APPLIED MATHEMATICS AND MECHANICS An International Series of Monographs
EDITORS FRANCOIS N. FRENKIEL
G. TEMPLE
Washington, D. C.
1. K. OswATITSCH : Gas Dynamics, English version by G. Kuertt
(1956) 2. G. BIRKHOFF and E. H. ZARANTONELLO: Jet, Wakes, and Cavities (1957 )
3. R. voN MISES: Mathematical Theory of Compressible Fluid
Flow, Revised and completed by Hilda Geiringer and G. S. S. Ludford (1958) 4. F. L. ALT: Electronic Digital Computers-Their Use in Science and Engineering (1958)
Volume 5A. WALLACE D. HAYBS and RONALD F. PROBSTEIN : Hypersonic
6. L. M. BREKHOVSKIKH : Waves in Layered Media, Translated from the Russian by D. Lieberman (1960)
7. S. FRED SINGER (ed.) : Torques and Attitude Sensing in Earth
Satellites (1964) 8. MILTON VAN DYKE : Perturbation Methods in Fluid Mechanics (1964)
9. ANGELO MIELE (ed.) : Theoryof Optimum Aerodynamic Shapes
(1965) Tolume 10. ROBERT BETCHOV and WILLIAM O. CRIMINALS, JR.: Stability
of Parallel Flows (1967) Volume 11. J. M. BURGERS: Flow Equations for Composite Gases (1969) Volume 12. JOHN L. LUMLEY: Stochastic Tools in Turbulence (1970) /olume 13. HENRI CABANNES: Theoretical Magnetofluiddynamics (1970) lolume 14. ROBERT E. O'MALLEY, JR.: Introduction to Singular Perturbations (1974)
INTRODUCTION TO SINGULAR PERTURBATIONS Robert E. O'Malley, Jr. DEPARTMENT OF MATHEMATICS
ACADEMIC PRESS New York and London 1974 A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT C 1974, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDINO, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
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O'Malley, Robert E Introduction to singular perturbations.
2.
(Applied mathematics and mechanics) Bibliography: p. 1. Boundary value problems-Numerical solutions. 3. Perturbation Mathematics. Asymptotic
expansions. QA379.04
I.
Title. 515'.35
ISBN 0-12-525950-6 AMS(MOS) 1970 Subject Classifications: 34E15, 34E05, 34E20, 34A10, 341105, 341115, 351130, 41A60, 49A10. PRINTED IN THE UNITED STATES OP AMERICA
Chapter 1. First Concerns 1.
2. 3. 4.
Examples of Singular Perturbation Problems Asymptotic Expansions Intuitive Approach of Matched Asymptotic Expansions Two-Variable Expansions
Chapter 2. The Regular Perturbation Method
Chapter 3. Linear Boundary Value Problems
1.
2. 3.
Chapter 4. Nonlinear Initial Value Problems 1.
2. 3.
The Basic Problem Two-Parameter Problems Differential-Difference Equations with Small Delay
Chapter 5. Nonlinear Boundary Value Problems
Some Second-Order Scalar Problems Second-Order Quasi-Linear Equations Quasi-Linear Systems An Extended Discussion of a Nonlinear Example
1.
2. 3. 4.
Chapter 6. The Singularly Perturbed Linear State Regulator Problem
Chapter 7.
Chapter 8. 1.
2. 3.
Boundary Value Problems with Multiple Solutions Arising in Chemical Reactor Theory
PREFACE Through this book the reader will become acquainted with certain fundamental techniques for obtaining asymptotic solutions to boundary value problems. It is intended for engineers and applied mathematicians, students of these disciplines, and others interested in the topic. It does not suppose many mathematical prerequisites. Some knowledge of ordinary differential equations, linear algebra,
and basic analysis is expected, however, as well as a belief in the inherent usefulness of the subject in important applications. Topics covered are restricted partly by the experiences of the author and partly because it is his intention not to be too technical. Notably
absent is any discussion of partial differential equations and of problems in unbounded domains. Without doubt, the underlying philosophy and methodology expressed here carries over with some modification to these and many other related areas of asymptotic analysis. Further, it should be directly stated that the presentation is admittedly distorted by the current opinions and biases of the author.
Instead of using inner and outer approximations, for example, outer expansions are corrected in regions of nonuniform convergence by the addition of "boundary layer corrections." This works most satisfactorily for the problems discussed. No claim of universal applicability is intended, nor is there any intention to denigrate the common practice of using local expansions to study complicated physical problems.
One of the most exciting features of singular perturbations is its unexpected appearance in many varied areas of application, often disguised among certain intuitive practices which are part of the folkways of the specific field. Quite naturally, then, singular perturbation techniques are among the basic tools of many applied scientists, but the common content of their experiences is not well known. vii
Among the literature of singular perturbations are the books of Van Dyke, Wasow, Kaplun, Cole, Nayfeh, and Eckhaus. Each has a different outlook, as does this volume. Nearly two hundred references have been cited, but this represents only a small portion of the relevant literature. Several hundred more items are to be found in the literature. Earlier versions of this work have been used several times as the basis of a one-semester course for applied mathematics and engineering students at New York University. Students were encouraged to work through the detailed calculations indicated and to obtain solutions to examples illustrating the results obtained here or in references cited. A second course was then offered as a seminar with students giving presentations on singular perturbation methods for partial differential equations and on applications of interest to them. Thanks are due to many individuals who have taught me about singular perturbations through their publications, lectures, conversations, or questions. Many more should be thanked for their interest in this and the encouagement they provided. The writing of this manuscript was supported by the Science Research Council at the University of Edin-
burgh, by the National Science Foundation (contract number GP-32996X2) and the Air Force Office of Scientific Research (contract number AFOSR 71-2013) at New York University, and by the Office of Naval Research (contract number N00014-67-A-0209-0022 ) at the University of Arizona. It is hoped that these notes will be useful and interesting to a wide variety of scientists.
1.
Consider a family of boundary value problems P depending on a small parameter E. Under many conditions, a "solution" yy (x) of P can
be constructed by the well-known "method of perturbation"-i.e., as a power series in E with first term yo being the solution of the problem
J. When such an expansion converges as E -* 0 uniformly in x, we have a regular perturbation problem. When ye(x) does not have a uniform limit in x as E -* 0, this regular perturbation method will fail and we have a singular perturbation problem. We shall list examples of such problems.
In these examples and below, it will be convenient to use the Landau order symbols 0 and o which are defined as follows: Given 1
1. FIRST CONCERNS
two functions f (e) and g(e), we write f = 0(g) as e - 0 if If (e)/g(e)l is bounded as a - 0. We write f = o(g) as e --> 0 if f (e)/g(e) ---> 0 as e - 0. [For further discussion, see Olver (1973) or Erdelyi (1956).] EXAMPLE 1:
Y(0) = 1.
obtained by setting e = 0, has the solution yo(x) = sinx. A power series expansion Ye (x) _ I00Yj (x)ej 0
can be easily constructed to formally solve (1.1) termwise. Equating coefficients of ej successively in the differential equation and the initial conditions, each yj for j > 1 will satisfy
Yj(0) = y(0) = 0. Any number of terms yj can obviously be constructed recursively. [For example, yi(x) = J(x cos x - sin x).] The actual solution ye(x), however, is given by
y,(x) = (1 + e)-t/2sin ((1 + e)h/'2x).
In order to compare ye with the partial sums of ye formally obtained, we expand 00
1. EXAMPLES OF SINGULAR PERTURBATION PROBLEMS
whenever e^'+1 x = o(l) as e -- 0. Thus, ordinary Taylor series expansion of the right-hand side yields N
whenever x is restricted such that eN+1x = o(l) as a - 0 (and, in particular, on bounded x intervals for a sufficiently small). (The reader
should do the expansion to check that the coefficients yj are those found previously by using the differential equation.) Applying the mean value theorem, however, shows that (1.2) is no longer valid when x is so large compared to a-1 that 1/x = o(e^'+1). Thus, we have a regular perturbation problem if we restrict x to bounded intervals. If we let x become appropriately unbounded as e - 0, however, the initial value problem can no longer be solved by the regular perturbation method since ultimately (1.2) no longer holds. The reader might have expected this breakdown in uniformity since y1 (x), for example,
does not remain bounded as x - oo. [Those familiar with twovariable expansions (cf. Section 1.4) will realize that an expansion using "times" t and et can be formally generated, but this too will fail
to remain valid for all t > 0.] Thus we observe that there is not necessarily any universal validity to formally obtained expansions. EXAMPLE 2:
0
y"-e2y=0,
y(oo) = 0,
Y(0) = 1,
has the solution ye(x) = C". Here 1
liye(x) = 8-0
e-`
to
ex = 0(1) if ex = c + o(1), c arbitrary if
if
ex - oo.
