer expansions V(~; e) and U(rl; e), (12.2.10) and (12.2.15). Hence it is spanned by a fundamental system of k solutions of the homogeneous equation Lk[w] = 0 and by the p + q = m - k exponential boundary layer functions { e t ' ~ } jq= l and { e ~ ' } P = l 9 this m e a n s t h a t we have at our disposition m degrees of freedom which corresponds with the number of boundary conditions (12.2.4)-(12.2.5). The boundary layer function V(~; e) can account for q boundary layer corrections at x = 0 and the other boundary layer function U(r/; e) for p boundary layer corrections at x = 1. This leaves for the outer expansion W(x; e) and hence for the reduced problem ( r - q) boundary conditions at x = 0 and ( s - p ) conditions at x = l . At x = 0 we have the conditions ~, : B i [ W + V](0) : Bi[W](0) + e ~-~'
dX, Q
d~Xi
,~,- 1 dj i7 (0; e ) + E 7 i J e ~ - J - ~ (0)' i = 1 2 , . . . r .
(12.2.21)
j=O
Expecting that W ( x ; E) should satisfy for e -+ 0 exactly ( r - q ) boundary conditions, not more and not less, gives a - At_q+1; further we obtain for wo(x) the conditions Bi[wo](O) - c~i, For r - q + l _ < i _ _ r
i = 1, 2 , . . . (r - q).
(12.2.22)
we have from (12.2.21)
)~i--1
dj f/
a~,d~, (" (o,, ~) = ~'-~-~+' { ~, - B,[W](O) } - ~ ~,j~'-J - ~ ( 0 ) , j=0
and therefore
dXr-q+lvo
d~Xr_q+l (0)
--
Olr_q+
1 --
Br-q+l[Wo](O),
(12.2.23)
and d Xi VO
~
(o) - o,
i -
(~ - q) + 2 , . . . ~ .
(12.2.24)
Expansion of W and V in their asymptotic series give the boundary conditons for the higher terms w,, and vn at x - 0 . In a similar way we have at x - 1 the boundary conditions ~ = B i [ W + U](1) = Bi[W](1) + ~ [(-r -x' dX'[Td@'(0; e)
+ A,-1 ~,j(-~l-J ~d J U ( 0 ; e ) ] , j=0
i = r + l, . . r + s = m.
(12.2.25)
2.1 Introduction
313
Because W ( x ; e) should satisfy for e --+ 0 exactly ( s - p) b o u n d a r y conditions, not more and rtot less, we must take ~- = At+s-p+1 - Am-p+1; further we get for w o ( x ) the conditions B i [ w o ] ( 1 ) = /3i, i - r + l , . . . , r + (s - p) -- m - p. (12.2.26) For m - p + l < i < m
dAi~f ( 0 ; drlAi
~) =
we get from (12.2.25)
(--1)AI~)'i-)'m-p+l
{ / 3 i --
Bi[W](1)} - ,x,-1 E ~iJ(--~))ki--J dJ U j=O
(0; e)
and therefore the boundary conditions d),m-p+ l Uo d77Am_p+1 (0) = ( - 1 ) Am-v+1 { f l m - p + l
- Bm-p+l[Wo](1)
(12.2.27)
}
and d Ai u 0
dr/A' ( 0 ) = 0 ,
i=m-p+2,...,m.
(12.2.28)
Expanding W and U in their asymptotic series we obtain the b o u n d a r y conditions for the higher terms w,~ and v~ at x - 1 . Summarizing we have constructed in principal an "Ansatz" for a formal approximation (x)
oo
oo
n=0
n=O
1 - x),
x where w0(x), v0(~)
].--x and u0(---/-)
E
n=0
(12.2.29)
E
satisfy the following boundary value problems:
1. Lk[Wo](x) = f(x),
0 < x < 1
with Bi[w0](0) - ai, i - 1, 2 , . . . (r - q) and B i [ w o ] ( 1 ) - ~ i , i = r + 1, r + 2 , . . .
dmvo
d~----~
(12.2.30)
(m-p).
dkvo
+ b0(0)-=-.~ - 0,
~ > 0
with
d Aiv 0 d~Ai (0) -- ~r_q+l,i{o~r_q+l- Br_q+l[Wo](O)}, i - - ( r - q ) + lim v0(~) = 0, where
(~r--q+l,i
is the Kronecker symbol.
1,...
r, (11.2.31)
314
Chapter 12. Perturbations of Higher Order
(_l)m--k dmuo dkuo dr/----~ + b0(1)-~@ = O,
r/> 0
with d A' uo d@' (0) - 5m_p+l,i ( - 1 ) am-p+' {Zm-p+l - Bm_p+l[wo](1)}, i = m- p + 1,...m,
(12.2.32)
lim u0(r/) = 0.