4
1. FIRST CONCERNS
For x bounded, of course, ye - 1 as a - 0, but there is no uniform limit for x > 0. Note that setting e = 0 in (1.3) yields a reduced problem
y"=0 Y(0) = 1,
Y(oo) = 0,
with no solution yo(x). Further, if we introduce a = ex, the resulting problem
Yy - Y = 0,
Y(0) = 1,
Y(oo) = 0
has the unique solution
y(o) = e-*
for all a > 0.
As might be anticipated, introduction of appropriate new coordinates is a basic technique used in obtaining solutions to singular perturbation problems.
The nonuniform convergence exhibited in these two examples occurred because the independent variable ranged over the infinite interval x > 0. Moreover, the nonuniformities "occurred at x = oo." In many singular perturbation problems x is restricted to bounded domains. In the following, this shall generally be so. Instead, the problems are singular because the order of the differential equation defining P for a positive drops when a becomes zero. Nonanalytic dependence of the solution ye on a (as in the next example) often
results in markedly different behavior, depending on how a - 0. Henceforth, then, we shall take a to be a small positive parameter or a small complex parameter defined in a narrow sector about the positive real a axis. Before proceeding, we note that e
x+e
and
axle
are examples of functions which converge nonuniformly at x = 0 as
e - 0. For x > 0, both functions tend to zero as a - 0, while at
1. EXAMPLES OF SINGULAR PERTURBATION PROBLEMS
5
x = 0 both are equal to one. (See Figs. 1 and 2.) Functions featuring analogous nonuniform convergence are typical elements of solutions to singular perturbations problems.
FIGURE 1
FIGURE 2
The function f (x) = e/(x + e) for e = 0.2, 0.1, 0.01.
The function g(x) = e-x/e for e = 0.2, 0.1, 0.01.
1. FIRST CONCERNS
6
EXAMPLE 3:
The two-point problem
ey" + y + y = 0, Y(O) = a(E),
O<x<1 Y(1) = /3(E)
has the solution /3 - ae°2
ae° - /3
Y.(x)=(eP,-eP2)e Ptx +(eP,-
e P2
PZx
)
e
where Pi (e) =
Ii (-1 + (1 - 4E) V2) _ -1 + O(E)
and P2(e)
ZE
(-1
- (I - 4e)1/2)
E + 1 + O(E)
are roots of the characteristic equation
ep2+p+ 1 = 0. Suppose that for any positive integer N N
a(e) = 2j=0 aj eJ + O(eN+l )1 and N
N(E) =
j=0
/
Rj ej + O(eN+l)
as
a -* 0.
Then, since pi -* -band P2 -* -oo, we have $0e1-x + (ao y.(x) _ - loe)exe-x/e + O(e) throughout 0 < x < 1 as e -* 0. For x > 0,
(1.5)
e-x/e = O(EN)
for every N as the positive parameter E -* 0, while eo = 1. Thus, ye(x)
converges nonuniformly on the interval 0 < x < 1 as a -* 0 and ye(x) = $0el-x + O(e)
for
x > 0.
1. EXAMPLES OF SINGULAR PERTURBATION PROBLEMS
7
Note that the "limiting solution" Y(x) = /jo e1
satisfies the "reduced problem"
y'+y=0 Al) = Qo
and that ye (x) - Yo(x)
as
a -* 0
except at x = 0, where ye(0) = a. This nonuniform convergence as e -* 0 (unless ao = ao e) implies that we have a singular perturbation problem. (See Fig.//3.) In general, then, lim (lim ye (x)) = Y (0 ) X-0
\\e-
o
lim lim ye (x)) = ao. "0
FIGURE 3
Nonuniform convergence of the solution y of ey" + y' + y
0, y(0) - 0, y(1) - 1 to the solution Y (x) - ei_x of the reduced problem.
1. FIRST CONCERNS
8
Higher-order approximations may also be obtained. Specifically, we have
y(x) =
[P(E)e-(I-x)PI(1)]
+ [(a(E) - N(E)e
Pi(e))(e(P2(,)+I/e)x)]e-x/e
for every integer N > 0.
+O(EN)
Expanding the bracketed expressions as power series in E, we obtain approximations of the form
j=0
Aj(x)Ej) +
B,(x)Ejex/e + O(eN+I)
(Y. j0
(1.7)
uniformly on the closed interval [0, 1] while N
Aj(x)Ej + O(EN+I)
Y" (x) = j
uniformly on any interval 0 < S < x < 1 as E The limiting solution away from x = 0,
1-->
0.
00
(1.8)
I Aj(x)Ej, j=0
must be a formal power series solution of the equation Ey" + y' + y
= 0. (Thus, A' + A0 = 0, Ai + AI = -At, etc.) We shall call this sum the "outer solution" or "outer expansion," while we shall call
(i0 Bj (x)Ej
)e-x/e
(1.9)
the "boundary layer correction." Likewise the region of nonuniform convergence near x = 0 will be called a "boundary layer" in reference to Prandtl's boundary layer theory [cf. Prandtl (1905) and Meyer (1971)] for flow past a body at small viscosities. [The well-read should be warned that the boundary layer correction is not the same as the inner (or boundary layer) solution used by many authors, including Kaplun (1967) and Van Dyke (1964).]
1. EXAMPLES OF SINGULAR PERTURBATION PROBLEMS
9
Why the limiting solution Y(x) = Ao(x) = /3oe'-x retains the boundary condition at x = 1 (or, equivalently, why the nonuniform convergence occurs at x = 0) needs to be explained. We note, for now, that the limiting solution would satisfy the limiting boundary condition at x = 0 if, instead, E were small and negative. EXAMPLE 4:
We now consider the nonlinear problem
-1<x<1
Ey"=YY,
Y(-1) = a,
(1.10)
Y(l) = a,
as the real boundary values a and /3 vary remaining independent of E. Before proceeding, note that any solution of this two-point problem will be unique by the maximum principle [cf. Protter and Weinberger (1967) and Dorr et al. (1973)]. Six cases will be considered separately. CASE 1: a > 0, a + /3 > 0. solution is given by 1+
Ye(x)=a
Here the reader can verify that the
1-
1- 1-
2a
)(a/eXX_1)
a+/3 a'+ 0)
(1.11) e(a/exx-')
up to asymptotically exponentially small terms, so
ye(x) -tea
as E- 0 for x< 1.
(1.12)
Since y(1) _ /3, nonuniform convergence generally occurs at x = 1. CASE 2: /3 < 0, a + /3 < 0. readily 'find that
Ye
Knowing the solution in Case 1, we
(x) = R
1-
1
- a+/3 1 e«/exx+Q
(1.13)
1. FIRST CONCERNS
10
up to asymptotically exponentially small terms and
ye(x) - /3
as
r: - 0
for
x > -1
(1.14)
with nonuniform convergence at x = -1. CASE 3:
a = -/3 > 0.
Here
Y. (x)
=
e -A 1 -a[ 1I +- e-1e.1
(1 .
5)
up to asymptotically exponentially small terms. Thus
-. ye(x)
a
-I <x<0
0
for for
/3
for 0<x<1
x=0
(1.16)
as E - 0. The nonuniformity at the interior point x = 0 might be considered roughly reminiscent of shock phenomena in fluid mechanics [cf., e.g., Cole (1951) and Murray (1973)]. CASE 4:
a < 0 < P. The function
Ye(x) = E711 I +
E(/3 - a) C(E)
\ a/3 xtan[Zr(1+E(aaa C(E)lx_E(a aa )k(E)1
(1.17)
solves the problem for unique constants C(E) = 1 + O(E) and k(E) = 1 + O(E). Thus ye(x)-->0
as E-0 for -1 <x < 1
and nonuniform convergence occurs near both x = ± 1.
(1.18)
1. EXAMPLES OF SINGULAR PERTURBATION PROBLEMS
CASE 5:
a = 0,
11
> 0. Here
Ye(x) = 2 1 1 - E aE)
r /
tan f 41 1
1)] - % E)) (x + 1)]
(1.19)
for the appropriate m(e) = 1 + O(E). Thus
y, (x) - 0
as
a --> 0
for
-1 < x < 1
(1.20)
and nonuniform convergence occurs at x = 1. CASE 6:
/3 = 0, a < 0. Similar to ase 5, we have
Y. (x) = 2 I + e a(e) / tan
\
for some n(e) = 1 + O(e) and
y, (x) - 0
1 1 + e a(e)
(x -
1)]
(1.21)
\
as E-0 for -1 < x < l
.
(1.22)
a
YO (x) =
a, 0,
x<0 x=0
i 0=-a, x>0
The dependence of the limiting solution Yo(x), -1 < x FIGURE 4 < 1, of ey" - yy', y(-1) - a, y(1) = 0 on a and $.
1. FIRST CONCERNS
12
Note that in all cases the limiting solution of (1.10), away from the nonuniformities, is constant and therefore satisfies the reduced equation yy = 0. This limiting behavior within -1 < x < 1 is summarized in Fig. 4. Similar analyses of the limiting behavior of the solution to
0<x< 1
Ey"+yy'-y=0, y(0)=a,
y(1)=R
have been done by several authors [see, especially, Dorr et al. (1973)]. The problem is more difficult than (1.10), but the asymptotic results are similar. Further examples which illustrate "the capriciousness of singular perturbations" are contained in the article by Wasow (1970), which is recommended reading. More complicated examples, treated heuristically, are discussed by Carrier (1974).
2. ASYMPTOTIC EXPANSIONS
Before proceeding, the reader should know some basic results involving asymptotic expansions. They will be briefly discussed below,
but the reader should consult Erdelyi (1956) or Olver (1973) for details.
A sequence of functions quence as E --> 0 if for every n > 0 DEFINITION:
-on+l = o(4n)
is an asymptotic se-
as r: --> 0.
An example is provided by the power sequence where 0.(E) = En. DEFINITION:
The series
I 0
(1.23)
2. ASYMPTOTIC EXPANSIONS
13
is an asymptotic approximation (to N terms) of the function f(E) as E - 0 with respect to the asymptotic sequence (¢n) if .f(E) = J An-0n(E) + o(4N(E))
as
r: - 0.
(1.24)
R
If (1.24) holds for every N > 0, we write .f (E)
(1.25)
I A. -On (e)
R-0
and call (1.25) the asymptotic expansion of f (e) as e ---> 0.
The reader should realize that convergent series expansions are asymptotic, while asymptotic series expansions are, in general, divergent. Only the finite sums of (1.24) need be defined for small enough E; no infinite series need be summed.
The functions f (E) and g(E) are asymptotically equal with respect to the asymptotic sequence (4(E)) if DEFINITION:
f (E) = g(E) + O(,Sn (E)) as
r: - 0
for all integers n > 0. In particular, we write f(E)
0
if f (E) = o(-O. (E)) as r: - 0 for all n, and we then call f (E) asymptotically negligible.
As an example, note that a 11 - 0 with respect to the power sequence (En) as r: - 0 through positive values. Note further that the coefficients of the asymptotic expansion (1.23) are uniquely determined since, for each m, (1.26) A.m = 1im .f (E) - 2n=E J ( 4m() Linear combinations and integrals of asymptotic expansions are
defined in the obvious manner. Differentiation of asymptotic expan-
1. FIRST CONCERNS
14
sions is not always possible since, for example, the function
f (e) = e '/,sin ellsatisfies f (e) ^- 0 for the asymptotic sequence {ss,,(e)) = {e') as e tends to zero through positive real values. However, 1
f'(e) = Z[e-l/esin e'1° - cos el/°] e
does not have a limit as e - 0 and does not possess an asymptotic expansion with respect to {e'). Likewise, multiplication of asymptotic expansions is generally not possible because elements of the double sequence cannot always be expanded with respect to the single asymptotic sequence (4p). We shall usually restrict attention to asymptotic power series 00
7, A,e'.
(1.27)
r=0
Then multiplication is always possible and termwise differentiation of
the expansion of any function f (e) is permitted provided f (e) is holomorphic in any complex sector
S = (e:0
Given any formal power series (1.27) and any sector S, there is a (nonunique) function f (e) holomorphic in S for lei arbitrarily small so that 00
as a - 0 in S. r-o This basic result will be important in following chapters for summing asymptotic series which are formally obtained. For generalizations of this theorem, see Pittnauer (1969, 1972).
f (e) ^- I A, e'
Often it is necessary to use asymptotic expansions more general than the Poincar6 expansions introduced above. Alternative defini-
3. INTUITIVE APPROACH OF MATCHED ASYMPTOTIC EXPANSIONS
15
tions have been given by Schmidt (1937), van der Corput (1955-1956), Erdblyi (1961b), Erdblyi and Wyman (1963), and Riekstins (1966).
The formal series
DEFINITION (ERDELYI AND WYMAN): w
(1.28)
I fk(e) k=0
is an asymptotic expansion of the function f(e) with respect to the asymptotic sequence {,pj} as a - 0 if for every N > 0 N
as
II,,
AO (e) = I0 fk (e) + 0(`YN (e))
e - 0.
(1.29)
We write 00
f (e) ^' I0fk(e)
(4,j)
as
e ---> 0.
(1.30)
Note that expansions of this general form are not unique. The nonuniqueness allows greater flexibility and extends considerably the
class of functions whose asymptotic behavior can be studied. The unique Po''incarb expansions correspond to the special case where fk(e) = Ak`Yk(e)
3. INTUITIVE APPROACH OF MATCHED ASYMPTOTIC
EXPANSIONS
The method of matched asymptotic expansions (or of inner and outer expansions) has been successfully applied to numerous physical problems which involve singular perturbations. The basic ideas involved were present in Prandtl's boundary layer theory and in other even earlier work [cf. the historical survey of Van Dyke (1971)]; they occurred irl Friedrichs' work in elasticity [see Friedrichs and Stoker (1941) and Friedrichs (1950)] and in his mathematical lectures (Friedrichs, 1953, 1955); but they became best known through the work of
Kaplun and Lagerstrom [cf. Kaplun (1967) and Lagerstrom and
1. FIRST CONCERNS
16
Casten (1972)] and through the book of Van Dyke (1964). Despite some uncertainties [cf. Fraenkel (1969)], the variant technique which Van Dyke called the asymptotic matching principle has been most widely accepted by users. Meanwhile, work continues toward the development of a theory based on the concept of overlap domains and Kaplun's extension theorem [see, e.g., Meyer (1967) and Eckhaus (1973)]. Among the most significant mathematical singular perturbations theory, Vasil'eva's work [cf. the survey articles by Vasil'eva (1962, 1963a,b), and Wasow (1965)] closely parallels the popular matching techniques. Matching methods have generally superseded
earlier techniques of patching solutions (often numerically) at a somewhat arbitrary point, presumably in an "overlap domain." In any perturbation problem involving a small positive parameter e, it is natural to seek a solution of the form 00
Y.0 (x)
I aj (x) A (e)
as
a --> 0,
(1.31)
where x ranges over some (usually bounded) domain D and (/3j(e)) is an asymptotic sequence (often the power sequence {ei}) as a -f 0. In a singular perturbation problem, such an expansion cannot be valid
uniformly in x. (For example, this solution may fail to satisfy all boundary conditions. In applications, moreover, physical considerations will often indicate which boundary conditions are so omitted.) Instead, the expansion y1 will be generally satisfactory in the "outer region" away from (part of) the boundary of D. It will be called an outer expansion (or outer solution). In order to investigate regions of
nonuniform convergence, one introduces one or more stretching transformations
t _(x,e)
(1.32)
which "blow up" a region of nonuniformity (near a part of the boundary with neglected boundary conditions, for example). Thus = x/e might be used for nonuniform convergence at x = 0. Then if is fixed and e -- 0, x -* 0, while if x > 0 is fixed and a - 0, -4 oo. Selection of "correct" stretching transformations is an art sometimes motivated by physical considerations (cf. Van Dyke's discussion
3. INTUITIVE APPROACH OF MATCHED ASYMPTOTIC EXPANSIONS
17
on reference lengths), or mathematically (as in Kaplun's concept of principal limits). In terms of the stretched variable J, one might seek an asymptotic solution of the form 00
y,i(j) - I b;(j)aj(e)
as a -* 0
(1.33)
(where the sequence {al(e)) is asymptotic as e --> 0) valid for values of j in some "inner region." This will be called an inner expansion. (The inner expansion often accounts for boundary conditions neglected by
the outer expansion.) [We note that this generally accepted "innerouter" terminology is natural to boundary layer flow problems in fluid
dynamics. In the theory of thin elastic shells, however, exactly the opposite terminology is appropriate and used; cf. Gol'denveizer (1961) and Reiss (1962).] The inner region will generally shrink completely as e - 0 when expressed in terms of the outer variable x. Hence, the inner expansion is "local." In most problems, it is impossible to determine both the outer and the inner expansions y° and yi completely by straightforward expansion procedures. Since both expansions should represent the solution of the original problem asymptotically in different regions, one might attempt to "match" them, i.e., to formally relate the outer expansion in the inner region (y,)' and the inner expansion in the outer region
(yy)° through use of the stretching J = ip(x,e). The rules for even formally accomplishing this, in all generality, can be very complicated (cf. Fraenkel). Justification in particular examples through use of an
overlap domain and intermediate limits is difficult (cf. Lagerstrom and Casten) and, in general, there is no a priori reason to believe that an overlap domain (where matching is possible) exists. Once matching
is accomplished, however, the asymptotic solution to well-posed problems becomes completely known in both the inner and outer regions.