~/--4oo
At this point we have to make the important remark that this construction makes only sense whenever q < r and p < s, (12.2.33) and the boundary value problems (12.2.30)-(12.2.32) have a unique solution. The condition (12.2.33) is rather severe for the generality of our procedure, because the 1
number of roots (-b0(0)) k with positive or negative real part has nothing to do with the distribution of the boundary values over the endpoints of the interval [0, 1]. The condition concerning the unique solvability of the boundary value problems (11.2.31) and (11.2.32) amounts to the unique solvability of the sets of algebraic equations q
ECjo#jAi
=
6i,r_q+l{~r_q+l
i= (r-q)+
-- B r - q + l [ W o ] ( O ) }
1,...,r.
j=l
and
P E d j o @ ' = 5i,m-p+l ( - 1 ) x=-p+I {]~rn-p+l
-- Bm-v+l[w0](1)},
i=(m-p)+l,...m.
j=l
The coefficient determinants are related to the Vandermonde determinant and these equations have a unique solution if and only if the integers Ar-q+l, Ar_q+2,..., A~ are distinct modulo ( m - k) and the same for the integers A,~_p+l, Am-p+2,...A,~. Omitting the proof that our formal approximation is also a good asymptotic approximation we arrive at Wasow's theorem THEOREM
I
Assume that the data of the boundary value problem (12.2.1)-(12.2.6) satisfy the following conditions i) q<_r, p<_s and p + q = r a - k , ii) the reduced problem defined by (12.2.30) has a unique solution wo(x), iii) None of the integers A~-q+l, A~_q+2,... A~ are congruent to each other modulo ( m - k) and the same for the integers A,,~-p+l, Am-p+2,... Am. Th~n th~ boundary ~lue p~oblem (~2.2.~)-(~2.2.~) po~e~e~ ~o~ ~ ~umci,nt~y ~m~ll
unique solution y(x; e) with the property limy(x;e)=wo(x) inS<x< ~$0
--
1-5, --
(12.2.34)
315
3.2 Elliptic Partial Differential Equations
with 5 arbitrarily small positive, but independent of e. See also Wasow, Ref. [142, 139] or Vi~ik-Lyusternik [138]. The relevance of the conditions in this theorem is illustrated by the following examples, which we have taken from [139]
d4y
dy
~ x 4 + dx = O, with y(O) = y'(O) = y" (0) - 0 and y(1) = 1,
d2y e dx----5 d4y -----~ dx
dy dx = O, with y' (0) = 1 and y.(1) - O, dy dx = O, with y(O) = y'"(O) = 0 and y(1) = O, y'(1) - 1.
It is not difficult to show that the solution of each of these boundary value problems has the property that y(x) diverges when e --+ +0. The reader can easily check that in each example one of the conditions of Wasow's theorem is violated. An interesting and relatively simple example is given by the bending of a loaded elastic beam clamped at both ends.
3.
Elliptic Perturbations of Elliptic Equations
3.1
Introduction In this section we study singular perturbation problems involving equations of the
type eL2m[u] + L2k[u] = f(x).
(12.3.1)
These problems are in a certain sense a generalization of the theory of the preceding section in so far as we consider now partial differential equations for functions depending on an arbitrary number of independent variables; however on the other hand the orders of L2k and L2m are restricted because we assume that both operators are uniformly strongly elliptic in their domain of definition. We include this type of problems in our discussion of higher order perturbations because it provides a simple and elegant application of the theory of elliptic partial differential equations. Because we need some functional analysis and some concepts which may not be familiar to the general reader, we give first a concise introduction to the theory of elliptic partial differential equations however without proofs; for these we refer the reader to the excellent treatise by Friedman [43] and also to the book by S. Agmon [2].
3.2 Elliptic Partial Differential Equations 3.2.1 Sobolev Spaces Let gt be an open set in ][~ and 0~t its boundary; cm(Ft) is the set of all functions m times continuously differentiable in ~t and we denote by (~m(~t) the subset of cm(f't) consisting of all functions ~o E Cm(gt) with the finite norm:
{
1
1
9=
f~
I~lj
(12.3.2)
316 where a
Chapter 12. Perturbations of Higher Order is a multi-index a = ( a l , a 2 , . . . ,
0 , Dj = ~Oxj
an);
we denote
D s - D~'D~2...D~",
lal = ~
a j and
j=l
F u r t h e r we write x s = x~ ix s2 2 . . . x sn, and a! . a.l ! a 2. ! .. . a , , w T h e completion of the space (?m(~2) with respect to the n o r m (12.3.2) is called the Sobolev space H m ( f l ) . A oo Cauchy sequence {~/}1 in (~m(fl) satisfies for 0 _ 1hi <_ m the relation
/
IDS~j - DS~kl2dx ~ 0 for
j, k --+ co,
~2
and so, because that
L2(~)
is a complete space, there exists a function lim
j-+oo
flVS~oj - uSl2dx
= O,
u s 9 L2(~)
such
(12.3.3)
and this is valid for all a with 0 _< lal _< m. T h e function u s is called the strong derivative of order a of the function u 9 H m ( f l ) and we write u s = (DSu)8. Also the concept of weak derivative is useful. W h e n e v e r u and v are b o t h locally integrable in then v is called the weak derivative of u iff
/
uDS
dx
~2
-
(-1)J Jfv d ,
e
(12.3.4)
~2
where C ~ is the subset of C~176 consisting of functions with compact s u p p o r t in Q. We write v = (DSu)~. It can be shown t h a t if u E L 2 ( ~ ) with a strong derivative (DSu)8 then also (DSu)~ exists and they are equal. Also the converse is true: If u 9 L2(Q) has a weak derivative (DSu)~o then also (DSu)8 exists and again they are equal. According to the particular situation b o t h concepts m a y be used and we omit henceforth the suffix w or s. The space H m ( Q ) is a Hilbert space with scalar product
(u,v)m= ~
(DSu, DSv)i2= ~
Isl<_m
(12.3.5)
Isl_<ma
where the bar denotes complex conjugation. Similarly the space H ~ ( f l ) is defined as the completion of the space C ~ ( f l ) with respect to the norm (12.3.2) and also H ~ ( f l ) is a Hilbert space with scalar p r o d u c t as defined by (12.3.5). Finally we r e m a r k t h a t we have the obvious inclusions Hm(~) C H k ( ~ ) and H~(~2) C H0k(~), k _< m.