Frequently, it is convenient to obtain a composite expansion y` uniformly valid in D. One method of doing so is to let
y. = x° + Al - (y:)°,
1. FIRST CONCERNS
18
making the appropriate modification if several regions of nonuniform
convergence (several inner regions) are necessary. [Note that (yi)' = ye, so that (y`)' = yi. Likewise, in the outer region, (y`)° = y.'.] In certain problems, as Van Dyke observes, other composite expansions ("multiplicative," for example) may be preferable. We refer the reader to the cited references for more details of this important technique and many examples of its application. Most of
the techniques used in following chapters to construct asymptotic solutions to singular perturbation problems can be interpreted as specialized matched expansion methods, although we shall not generally so identify them. ExAMPLE:
Let us reconsider the simple problem
ey"+y'+y=0 y(O) = ao,
Al) = Q0
of Section 1.1 and obtain an asymptotic approximation to the solution by the method of matched asymptotic expansions. We shall seek an outer expansion as the asymptotic power series 00
I aj(x)ej
ye (x)
j0
valid for 0 < x < 1, and an inner expansion 00
I bj(f )ej j=0
ye (5)
in terms of the stretched variable
f = x/e valid as e ---> 0 near x = 0. The outer expansion must satisfy the differential equation and the terminal condition asymptotically; i.e., e(a" + ea" +
.
- ) + (a0 + eai + e2 a' +
+ (ao + eal + e2 a2 +
0
3. INTUITIVE APPROACH OF MATCHED ASYMPTOTIC EXPANSIONS
19
and
ao(1) + eal(1) + e2a2(1) + ... = 00Thus, equating coefficients,
ao + ao = 0,
ao(1) = Qo
ai+al+a'o=0,
a,(1)=0,
etc., so that ao(x) =
/3oe'-x
and
al(x) = (1 - x)/3oel-x. In terms of f, note that the differential equation becomes y££ + y£ + ey = 0 since
d _ 1d dx
a df-
and
d2
dx2
1 d2
= e2d
2
Thus, the inner expansion ye (j) must asymptotically satisfy this differential equation and the initial condition; i.e., (bog +eblg +e2b, e(bo + ebl +
)=0
and b0(0) + ebl(0) + e2b2(0) +
= ao.
Then bog + boE = 0,
b0(0) = ao
big + bl£ + bo = 0,
b1(0) = 0,
1. FIRST CONCERNS
20
etc., so integration yields bo(b) = Yo + (ao - Yo)e£ and
b1(f) = -(Yo + Yi) + [(ao - Yo)( + Yi ]eE with Yo and yi being undetermined constants. Hence, Y° (x) = /3o el-x + e(1 - x)Qo el-x + O(e2)
while y. '(x) = [Yo + («o - Yo )e E ] + E[-(-yo f + Yt )
+ ((«o - Yo)j + Yi)e{] + 0(e2). Writing the outer expansion in terms of the inner variable J, then, we have y. '(x) = /3oee'11 + e(1 - ej)/3oee 4 + O(e2) which leads to the 0(e2) approximation (Y: (x))'
/3oe(1 + e(-t + 1)).
Analogously, writing the inner expansion in terms of the outer variable x for x > 0, the e -f terms are asymptotically negligible and we have the O(e2) approximation (Y'(x))° ^ Y0(1 - x) - ey1. Since = selecting
x/e, matching (to this order) will be accomplished by
yo =-Yi=Poe Expressing all approximations in terms of the outer variable x, then,
3. INTUITIVE APPROACH OF MATCHED ASYMPTOTIC EXPANSIONS
21
we have the composite approximation y, (x)
[f3oe'-x + e(1 - x)/3oei-x] + [aoe + (ao - 13oe)e-xl`
+ e(-Poe(e
- 1) + (a. - 00e) - 0oe)e-x1tl J
- [/3o e + e/3o e(
+
or
yy (x)
[ /3o
el-x + (ao
- 00e)(1 + x)e xl ` ]
+ e[(1 - x)1oe'-x - 10eex19],
which should be compared with the exact solution previously given. Higher-order approximations could also be obtained analogously. The inner and outer expansions are depicted in Fig. 5. The asymptotic
solution (cf. Fig. 3) follows the inner solution near x = 0 and the outer solution for x > 0.
0
ye
x
FIGURE 5
The inner expansion yi and the outer expansion y0 for
ey"+y'+y-O,y(0)-O,y(1)= 1,e=0.1.
1. FIRST CONCERNS
22
4. TWO-VARIABLE EXPANSIONS
Two-variable expansion techniques have frequently been used to solve initial value problems with a small parameter on semi-infinite intervals [cf. Mahony (1961-1962), Kevorkian (1966), Cole (1968), Kollett (1973), and Nayfeh (1973)]. For such problems, "two-timing" is closely related to well-known averaging methods [cf. Morrison (1966), Perko (1969), and Whitham (1970)]. Analogous techniques have also been applied to obtain asymptotic solutions to singularly perturbed boundary value problems for ordi-
nary differential equations on finite intervals, say 0 < x < 1
[cf.
Cochran (1962), Erdelyi (1968b), O'Malley (1968a, 1970a), and Searl (1971)]. In addition to the ("slow-time") variable x, one introduces another ("fast-time") variable q ranging over an unbounded interval. [If, for example, nonuniform convergence occurred at x = x0, the fast variable q = (1/e) fzo g(s) ds for some positive g might be appropri-
ate.) Selection of a proper fast variable may be based on different physical time scales occurring, may be motivated by simple model equations, or may be left somewhat arbitrary initially. One seeks a solution y(x, rt, e) which is a function of both the slow and fast variables. The original differential equation becomes a partial
differential equation in the variables x and n and an asymptotic solution of it is sought having a power series solution 00
y(x, rj, e)
I y, (x, i )ej
j=0
as
a
- 0,
(1.34)
where the coefficients are bounded for 0 < x < 1 and for all q > 0. Substituting this expansion into the differential equation and equat-
ing coefficients of like powers of e, we obtain partial differential equations for the coefficients yj. Applying the boundary conditions will generally not suffice to determine the terms yj successively. In addition to the boundedness condition on the yj's, additional conditions must be imposed. (We do not expect to determine the yj's uniquely, since the expansion sought is a generalized asymptotic expansion, but we need to eliminate some arbitrariness.) Following
4. TWO-VARIABLE EXPANSIONS
23
Poincare, one usually asks that certain "secular terms" (like, e.g., rake ", k > 0) be eliminated. These somewhat arbitrary requirements can be motivated mathematically. For example, Erdblyi and Searl seek approximations whose correctness as e - 0 can be ascertained by applying their previously obtained theorems. Analogously, Reiss (1971) seeks to minimize the error at each step in the approximation.
A special, but frequently occurring, situation is when the twovariable expansion has the form y(x,'n, e) = Yi (x, e) + Y2 (n, e),
i.e., when there is an additive decomposition of the asymptotic solution as the sum of functions of the slow and fast variables separately. Indeed, the ("additive") composite expansions resulting from matched asymptotic expansions generally have this form. It is important to also observe that equations with slowly varying coefficients arise in a variety of physical applications and engineering approximations. Two-time techniques are well suited to such problems.
EXAMPLE: As an illustration of the two-variable method, consider the two-point problem
ey" + a(x)y' + b(x)y = 0 Y(0) = ao,
y(l) = Qo,
where a and b are infinitely differentiable functions on 0 < x < 1 and a(x) > 0 there. By analogy to the constant coefficient problem treated previously, one might expect nonuniform convergence as e - 0 near x = 0 and that the fast variable 1
a
Jox a(S)dd
might be appropriate. (Many other choices are also possible.) Proceeding with this q requires the solution y(x,,q, e) to satisfy the partial
24
1. FIRST CONCERNS
differential equation ae (x)
[ y, + y,, ] +
[2a(x)Yxq
+ a'(x)Y,, + a(x)Yx + b(x)Y] + eYxx = 0
since d
_
TX
a(x) a
a
ax
a7,
e
and
+
de
ae
dxz
axe
2a(x)
a
+
a2
ax a q
a(x) a
+
ae(x) ae
aq,
a
ee
Substituting the expansion Y(x,'n, e) ,,, 00I Yj (x,'n)ej
jo
into this differential equation, we formally equate coefficients of each power of e separately to zero. Thus, from the coefficient of a-1, we have a2 (x) [Yo,m + Yo,,] = 0
and, integrating, we obtain Yo (x, 'n) = Ao (x) + C0(x)e",
where A0 and Co are undetermined, Likewise, from the coefficient of eo, we have ae (x) [ Y1, + y1,, ] + 2a(x)Yoxq + a`(x)Yo,, + a(x)Yox + b(x)Yo = 0.