317
3.2 Elliptic Partial Differential Equations
3.2.2 Elliptic O p e r a t o r s , B i l i n e a r F o r m s a n d G h r d i n g ' s I n e q u a l i t y It will appear very useful to write our differential operators in the so-called divergence form L[u] = E (-1)IplDP(aP~ ' (12.3.6) 0
E ~PaP~ Ipl=l~l=m
>- c~
:= c~
~2 + ' " +
~)m,
(12.3.7)
for all ~C]R~ and all x C ~ . In case co(x) can be chosen independent of x then L is uniformly strongly elliptic and co is called the molulus of ellipticity. The formal adjoint L* of L is defined by L*[v] =
E (-1) I~ID~ (aP~(x)DPv) 0
(-1)IplDP(a~P(x) D~v)"
(12.3.8)
O<[p[,[a[<m
For u, v C C ~ (gt) we have
with B[v, u] =
E (DPv, aP~D~u) O<_lpl,l~l<~
(12.3.9)
Henceforth we write (v, u)0 - (v, u), the usual L2 scalar product. The expression B[v, u] is called the bilinear form associated with L; this form is defined for ap~ bounded in ~ and for v and u belonging to Hm(Ft). This form satisfies the following theorem, important in the theory of elliptic partial differential equations, particularly for the proof of the existence of solutions of boundary value problems. THEOREM A
Suppose that 1. L is strongly elliptic in a bounded domain f~ with modulus of ellipticity co, independent of x. 2. The coefficients a p~ are bounded in f~ for 0 <_ [p[, [a[ <_ m with upper bound r
for I P l - I o l - m, for all x and y in f~, and
3. laP~(x)-ap~(y)l <_ c 2 ( l ~ - y l ) c ~ ( l ~ - yl) ~
0
for
I~ - yl ~
0.
318
Chapter 12. Perturbations of Higher Order
Then there exist constants
c and ko d e p e n d e n t on co, cl, c2 and ft such that
ReB[u, u] > c]lul] m2 - ko]lu]] 2 ~
Vu 9 H g ( a )
"
(12.3.10)
This inequality is known as Gg~rdings inequality and a bilinear form satisfying (12.3.10) is called coercive. In the following we will suppose that the elliptic operators to be considered satisfy the three conditions stated above. 3.2.3
Generalized
Dirichlet
Prolems
We are interested in Dirichlet problems L[u] = f in 12 C IR'~
(12.3.11)
with OJu 0uJ = gj in 09t,
j = 0, 1 , . . . , m - 1,
(12.3.12)
where L is an elliptic operator of order 2m and 12 is a bounded domain with boundary 012 of class C m-1. f and 9 are functions continuous in ~ and Oft respectively and o denotes differentiation in the direction of the outward normal of 012. If u E C2m(12)N C m - l ( f i ) and if u satisfies (12.3.11)-(12.3.12) then u is a classical solution of the Dirichlet problem. In case 012 E C 2m+1 and 9j E C2m(012) the classical solution u is also determined by the classical solution of the h o m o g e n e o u s Dirichlet problem (12.3.13)
L[v] = f* E 12, OJv =OinOft, OuJ
j=O, 1,...,m-1,
(12.3.14)
with v = u - 9 and f* = f - L[~], while ~ is an appropriate function belonging to C2m((~) with the property ~O J ~ - 9j, J = 0, 1 , . . . , m - 1. It follows that it is not a too strong restriction to discuss only homogeneous Dirichlet problems. We shall now generalize these problems in such a way that under certain conditions the solution of the generalized problem is also a classical solution; the advantage is that the proofs for existence and regularity of the solution become more transparant and elegant. The generalization of the differential equation L[u] = f is obtained by the associated bilinear form B[~o, u] = (~o, f), V~o G C ~ ( a ) , where the left-hand side is defined for u E H'~(ft)
and derivatives up to only order OJu
m are required. The generalization of the homogeneous boundary conditions ~ Ion = 0, j = 0, 1 , . . . , m - 1, is given by the assumption u E H~(gt). This is justified by the following lemma. LEMMA
Suppose
Oft
0 <_ j <_ r n - 1 ;
is o f class
C m.