Integrating with respect to q, then, n
ae (x) [Y1,, + Y1 ] + 2a(x)Yox + a'(x)Yo +f
[a(x)Yox + b(x)Yo] d'n = 0
25
4. TWO-VARIABLE EXPANSIONS
and, substituting for yo, we have a2 (x) [Y1" + Yi ] + (a(x)A' + b(x)Ao),q
+ (a(x)Co' + a'(x)Co - b(x)Co)e-" = a2(x)A1(x)
for Al arbitrary. We then integrate with respect to q. Since yl must remain bounded as q --* oo (i.e., as e 0 for x > 0), we must have
a(x)A' + b(x)Ao = 0. Likewise, to avoid a secular term in yl which is a multiple of rye-", we must have
a(x)Co' + a'(x)Co - b(x)C0 = 0. Thus, integration implies that yl has the form Yl (x, ,q) = A1(x) + C1(x)e",
where Al and C1 are so far arbitrary. Applying the boundary conditions toy = Ao (x) + Co (x)e-" + e( ), noting that a-" is asymptotically negligible at x = 0, implies that we must select Ao(1) = /3o
and
Co(0) = ao - Ao(0).
Summarizing, then, we have begun to develop an asymptotic solution of the form
y(x,n,e) = (Ao(x) + Co(x)e ") + e(A1(x) + Ci(x)e-'Q ) + O(e2), where A o (x) = Qo eXp (- ( x a(s) ds>
Co (x) =
a(0)
(a0 - AO(O) ) exp
(x
a
s)s ds)
26
1. FIRST CONCERNS
and A, and Cl are, as yet, undetermined. We note that it was necessary to determine the form of the second-order coefficient y, (by boundedness and secularity conditions) in order to completely obtain the first coefficient yo. This is typically the case for two-time methods. This formal result can be continued to higher-order terms and can be easily shown to be asymptotically correct by comparison, say, with the asymptotic solution constructed in Section 3.1 by other methods.
CHAPTER
2
THE REGULAR PER TURBA TION METHOD
To illustrate the regular perturbation method (which will be basic to the singular perturbation techniques developed later), we consider the nonlinear initial value problem:
d =f(x,Y,e) Y(x0) = c(e),
where f and c have asymptotic power series expansions 00
f (x,Y, e) ,,, 7. f (x,Y)ei
j-0
27
28
2. THE REGULAR PERTURBATION METHOD
and 00
C(E) - 2 cj ei
j0
as the small parameter e tends to zero and the f;'s are infinitely differentiable in x andy. (In this discussion, there is no need to restrict e to any sector.) We shall assume that the "reduced problem" fo(x,Y)
ddxx =
(2.2)
Y(x0) = co
has a unique solution y0(x) on the bounded interval Ix - x0 < B, B > 0, and we shall seek a solution y(x, e) of the "full problem" (2.1) having an asymptotic expansion of the form y(x,e)
200 yy(x)eJ J=0
as
a - 0.
(2.3)
Formal termwise differentiation implies 00
f(x,Y(x,e),e)
a
(x,e) _
aje',
j-o
while substitution and termwise rearrangement yields f (x,Y(x, e), e)
f'(x,Y0,Y1, ... , yj)ej
j-o
where f°(x,Yo) = f0(x,Y0) f'(x,Y0,Y1) = Y1f0y(x,Y0) + fi(x,Yo)
and, generally, for each j > 1, fJ(x,Y0,Y1, ... Yj) = Yyfoy(x,Y°) +j,-I(x)+
29
THE REGULAR PERTURBATION METHOD
... , y,-1(x) only. Likewise,
where fj-1 depends on x, yo(x),
00
c(e) ,,, 7.Y;(xo)eJ
Equating coefficients of like powers of e in the differential equation and the initial condition, then, we ask that yo satisfy the nonlinear initial value problem o
= fo (z, yo)
Y0(X0) = co,
while successive yy's satisfy the linear initial value problems
d' = f'(x,Yo,
. ,Yj)
= Yjf0y(x,Y0) +fj-i(x) Yj (xo) = c;.
Hence, yo(x) is the unique solution of the reduced problem (2.2) on I x - x0 I < B and integration implies that the yj's are successively given by
y; (x) = c, exp[ fx'foy(S,Yo(S)) ds] X
+ 1xo exp [ f
f oy (S,Yo(S))
d s ] f j-1(t) dt.
Thus, a unique expansion (2.3) has been formally obtained. To prove the asymptotic correctness, we set N
Y(x, e) = I Yj(x)e' + R(x, e),
jo
(2.4)
where N is any nonnegative integer. We shall show that the remainder R is unique and O(e N+i) throughout I x - xo I < B as e --> 0. This will prove that the problem (2.1) has a unique solution y(x, e) on I x - x0 I
< B for e sufficiently small whose asymptotic series expansion is given by (2.3) there.
2. THE REGULAR PERTURBATION METHOD
30
Differentiating (2.4), we first note that dy
=
dx
N
dR
j=o
TX
2 fj(x,Yo, ... ,Yj)Ej +
However,
f
(X,
Y, Yy(x)eJ + R(x, e), e) _ Y. fj(x,y0,... ,Yj)EJ + J(x, R, e),
jao
j=o
where f is of the form f (x, R, e) = Rf o (x, E) + e N+lfl (x, E) + R 2j2 (x, R, E),
where fo, Al and f2 are smooth functions of their arguments. Since y satisfies (2.1), it follows that
dR_ dx
for
-f(x,R,e)
Ix-xoI
N
R(xo, e) = c(e) -, I Cj ej v eN+l
00
CN+1+j ej.
Integrating, R must satisfy the nonlinear Volterra integral equation R(x, e) = R°(x, e) + f x R2 (t, e)12 (t, R(t, e), e) xo
x exp [ f x fo(s, e) ds] dt on I x - xo I < B, where R° (x, e) = R(xo, e) eXp [
f xxo fo (s, e) ds]
[
+ eN+l f fl (t, e) exp f x fo(s, e) ds] dt = o(eN+1).
This equation, however, can be uniquely solved throughout I x - x0
I
THE REGULAR PERTURBATION METHOD
31
< B for e sufficiently small. We merely recursively define x
RJ(z,e) = R°(z,e) +1xp (RJ-'(t,e))212(t,Ri'(t,e),e)
[x
x exp 1r
]0 (s, e) ds
dt = 0(E N+1 )
for each j > 1 and the unique solution is given by
R(x,e) = limRJ(z,e) = O(eN+1), J-,,,
Ix - xoI < B.
This follows by writing
RI (z, e) = j (Rk(z, E) - Rk-' (z, e)) k=1
and by proving convergence to R(x, e) in the space of continuous functions on I x - xo I < B by estimating the differences Rk (z, e) - Rk-'(x,e) [cf. Erd6lyi (1964) and Willett (1964)]. Summarizing, then, we have THEOREM 1:
Suppose the initial value problem
d = fo (x, y) y(x0) = co
has a unique solution yo(x) throughout I x - x0 < B, B finite. Then, for each e sufficiently small, the problem
-
= f (z,y, e) ,,, If J=0
(z,y)e.i
00
00
y(x0) = c(e) ,,,
I ciEi
j0
with infinitely differentiable coefficients f (z, y) has a unique bounded solution for I x - x0 I < B such that for each N > 0 N
y(x, e) _
yy(z)ei + O(eN+1)1 j=U
32
2. THE REGULAR PERTURBATION METHOD
there. Further, the coefficients y,, j > 1, may be uniquely obtained recursively as solutions of linear initial value problems.
NOTE 1. Less differentiability of the coefficients f; requires the expansion for y(x, e) to be terminated after an appropriate number of terms.
2. The theorem also holds when y is a vector.
3. The theorem is false on unbounded intervals (cf. Example 1, Section 1.1) without additional hypotheses.
The reader should be aware that regular perturbation techniques find applications in a wide variety of settings and levels of sophistication [cf., e.g., Bellman (1964), Rellich (1969), Morse and Feshbach (1953), and Kato (1966)]. EXAMPLE:
Consider the concrete example
d = m(x, e)y2 + n(x, e)y + p(x, e) A) = c(e)
I g6',
00 >o
where coefficients m, n, and p have asymptotic series expansions as e - 0 with infinitely differentiable coefficients; e.g., 00
m(x, e) - 7. mj(x)ei. >o
Suppose further that the reduced problem
d~x
= mo (x)y2 + no (x)y + po (x)
y(0) = co
33
THE REGULAR PERTURBATION METHOD
has a unique solution yo(x) on some interval jxI < B, B finite. Seeking an asymptotic series expansion for the solution, 00
AX, E) - I yj (x)Ej, jo and formally substituting into the differential equation, we have
dx
+Edx1+e2dx +... _ (mo + EMI + e2m2 + x (yo + 2eYoYi + e2[2Yoy2 + yfl + ... ) + (no + en1 + e2n2 + )(yo + eyl + E2Y2 + ...)