If
u E g ~ ' ( f t ) N cm-l(~'~),
also conversely, if u E c m ( f i )
oJ~ =0 and -5-~]o~
then
0~ ~ = 0 -5-~[o~
for
for O -< j -< m - 1 ' then
319
3.2 Elliptic Partial Differential Equations
u C H~(gt). These results lead to the following definition of a generalized solution of a homogeneous Dirichlet problem. DEFINITION u is a generalized solution of the Dirichlet problem boundary conditions if u C H ~ (f~) and
B[~, ~] = (V,/),
L[u] = f
with homogeneous
V~ e C3~
(12.3.15)
where B is the bifinear form associated with the differential operator L. The problem to find a function u E H~(f~) that satisfies (12.3.15) is called the generalized Dirichlet problem The relation between a classical and a generalized solution is given by the following theorem THEOREM B Let Oft be of c/ass C m and ap~ E clpl(~t). I f u is a classical solution of a homogeneous Dirichlet problem of order 2m with the property u E cm(~t) then u is also a generalized solution. Also conversely, if u is a generalized solution belonging to Hg'(a) . n d with th~ p ~ o p ~ t y ~ c C 2 ~ ( ~ ) n C ~ - ' ( ( ~ ) t h e . ~ i~ ~1~o ~ c l ~ i c ~ l solution of the Dirichlet problem with homogeneous boundary conditions. The consequence of this theorem is that the generalized solution yields also the classical solution whenever the data of the boundary value problem are sufficiently smooth. 3.2.4 E x i s t e n c e o f G e n e r a l i z e d S o l u t i o n s The existence of generalized solutions of Dirichlet problems with homogeneous boundary conditions may be proved with the aid of the Lax-Milgram theorem, a generalization of the well-known representation theorem of Riesz. The latter reads as follows
THEOREM OF RIESZ Let H be a Hilbert space and let F[v] be a linear bounded functional on H, there exists a unique element u C H such that
then
F[v] = (v, u). The Lax-Milgram theorem is a generalization in so far as the scalar product replaced by the bilinear form B[v, u].
(v, u) is
LAX-MILGRAM THEOREM Let there be defined on a Hilbert space H with norm [1" [[ a bilinear form B[v, u] with the properties. i ) There exists a constant cl independent of v and u such that
IB(v, u)l < clllvll. Ilull, v,,,,., e H. ii) There exists a constant c2 independent of v such that [B(v,v)[ ~ c~ll~ll ~-,
Vv e H.
320
Chapter 12. Perturbations of Higher Order
Then every bounded functional F[v] on H may be represented as
F[v] = B[v, ~}, with u uniquely determined by F. An immediate consequence is the following existence theorem THEOREM C If the bilinear form B[v, u] constant c such that
is bounded on
ReB[v, v] >
r
and if there exists a positive
H~(~t)
w
e Hg(f~)
(12.3.16)
then the generalized Dirichlet problem B[~a, u] = (~, f),
V~ E C~(~t),
u e H~(f~)
(12.3.17)
has for any f E L2(f~) a unique solution u with
I1~11,-,,< 1-11filL=.
(12.3.18)
C
The existence of the generalized solution follows from the Lax-Milgram theorem by taking F[v] = (v, f). The estimate (12.3.18) is obtained from (12.3.16) and the existence of a sequence { ~ } C C ~ ( f t ) with lim ~ = u in H ~ ( f t ) ; we have the inequalities n-~oo
cllull~ ___ Re B[u, u] < IB[u, ~]1 = n -lim IB[~o., ~]1 - - # or = n -lim [(So,-,,f ) [ - - (u, f) < [[UiiL=- II/IIL= _< Ilu~ll. II/IIL= ' - + O0 and so the estimate (12.3.18) is obvious. A bilinear form that satisfies the inequality (12.3.16) is called strongly coercive in constant with coercive as defined by the Gs inequality (12.3.10). Regarding applications to elliptic boundary value problems it is important to give some examples of strongly coercive bilinear forms. 1. Let 12 be a bounded domain and L = - A + k with A the Laplace operator and k some positive constant. We have B[v, u] = E (DPv' DPu) + k(v, u) Ipl=l -
~-5-&~_.dz+ k _ _ , _ _ .
Svudx,
[1
and therefore
B[u, u] ~ rnin(1, k)llul[ 2.
3.2 Elliptic Partial Differential Equations
321
It follows t h a t the b o u n d a r y value problem
-~(~)
+ k~(~) - f ( ~ ) ,
~ e a,
~1o~ - 0
has a unique generalized solution in H 1 for all f C L2(fl). This solution is also a classical solution if 0~t is of class C 1, u C C 2 ( ~ ) N C ( ~ ) and f C C(Ft) A L2(~t). 2. Let ~t be b o u n d e d and L = - A . To show t h a t we have also in this case a strongly coercive bilinear form we need the Poinca% inequality which may be applied to domains only b o u n d e d in one direction. This inequality reads n
i1~11~ < d 2 -
I 112 ; d 2 ~ - ~ '
[lO~
V~ C C~(Ft)
(12.3.19)
i--1 where d is the m a x i m a l diameter of ft. Suppose ft C {x C 1[~ []x~I < c}, then we have for ~ C C ~ ( f t ) Xn
f ~-:~ O~ (x 1, x2 , . . . . , x,~
~(~)=
~, ()d(.
~C
Using Cauchy-Schwarz we get +c
I~(x)l~ _< 2,
+c
Ib--~-~-(xl,x~.,...,-,~_l,e)l'd~=d --C
Ib--~ 1
,...,x,_l,
--C
therefore
+c
+c
_< --C
~C
integrating once again with respect to XliX2,...,xn_ 1 we obtain (12.3.19). The bilinear form B[u, u] associated with L - - A satisfies according to (12.3.19) Re
B[u,
n/
Ou]2 1 2 1 ~ d~ = I-I~ > ~1-1~ + 7~ I"~
u] = E i=0
E
1 1 _> min (~, ~-d--ff)ilull2,
Vu e
and so - A is strongly coercive. We remark t h a t coercive bilinear form.
L2
H~(f~), +A
3. Let f~ be b o u n d e d and L = A 2. For any qo E C~(f~)
i
a
n
"= J = l a n [
i=1 j = l f~
02 ~
2
~
does not yield a strongly we have the relation
~ 2
322
Chapter 12. Perturbations of Higher Order Applying the Poincar~ inequality to ~cOx l we have 2
_< d 2 L2
and hence n
0~O ]2
n < d2 ~"~ --ixi L2 -i=1
i=1
2
1
n
n
02~
=d2EE OxiOxj i=l j=l
L2
Therefore we get for any q0 9 C ~ (f~) I1 11 =
+
+
-< (d~+ x)l~~ ~1+ I1,~,~. 2
_< {(d 2 + l)d 2 + l}l~,l~ = {(d ~- + 1)d 2 + 1}B[~o, ~o] and we obtain by u = lim ~0,~ in Ho2(f~) the inequality n--~ r
Re B[u, u] = (Au, Au) > cllul[ 2,
Vu 9 Ho2(f~),
from which the strong coercivity follows. To give an example of a coercive but not a strongly coercive bilinear form we consider for U(Xl,X2) the boundary value problem Au+#u--0,
0_r=v/x2+x~
#>0,
withu(ro)=0.
This problem has a nontrivial solution
= g0(v~), if V/-fir0 is a zero of the Bessel function J0. It follows that the boundary value problem - - A u - - #u = f (xl,X2), 0 ~ r < to, u(ro) = 0,
# > 0
does not have a unique solution if v/-fir0 is a zero of J0 and therefore the strongly elliptic operator - A - # with J0 (v/-fi r0) - 0 cannot have a strongly coercive bilinear form. On the other hand it has a coercive bilinear form; this follows from Theorem A or by direct computation. As to existence of generalized solutions of coercive elliptic operators we have the following Fredholm alternative THEOREM
D
I f the differential operator L satisfies the conditions 1, 2, 3 of Theorem A and f E L2(~) then we have the next two possibilities 1. B[~o, u]-- (~o, f ) with ~o e C~(f~) has a unique solution u 9 H~(12), or
323
4.1 The Boundary Value Problem 2. there exists a finite number of linear independent solutions the homogeneous equation
U[v, ~] = o, In this case B[qo, u] - (qo, f) This solution is not unique.
vj, j - 1, 2 , . . . g
of
v~ e c g (a).
has a solution if and only if (f, vj) = O, j - 1, 2 , . . . ,g.
4 Elliptic Singular P e r t u r b a t i o n s
of Higher Order
4.1 The Boundary Value Problem We study the following Dirichlet Problem for a real valued function u that satisfies the partial differential equation L~[u(x)] = ~L2m[u(x)] + L2k[U(X)] = h(x),
x e Ft C
(12.4.1)
with the boundary conditions
~gu On t (x) = O,
g = O, l, . . . m - 1 ,
x e OFt,
(12.4.2)
where Ft is a bounded domain with boundary of class C ~176The differential operators are given by L2k--
E (-1)IplDP(aP~(x)D~) ' o
(12.4.3)
L2m :
E (-1)IplDP(bP"(x)D~) ' O<_lPl,l~l<m
(12.4.4)
and
with 1 < k < m and where the coefficients a p'~, bp" are assumed real valued and of class C~ Moreover, L2k and L2m are uniformly strongly elliptic in ~ while L2k has a strong coercive bilinear form on the Sobolev space H0k(f~). Finally h(x) is also real valued, h E C ~ ( ~ ) and o denotes differentiation in the direction of the outward normal on 0Ft. Since all functions involved are real valued we consider in H0k(Ft) and H~(Ft) only real valued functions. The strong regularity conditions with respect to the data of the problem may be weakened; they have only been introduced to keep the exposition simple and a generalization is of course possible. Instead of the boundary value problem (12.4.1)-(12.4.2) we consider the generalized Dirichlet problem eBm[~, u] + Bk[~, u] = e +
E (DP~~ bP~D~u) o
E (DPcp, aP"D"u) = (~, h), gqo 9 C~(f~), 0
(12.4.5)
324
Chapter 12. Perturbations of Higher Order
and it is our task to approximate the solution u C H~(f~) for e -+ O. This approximation involves the solution w E H0k(f~) of the reduced problem
Blc[cp, w] =
Z
(D~
a~
- (~a, h), V~a e C ~ ( a ) .