+ (po + ep + e2p2 +
...)
= (moyo + noyo + po) + e(2moy0y1 + m1Yo + noy1 + n1Yo + Pi)
+ e2(2m0YoY2 + moyi + 2m1yoy1 + m2Yo + noy2 + n1Y1 + n2Yo + P2) + ....
Thus, equating coefficients here and in the initial condition, we have
d
oo =
moyo + noyo +po,
yo(O) = co
d l = (2moyo + no)y1 + (m1Yo + n1Yo + A),
Yi (0) = cl
22 = (2moyo + no)Y2
+ (moyi + 2m1Yoy1 + m2Yo + n1Y1 + n2Yo + P2),
Y2(0) = c2,
ctc. Takingyo as the unique solution of the nonlinear reduced problem
34
2. THE REGULAR PERTURBATION METHOD
and integrating the linear equations for y, and Y2, we have Yi (x) = c1 exp [ fox (2m0 (s)Yo(s) + no(s)) ds]
+ fx (mt (t)y2(t) + ni (t)Yo(t) + Pi (t)) x exp [ f x (2mo(s)yo(s) + no(s)) ds] dt and
Y2 (x) =
c2exp [fox (2mo(s)Yo(s) + no(s)) ds] x
+f
(mo(t)yi (t) + 2m1(t)Yo(t)y1(t) + m2(t)Yo(t)
+ nl (t) yl (t) + n2 (t) yo (t) + P2 (t))
X exp [ f x (2mo(s)yo(s) + no(s)) ds] dt.
Further terms can likewise be easily obtained recursively on jxj < B (where yo is defined). A specific example is pictured in Fig. 6.
Y
-1
(0,-2)
FIGURE 6
.5
1
x
The solution y(x, e) = -2 + 5ee 2x + O(e2) of dy/dx = y2
+ 2y, y(0) - -2 + 5e.
CHAPTER
3
LINEAR BOUNDARY VALUE PROBLEMS
1.
SECOND-ORDER PROBLEMS
In this section we shall consider boundary value problems for the linear equation
ey" + a(x)y' + b(x)y = 0
(3.1)
on the interval 0 < x < 1 as the small positive parameter e tends to zero. We suppose that a(x) and b(x) are infinitely differentiable and that a(x) > S > 0 35
(3.2)
36
3. LINEAR BOUNDARY VALUE PROBLEMS
there. [If a(x) < 0, the substitution z = 1 - x will allow this condition to be satisfied for 0 < z < 1.1 Due to the linearity, all solutions will be linear combinations of any two linearly independent solutions
e.g., Coddington and Levinson (1955)). Analyzing constant coefficient examples (like Example 3 of Section 1.1) prompts us to seek linearly independent asymptotic solutions of the form [cf.,
y, (x, e) = A(x,e)
(3.3a)
and
/' Y2 (XI e) = B(x, e) exp
L-
EJ
a(s) ds],
(3.3b)
0
where 00
I aj(x)eJ,
A(x,e)
j=o
A(0,e) = 1
(3.4a)
B(0, e) = 1.
(3.4b)
and w
B(x, e) - I b, (x)eJ,
jo
Note that y, (x, e) will be bounded throughout 0 < x < I while y2(x,e) will be exponentially small as e -* 0 for x > 0. Substituting (3.3) into the differential equation implies that we must have
a(x)A' + b(x)A = -eA" and
a(x)B' + a'(x)B - b(x)B = eB" throughout 0 < x < 1. Then, using the expansions (3.4) and equating
coefficients of like powers of e, we find that the ad's and bb's are uniquely determined recursively as follows: ao (x) = exp C-
aj (x) - -
f
x
a(s)
ds]
a-' )t) exp
[-
f
d..] dt, s)
j>l
37
1. SECOND-ORDER PROBLEMS
and
bo (x) = a(0) exp i
f
b x a(s) ds]
Recall the Borel-Ritt theorem of Section 1.2 which lets us define (e) and q(e) as analytic functions of e such that 00
j0 and W
ry(e) ^- -a(O) + e,I bj(O)j. Thus, we have constructed a formal solution A(x, e) of the initial value problem
0<x<1
ey" + a(x)y' + b(x)y = 0,
y'(O,e) = (e)
A0, 6) = 1, and a formal solution B(x, e) exp
L
e J0
x
a(s)
d]
of the initial value problem
ey" + a(x)y' + b(x)y = 0, Y(O, e) = 1,
0<x<1 Y'(o, e) = rle)
(3.8)
To proceed, we prove the asymptotic validity of these formal solutions:
38
3. LINEAR BOUNDARY VALUE PROBLEMS
The initial value problem (3.7) has a unique solution y1 (x, e) such that, for each integer N > 0, LEMMA:
N
Y1 (x, e) _ 2 aj (x)ej + eN+1 R(x, e),
j-0
where R(x, e) is bounded throughout 0 < x < 1 for each e > 0 sufficiently small. Likewise, the initial value problem (3.8) has a unique solution such that, for each integer N > 0, N
Y2 (x, e) = (jI bj (x)ej)
exp
[-
x e
a(s) ds]
J
(3.10)
+ eN+1 S(x, e),
where S(x, e) is bounded throughout 0 < x < 1 for each e > 0 sufficiently small.
PROOF: We shall prove the assertions about Y2 only, the results for y1 being obtained in analogous fashion. Substituting (3.10) into (3.8), we find that S = eN+l S must satisfy the initial value problem
0<x<1
eS" + a(x)S' + b(x)S = -eN+1bl,,(x), S(0, e) = 0 S'(0,e) = eN9(e)
N ( e
\'q(e) + a(0) -
j-0
b'(0)i
l
- e N I00bN+1+j (0)ej
j-0
Integrating twice and reversing the order of integration, we find that 9(x, e) will satisfy the linear Volterra integral equation
fX 9(x, e) = so (x, e) -
(1f e
x
exp E- 1 J
a(s) t
dsdr)
b(t) (t, edt,
39
1. SECOND-ORDER PROBLEMS
where S°(x, e) = ENO(e) fo x exp L
E
J
r
a(s) ds dr
0
-EN+I f x
dr) bl;,(t) A
Note that, since a(x) > 8 > 0, X
J
r
exp L- E r
f
a(s) dsI
dr < 1 [i - exp (- s (x - t))
r
for x > t. This implies that S°(x,e) is O(EN+1) throughout [0, 1] and
that the integral equation can be uniquely solved by successive approximations [cf. Erdelyi (1964)]. One defines Si (x, e) = S ° (x, e) - J x (E J
x
exp I - E f r a(s) ds] dr)
1
X b(t) Si-I (t, e) dt
for each j > 1, and the solution is given by S(x, E) = lim SJ (x, e) = O(EN+1)j-+rI
Alternatively, using the resolvent kernel W(x,t,E) corresponding to the Volterra kernel K(x, t, e)
Jt
exp L- 1 J
note that we can represent the solution
t
a(s) ds] dr,
in the form
S(x,e) = S°(x,e)+ fxW(x,t,e)S0(t,e)dt = O(EN+1) [cf. Cochran (1968)]. Since S(x,e) = O(EN+I) is uniquely determined, the solution y2 (x, e) of (3.8) is also unique. This completes the proof.
3. LINEAR BOUNDARY VALUE PROBLEMS
40
Clearly, the two solutions y, and y2 are linearly independent on [0,1] for a sufficiently small. Since we have constructed a fundamental set of asymptotic solutions of (3.1), the general solution will be of the form y(x, e) = kl (,-)A (x, e) + k2 (e) B(x, e) exp L- E
a(s) ds] (3.11)
J
for arbitrary functions k;(e). Moreover, if any solution of (3.1) converges to a bounded limit in 0 < x < 1 as e - 0, it must converge to the solution k1 (0) ao(x) of the reduced equation a(x) y'(x) + b(x) y(x) = 0
for x > 0, and it will feature nonuniform convergence at x = 0 unless k2(0) = 0. Derivatives of the solution will then also converge to the corresponding derivatives of k1(0)a0(x) for x > 0. Suppose, as a first example, that we wish to find the asymptotic solution of the boundary value problem ey" + a(x)y' + b(x)y = 0, y(O) = a(e),
0<x<1 y(1) = 8(e),
(3.12)
where a and 8 have power series expansions as a - 0. The solution must be of the form (3.11) where kl(e) and k2(e) satisfy the linear equations a(e) = kl (e) + k2(,-)
r
/3(e) = k1(e) A (1, e) + k2 (e) B(l, e) exp L-
f
a(s) ds].