(12.4.6)
0
E x i s t e n c e a n d A Priori E s t i m a t e We start with the following lemma.
LEMMA
Whenever L2k has a strong coercive bilinear form on H~ then cL2m + L2k has also a strong coercive bilinear form on H ~ for e sufficiently small. PROOF Gs
inequality (12.3.10) gives for any v E Hy(f~)
Bm[v, v] =
E
(DPv' bP" (x)D" v)
0
>_ c~llvll~ - k~llvllo=, where cm and km are constants with coercivity of Bk follows
Bk[v, v] =
c,,, > O; [[vl[0 = [[Vl[L,(a). From the strong
( Dpv, aP"(x)D"v) > ckllvll~, O_
valid for all v E H0k(Ft). Therefore we obtain for v e H ~ ( ~ ) C H~(fl) the inequality
sB,~[v, v] + Bk[v, v] >_ ~ l l v l l = ~ - ~k~ll~llo= + ~kll~ll~ >- ~11"11~ + ( ~ - ~k~)li~llg
1 _> ~ l l v l l ~ + ~ckllvll~ for e sufficiently small. Hence
~B.,[v, v] + Bk[v, v] > ~ l l v l l ~ , 1 c~ which proves the lemma. for O < e < e o - _ ~km,
(12.4.7)
4.3 The Approximation of the Solution
325
From the inequality (12.4.7) it follows t h a t there exists a constant C = { min(cm, ~ck) 1 } -1 , which is independent of e, such t h a t
Ivll~m+ Ilvll~ ~ C{eBm[v,v] + Bk[v, v]},
(12.4.8)
valid for all v E H~(f~) and for 0 < e < e0. Henceforth we use the symbol C as a generic constant, independent of e, which may change its value without violating earlier results, where C has appeared. The l e m m a has the following consequences 1. According to T h e o r e m C the generalized Dirichlet problems (12.4.5) and (12.4.6) have unique solution in H~(f~) and H0k(f~) respectively and due to the regularity of the d a t a we have u e H ~ (a) N C a (~) and w e H0k (f~) M C a (~). 2. W i t h u =
lim ~
n--4~
in H~(f~)
and ~,~ e C~(f~)
we have from (12.4.8)
~ll~ll~ + II~ll~ _< C{~Bm[~,~] + B~[~, ~]} = Cli2~ {~B~[~, u] + B~[~, ~]} 1 2 = C n---+cx) lira ( ~ , h) = C(u, h) < wllull 2 + C 2 IihiI L2 L2 and therefore ellull ~m+ lull~ _< C l l h l l ~ ,
(12.4.9)
where C is the generic constant independent of e. This relation yields an a priori estimate for u and it appears t h a t u and its strong derivatives up to and including order k are bounded in L2(f~) uniformly with respect to e. This is not necessarily true for the higher derivatives.
4.3
T h e A p p r o x i m a t i o n of the S o l u t i o n It is natural to put
u(x) = w(x) + z(x)
(12.4.10)
where w(x) is the solution of the reduced problem (12.4.6) corresponding with e - 0. Hence z C H0k(~t)N C ~ ( F t ) and z satisfies the equation
L~[z] =
-~n~[~],
or in bilinear form eBm[~V, z] + Bk[~, z] = - e ( ~ , L2m[w]),
V~ e C ~ ( a )
(12.4.11)
It is impossible to estimate z(x) with the aid of (12.4.9) because z does not belong to H ~ (f~). This follows from the loss of b o u n d a r y conditions in w; therefore we introduce a correction t e r m v with the p r o p e r t y t h a t w + v belongs to H~(Ft). In case we succeed we put u(x) = w(x) + v(x) + 2(x), (12.4.12)
326
Chapter 12. Perturbations of Higher Order
where the remainder 5 belongs to H~(f~). An estimate of 5 in H~(f~) gives an estimate of the precision of the approximation of u(x) by w(x)+v(x) in H~(Yt) and an estimate of v in H0k(f~) together with that of ~ yields the precision of the approximation of u(x) by w(x) in H0k(f~). The function v(x) is a boundary layer function and therefore we introduce a local coordinate system (p, 01, t92,..., 0 n - l ) ; (v~l, 0 2 , . . . , 0,,-1) are the coordinates in 0f~ and p denotes the distance from an interior point in f~ to 0ft. Due to the regularity of the boundary, which is compact, there exists a positive number 5 such that the new coordinate system is well defined in the neighbourhood 0 < p < 5 of the boundary 0f~, see Figure 12.1. We introduce also the inner domain fl~o C fl defined as f~o = {x; x c f~, dist(x, 0f~) > 50 } with 0 < 5 0 < 5
and we fix the value of 50.