Since A (1, e) # 0 and a(x) > S > 0, the system is nonsingular for e sufficiently small and we have kl (e)
A IE)) + O(e a1` ),
41
1. SECOND-ORDER PROBLEMS
and
k2(E) = a(e) -
(E)) + 0(e-11e).
Thus the unique solution of (3.12) is of the form [_.fX
y(x, e) = A(x, e) + B(x, e) exp
e
a(s )
ds](3.13 )
where A(x, e) = /3(e) (1' + 0(e ale) A e) and
'9(X' e) _ (a(e) - A (E)) ) B(x, e) + 0(e-sie). l, e
Further, since A(x, 0) = 8(0) exp L- f x b(s) ds], a(s)
J
the limiting solution for x > 0 satisfies the reduced problem
a(x)y' + b(x)y = 0,
0<x<1
Y(l) = Q(0). Moreover, A(0,0) a(0), in general, so we must expect nonuniform convergence at x = 0. Crudely, then, we cancel the boundary condition at x = 0 and integrate the resulting terminal value problem for
the reduced equation to get the limiting solution for (3.12). Before going on, we note that a direct (but more sophisticated) proof of the asymptotic validity of (3.13) has been given by Cochran (1962) [see also O'Malley (1969a)]. Our results are pictured in Figs. 7 and 8.
42
3. LINEAR BOUNDARY VALUE PROBLEMS
FIGURE 7
The linearly independent solutions y, and y2 of ey" + y + y
=0,e=0.1.
X
FIGURE 8 small.
The solution of ey" + y + y = 0, y(0) = 0, y(l) = 1, e
43
1. SECOND-ORDER PROBLEMS
Analogously, we find that, for the initial value problem
ey" + a(x)y' + b(x)y = 0,
0<x<1 (3.14)
Y'(O) = y(e),
y(O) = a(e),
we can also determine the coefficients kl and k2 uniquely and we have
y([
+ e(y - a B'(0, e))
x, e) =
a(O) (0) + e(A'(O,e) - B'(O,e))I
+e
r
A(x, e)
aA'(0, e) - y
L a(O) + e(A'(O, e) - B'(O, e))
x exp L- E
J0
J B(x, e)
a(s) d ]
(3.15)
up to asymptotically exponentially small terms. Here, then, the limiting solution a(0) ao(x) for 0 < x < 1 satisfies the reduced problem a(x)y' + b(x)y = 0
y(O) = a(O); Y
1
.5
0
FIGURE 9
e - 0.1.
.5
1
x
The solution of ey" + y' + y = 0, y(O) = 1, y'(0) = 2,
3. LINEAR BOUNDARY VALUE PROBLEMS
44
i.e., the derivative boundary condition is canceled in defining the appropriate limiting problem. Thus, y(x, e) will converge uniformly on [0, 11 while y'(x, e) will converge nonuniformly at x = 0, and higher derivatives will be initially unbounded as e - 0. (See Fig. 9.) Finally, for the terminal value problem ey" + a(x) y' + b(x) y = 0,
0<x<1 (3.16)
Al) = $(e),
Y'(1) = K(e),
we can again determine the coefficients kl and k2 of (3.11) uniquely. Substituting, then, we obtain
y(x,e) _ {[$a(l)B(l,e) +
e) - cA(l, e)] B(x, e) exp [ I J a(s) ds] x
} J
x (a(1)A(l,e)B(l,e) + e(B(l,e)A'(l,e) - A(l,e)B'(I,e)))(3.17)
This solution, however, blows up exponentially for x < I as e - > 0 [unless perhaps /3A'(1, e) = tcA(l,e)] since exp[(l/e) fX a(s) ds] does so.
It seems from these examples that, whenever the solution of Eq. (3.1) subject to separated boundary conditions has a limiting solution
as e - 0, a boundary condition at x = 0 is canceled to define the reduced problem satisfied by this limiting solution for x > 0. This is because we assumed that a(x) is positive. If a(x) were negative on [0, 1], the transformation z = 1 - x shows that a boundary condition at x = 1 would then have to be canceled, in general, and nonuniform convergence would occur there if a limiting solution within [0, 1] could be defined. If a(x) had a zero within [0, 1], we would be faced with a turning point problem-something far more complicated than
the examples discussed here [cf. Chapter 8 and Wasow (1968), however].
45
2. HIGHER-ORDER PROBLEMS
2. HIGHER-ORDER PROBLEMS Following Wasow (1941, 1944), we will consider the boundary value problem consisting of the linear differential equation em-n (y(m) + a,
(x)y(m-1) + ... +
+ Q(x) (y(") +
am (x)Y)
Q1(x)Y("-1) + ... + Rn (x)Y)
=0
(3.18)
on the interval 0 < x < 1 and the boundary conditions y(k)(0) = 1,,
i = 1, 2,
Y ()_ .
i=r+1,.,m.
(a , )
1
1
-
,r (3.19)
We shall take m > n so that the order of the differential equation drops when e = 0, and shall assume that the coefficients of (3.18) are real and infinitely differentiable throughout [0, 1] and that (3(x)
0
(3.20)
there. (The last assumption eliminates the possibility that the limiting nth-order equation is singular within [0, 1].) Further, we will order the boundary conditions so that
m)'A1)'A2>...)'A,>0
(3.21 a)
and
(3.21b)
and we shall seek asymptotic solutions as the small positive parameter e - 0. (We introduce the parameter as em-" for later convenience. In physical problems, one would naturally expect more general parameter dependence. Analysis should then proceed analogously, however.) Since the differential equation is linear, any solution will be a linear combination of any set of m linearly independent solutions. Thus, we
46
3. LINEAR BOUNDARY VALUE PROBLEMS
shall first determine m linearly independent asymptotic solutions of the form Y(x, e) = Gj(x, e) exp [ E
x µf (t) dt],
J xj
j = 1, 2,
, m,
(3.22)
where xj is either 0 or 1 and where each Gj has an asymptotic series expansion G. (x, e) -- I Gj, (x)e
iO
as e ---> 0 with coefficients that are infinitely differentiable functions of
x such that Gjo(x) 0 0. Differentiating k times, we have
y(k)(x, e) = {Gy) +
ek-1
k(k2 1)Giµi-2µj) + Gek' }
X X
exp[l
fj µf(t)dt](3.23)
so substituting into the differential equation (3.18) and collecting terms we obtain e" 1
Gj (µj` + 0(x)µi )
+ e[G j(mµj -1 + n /3(x)µi-1)
+ Gj(m(m2
1)IL--2
µj +
n(n
+ ai (x)µj -1 + $i (x) N(x)µf -1 X [Gjn) +
X exp l e L
fx x
81(x)G(n-1) +
2
/J+
Q(x)IL -2,
C2( ... ) + e" $(x)
... + Rn(x)Gjl
µj(t) dt] = 0.
(3.24)
47
2. HIGHER-ORDER PROBLEMS
Thus, the leading term of (3.24) implies that we should ask that
µi + 8(x)µj = 0,
(3.25)
obtaining m - n distinct roots µj(x) _
(-/3(x))1/(m-n) which are nowhere zero in [0, 11 and n roots which are identically zero. Let these roots be ordered so that
Re µf(x) < 0
in [0, 1]
Re µ,(x)>0
for
j = 1, 2, ..., a
for
j=m-n-z+1,...,m-n
(3.26a)
(3.26b)
and
µ, (x)=0
for
j=m-n+1,...,m.
(3.26c)
Two cases arise: the nonexceptional case when a + r = m - n (i.e., when m - n µf's have nonzero real parts), and the exceptional case when a + r = m - n - 2 (i.e., µ,+1 = -µa+2 0 are imaginary), since the roots of (3.25) are distributed like the (m - n)th roots of either ± 1. [Determining the sign of the real parts of the µf's is clearly related to stability considerations. As such, analogous calculations were important in early control theory work (cf. Meerov, 1961) and in thin shell theory (cf. Gol'denveizer, 1961).]