X~
X~
Fig. 12.1 The neighbourhood of the boundary Using our experience with the exponential behaviour of boundary layer functions we try the correction
v(x)
aj-fi,
= r
e x p [ - - ~--~],
(12.4.13)
j=0 where u is a positive number and the r - r with r
aj
constants to be determined later on. Further
1
- 1 for 0 < p < ~ 0 ,
r
3
- 0 for p > ~ 0
and r e C~ c~). Since w e C~ (all data are regular), the function w + v belongs to C ~ (~) and by the lemma of subsection 3.2.3 we obtain as a sufficient
327
4.3 The Approximation of the Solution condition for w + v c H ~ ( g t )
the following set of equations
Oe(w +v)[o=o = 0 for 0 < i < m - 1. Ope - _
(12.4.14)
The function w belongs to H0k N C ~ (l)) and by the same l e m m a we get
O~w Ope ,Io=o = 0 for 0 _< s _< k - 1. Substitution of (12.4.13) into this equation gives immediately ak-1 - 0 and therefore v(x)=
r
(12.4.15) a0 = al = a2 . . . . .
( ~j:k l aj 7.1 pj ) exp [ -- ~--~]. P
(12.4.16)
Inserting this result into (12.4.14) yields a simple recursive set of ( r n - k) algebraic equations for the remaining coefficients ak, ak+l'"am-1. These equations read l
j~kaj(~) (-1)e-J ev(e-j) = -_.
Oew ~. = k~ k + l m-1. Ope Io=o' "'"
After solving we get the following estimates for
ak =
(12.4.17)
aj
Okw Opk [p=o = O(1) Ok+lw
ak Ip=o + (k + 1)~-V -- O(e - v )
ak+l =
0---~i
a~+~ =
Ok+2w 0p~+~ I,=0 + (k + 2)a~+~ ~ ......................
(k + 1)(k 2 + 2)
~a~
= O(~-~)
, etc.
W i t h the choice (12.4.17) we have (w + v) e H~(f~)
aj - O(e-(J-k)v),
and
j - k, k + 1 , . . . , m -
1.
(12.4.18)
It will appear useful to have an estimate of the norm ]]vl]p in HP(f~). Because derivatives of r are only different from zero for 1/250 < p < 3//450, with 50 > 0 and independent of e, we have for e sufficiently small
Ilvjl2 <_c E where C is independent of e.
~
dl2
328
Chapter 12. Perturbations of Higher Order
Putting p : e~T we get
OPeOev= r
Og"
~
rn-- 1
[~(x) Z
Tj
eJVaJ -j-(e-r] = O(e(k-e)u) ;
j=k
since the volume element df~ has a factor d p - r
we obtain (12.4.19)
Because w + v E H ~ (ft) and w e H0k (it) the boundary layer term v belongs also to H0k and it follows that lim v = 0 in H0k; (12.4.20) e--~0
therefore this term is not "observable" for e small in the H0k norm, but it becomes infinitely large in the H v norm for p > k. 4.4
The Estimate of the Remainder and Final Results We have defined two remainder terms z(x) and 5(x) by
u(x) = w(x) + z(x) and u(x) = w(x) + v(x) + 5(x), where w(x) is the solution of the reduced equation and v(x) the boundary layer term given by (12.4.16)-(12.4.17). In the following we give an a priori estimate of both remainders. From Le[z] = -eL2m[w] it follows that
eBm[~, z] + Bk[~o, z] = -e(~a, L2m[w]), Because C ~ ( a )
is dense in H ~ ' ( a )
and 5 e H~'(fl)
V~ E C~(12). we get
eBm[5, z] + Bk[5, z] = - e ( 5 , Lzm[w]) or
eBm[5, 5] + Bk[5, 5] = - e(5, L2m[w]) - eBm[5, z - 5 ] - Bk[5, z -- 5] = - e(5, n2m[w]) - eBm[5, v ] - Bk[5, v]
< ~11~110IIn~[~]ll0 + ~Cl]~ll~ I].11~ + CIl~llk I1,1]~With the aid of the estimate (12.4.19) there results the inequality
where C1 is again a generic constant independent of e. Because 5 e H g ( a ) we may apply the inequality (12.4.8)
zll~_
[s,
+
[s,s],
Remarks
329
and therefore
~11~11~ + I1~11~< c3{~11~110 + ~(k-m+ 89
+ ~ 89
1 2 -< { 4e2C2 + ~11~11~ + 2el+(k_m+~)2vC ~
2 + 4e~,C2+ ~11~11~}, 1 + 111~ lira
or after some reshuffling ell~ll~ + II~ll~ < c 4 { F + el+2r,(k-m+ 89 -4- er'}.
(12.4.21)
The exponent u > 0 is still free and we make now an optimal choice such that the left-hand side becomes as small as possible; this is the case when u = 1 + 2/2(k - m + 1), L
or
(12.4.22) 1
/]---
2 ( m - k)" So we have finally obtained the a priori estimate eli5]] 2 + 1]2,]]~ -- O(e2(:-k) ).
(12.4.23)
The estimate for z follows from
~llzll~m + Ilzll~ __<~11~11~+ I1~112+ ~llvll 2 + I1~112 and with the aid of (12.4.19) we get again e]]z[] 2 + [[z[]2 = O(e=(~-') ).
(12.4.24)
Summarizing our results we have proved THEOREM
2
The boundary value problem (12.4.1)-(12.4.2) has for e sumciently small a unique solution u(x) and this solution is approximated by the solution w(x) of the reduced equation with e = 0. This approximation satisfies the relations
Ilu- wllk - o(~<&~>),
l u- wllm -- o(~
~).
Remarks 1. This result is rather weak, because the discrepancies between u and w are only in the L2 norm; s h a r p e r results are given in [73]. Pointwise approximations may be obtained by applying Sobolev's inequalities, see [43], [2]; further we refer the reader to [10]. 2. Applications to thin plate bending problems are presented by Jiang Furu in [86]. 3. A generalization to certain quasilinear elliptic equations with elliptic-elliptic degeneration is possible, see [82].
330
Chapter 12. Perturbations of Higher Order
Exercises 1. Give a formal approximation of the solution of the following boundary value problems d3u
d2u
du
e~Sx3§
0,
0<x
with the boundary conditions a. u ( 0 ) = l , u ( 1 ) = 0 , ~d~(1) = 1 , b. u ( 0 ) = 0 , adu- ~ ( 0 ) = l , u ( 1 ) = l . 2. Give a formal approximation of the solution of the boundary value problem dau e dx 3
du d x + u = O,
0 < x < 1,
du u(O) = 1, u ( 1 ) = O, ~xx(1)= 1.
3. Give an approximation of the solution of the following boundary value problem, regarding the deflection of an elastic beam e2dau dx 4
d2u d x 2 = f (x),
O < x < 1.
with the boundary conditions u(O) = d u
du
~x(0) = u(1) = ~x(1) = 0.
Use as well the method of section 2 as that of section 4. 4. Consider the boundary value problem eL2,T,[u(x)] + u ( x ) = h ( x ) ,
with
x e ~ C R,~
~gu On e (x) - O,
g = O, 1, . . . , m - 1 ,
xE0~.
The operator L 2 m , the function h ( x ) and the domain fl are as defined in section 3.4.1.
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SUBJECT d'Alembert's paradox 4 asymptotic expansion 13 generalized expansion 22 sequence 12 multiplicable sequence 18 series 13 uniformly 13 asymptoticall~r convergent 13 equal 14 zero 14 Bachmann-Landau notation 10 barrier function 146, 178 bilinear form 318 boundary layer term 153, 188 parabolic 192 boundary layer flow 3 characteristic time like 211 space like 211 Chikwendu-Kevorkian approximation 123 coercive 318 strong 3:20 contraction mapping 119 dispersion relation 48 Dirichlet problem generalized 319 homogeneous 318 divergence form 317 energy integrals 140, 215 entrainment of frequency 83
INDEX equation autonomous 283 elliptic 176 quasilinear elliptic 293, 300 hyperbolic 209 quasilinear hyperbolic 250 Duffing 37 Hill 103 Korteweg-de Vries 260 Mathieu 105 Van der Pol 79, 247 fixed point theorem 230 Floquet's theory 102 formal adjoint 317 formal approximation 139, 151 Fredholm alternative 322 frequency response curve 83 Gs inequality 318 gauge function 12 Gronwall's lemma 34 Krilov-BogoliubovMitropolski theorem (K.B.M) 64, 69, 70 Laplace transformation 199 locking in phenomenon 83 magnetic hydrodynamic flow 199 maximum principle for ordinary differential equations 145 for elliptic partial differential equations 177 for nonlinear elliptic partial differential equations 292
340
Subject Index
method of averaging 61 local averaging 67 multiple scales 92 Lighthill 5O Lagrange 75 Poincar6 2 Temple 55 modulus of ellipticity 317 node attracting 203 repelling 204 operator elliptic 317 strongly elliptic 317 uniform strongly elliptic 317 order symbols 9, 10 oscillation linear 78 linear with damping 78 weakly forced 80 weakly nonlinear 75 perihelium precession 77 perturbation regular 28 singular 1 of boundary layer type 3 of cumulative type 1 perturbed wave equation 123 phase amplitude transformation 75 Poincar~ expansion 2 Poincar~ inequality 321 quasilinear initial value problems 232 boundary value problems 263 regularized boundary layer 194 relaxation oscillation 248 remainder term 142, 151 resonance 164 saddle turning point 205 secular term 45
skock wave 258 singular perturbation of boundary layer type 3 cumulative type 1 elliptic type 175 hyperbolic type 209 Sobolev space 315 solution classical 319 generalized 319 strained coordinate 43 strained parameter 47 theorem of Eckhaus-Sanches Palencia 74 Haber-Levinson 289 Krilov-Bogoliubov Mitropolski 64 Lax-Milgram 319 Poincar6-Bendixson 249 Riesz 319 Wasow 314 time scales 92 transition curve 106 transition layer 274 turning point 152 isolated 202 wave cnoidal 50 travelling 48 wave equation with linear damping 120 with cubic damping 121 of heat conduction 226 W.K.B. approximation 156