The coefficients Gk are determined successively so that higherorder coefficients of e' in (3.24) become zero. Thus, for j < m - n, setting the coefficient of el-" to zero implies that the Go must satisfy Gjo (mµ7 -1 + n/3(x) µ1-1'
+ Got m(m2 1)µj -2 µ, + ... + $i (x) $(x)µi -11 = 0. (3.27)
Noting that mµ7 -1 + n/3(x)µj-1 # 0, because the nonzero roots of
48
3. LINEAR BOUNDARY VALUE PROBLEMS
(3.25) are distinct, [ f X (Qi (S) - ai (s)) ,,j
8joexP J G'o
(m - n) j#(x)I(m-n+1)12(m-n)
where goo * 0 is undetermined. Successive GJk's satisfy nonhomoge-
neous forms of (3.27) and each is uniquely determined up to an arbitrary constant. Without later loss of generality, then, we will take
Gj(xj,e) = I for j < m - n. For j > m - n, µj = 0 so G. (x, e) must satisfy the original ("full") differential equation (3.18). Equating coefficients when e = 0 implies that Goo must satisfy the reduced equation Y(n) + $1(x)Y(n-1) + ... + /3n(x)Y = 0.
(3.28)
Likewise, successive G;;'s for i > 0 must satisfy nonhomogeneous forms of (3.28). Corresponding to the n linearly independent solutions of (3.28), then, we formally obtain n linearly independent asymptotic solutions of (3.18).
It is worth noting that the procedure used to obtain the m formal solutions 1 (x, e) of the form (3.22) is completely analogous to that used in the geometrical theory of optics [cf. Keller and Lewis (to be published)]. Equation (3.25) corresponds to the eiconal equations and (3.27) to the transport equations of that theory.
That the m asymptotic solutions formally constructed form a fundamental system (as e -* 0) follows from the results of Turrittin
(1936). His analysis, as might be expected, is based on integral equations. It also shows that these asymptotic solutions may be formally differentiated termwise.
Under appropriate conditions, the solution of the boundary value problem (3.18)-(3.19) will converge within (0, 1) to the solution of a reduced boundary value problem as e - 0. The reduced problem will consist of the reduced equation [i.e., (3.18) with e = 0] and n of the m boundary conditions (3.19). We tell which m - n boundary conditions are omitted in defining the reduced problem by applying a cancellation law. Before stating the somewhat complicated law, we note that
49
2. HIGHER-ORDER PROBLEMS
it could be motivated by considering a series of simple examples with
easily determined limiting behavior. We encourage the reader to examine such problems for himself and merely state: THE CANCELLATION LAW
(1) Cancel a boundary conditions at x = 0 and r boundary conditions at x = 1, starting from those with the highest derivatives (i.e., largest Aj's).
(2) In the exceptional case when a + r = m - n - 2 also cancel from the remaining boundary conditions those two with the highest order Aj of differentiation, requiring that they belong to the same endpoint, say x, and that their selection must be without ambiguity. Let S and T be the total number of boundary conditions canceled
at x = 0 and x = 1, respectively. (Thus, S = a and T = r when
a+-r=m-n.Otherwise, S=a+2 andT=-rwhen.9 =0and S = a and T = r + 2 when x = 1.) The cancellation law is well defined if and only if
S
and
T<m - r
(3.29)
and, in addition, in the exceptional case, AS > Ar+T+l
if m - r>T and x=0
(3.30a)
r > S and z = 1.
(3.30b)
or
X,-+T > A5+1
if
Under these conditions, the reduced problem is well defined as
z(")+/31(x)z("-1)+...+/3"(x)z=0, zCA,)(0)=1,, zCA,)(1) a 1j,
i=S+1,...,r i=r+T+ 1,...,m.
0<x < I (3.31)
50
3. LINEAR BOUNDARY VALUE PROBLEMS
NOTE:
If m - n is odd, (3.25) implies that the nonexceptional case
holds and IS- TI = Ia - -rI = 1. If m - n is even and a + ,r = m - n, then a = S = T = r, while if m - n is even and a + r = m - n - 2, then a = r but IS - TI = 2. In all cases, then, the total number of boundary conditions canceled at each endpoint differ by two or less. (For more general problems than (3.18){3.19), this is no longer true [cf. O'Malley and Keller (1968)].) Since the Gm-n+k(x,0), k = 1, 2, , n, form a fundamental set of solutions for the reduced equation (3.28), the reduced problem will have a unique solution provided the n X n matrix G;ks+k(0, 0)
(3.32)
m-n+k
is nonsingular [cf., e.g., Coddington and Levinson (1955)]. We are now ready to state our theorem. Before doing so, let us define the integers To and Ti:
Yo=As
and Yi = Xr+r
if
a + z = m - n. (3.33a)
Otherwise To = min (As, Jar+r)
and Ti = Xr+T
x=1
if
(3.33b)
and
To = A.
and Ti = min (As, Ar+r)
if
x = 0. (3.33c)
Then, recalling that p = q (modulo m - n) if m - n is a factor of p - q, we have Consider the boundary value problem (3.18)-(3.19), where the reduced problem (3.31) is well defined and has a unique solution z(x). Assume further that: THEOREM 2:
(a) A1, A2,
... , As are distinct modulo m - n, and
(b) Ar+t. Ar+z, . . . , Xr+r are distinct modulo m - n. Then, for e sufficiently small, the problem (3.18)-(3.19) has a unique solution. It has
2. HIGHER-ORDER PROBLEMS
51
the form
m-n
+ eY, k
I+1
fx
r
s
y(x, e) = G (x, e) + 6'0 1 Gk (x, e) exp [e
J
µj(t) dt]
r1 x Gk(x,e)expLEI µj(t)dtI ,
(3.34)
1
where the functions 0 and Gk have asymptotic power series expansions in e for 0 < x < 1 with coefficients which may be obtained successively by a scheme of undetermined coefficients. Further,
z(x) = G (x, 0).
(3.35)
Thus,
y(x, e) --* z(x)
as
e --> 0
in the open interval 0 < x < 1 (but for the exceptional cases when either
x = 0 and To = 0 or x = 1 and Ti = 0 when the limiting solution oscillates rapidly about z).
We note that the representation (3.34) can be formally differentiat-
ed repeatedly. In particular, then, in the nonexceptional case, the exponential factors are asymptotically negligible within 0 < x < 1 and we have dy U) (x, e)
dxj
-.> z(j) (x)
there as e ---> 0.
The theorem gives sufficient conditions for convergence to a limiting solution. That the hypotheses are nearly necessary for the problem is shown by the following list of constant coefficient examples whose solutions diverge as e - 0. e3y(4) + y, = 0 1.
y"(0) = Y'(0) = y(0) = 0, Here -r = 2, so we would need to cancel two boundary conditions at x = 1. Since only one boundary condition is given there, however,
52
3. LINEAR BOUNDARY VALUE PROBLEMS
the reduced problem (cancellation law) is not well defined.
ey(2) - y' = 0 2.
y'(0) = 1,
Al) = 0.
Here the reduced problem z' = 0, z'(0) = 1 has no solution. e3y(4) -y' = 0 3.
y'"(0)
= Y(0) = Al) = 0,
Y'(1) = 1.
Here the two boundary conditions to be canceled at x = 0 have orders of differentiation differing by m - n = 3. e2y(3) + y' = 0 4.
Y'(0) = 0, Y(0) = Y'(1) = 1. Here the exceptional case holds, but the reduced problem is not well defined since the last two boundary conditions to be canceled belong to different endpoints. e2
y() + y(2) = 0
5.
y(1) = 1. Y"(0) = Y'(0) = Al) = 0, Here, the boundary conditions to be canceled are not well determined. A SAMPLE PROBLEM:
Let us consider the boundary value problem
e2y(4) - y" = 0,
0<x<1
with Y,,,(0),
Y(0),
y'(1),
and y(l)
being prescribed constants. Here the exponents , , are roots of the polynomial µ4 - µ2 = 0 so a = z = S = T = I and Theorem 2 implies that there will be a unique asymptotic solution which is of the form y(x, e) = Z(x, e) + e3 L(x, e) e-18 + e R(x, e) e-0- 411 ,
53
2. HIGHER-ORDER PROBLEMS
where Z, L, and R have power series expansions in e and Z(x, 0) is the unique solution of the reduced problem
Z"=0 Z(l) = Y(l)
Z(O) = Y(O),
We will show how the first few terms are determined by using undetermined coefficients. The reader should be aware that such an expansion procedure is much more efficient than obtaining a fundamental set of asymptotic approximations (as in the preceding section) and then seeking the correct linear combination to satisfy the boundary conditions. Setting y(x,e) = Zo (x) + e[Z1(x) + Ro (x) e 0 - x)/e ]
+ e2[Z2(x)
+ R1(x)e
(1-x)/e]
+ e3[Z3(x) + R2(x)e_(l)i1+ 4(x)ex/e] + 0(e4) and differentiating formally, we have y'(x,e) = (Zo + Roe 0-x)/e) + e(Z1 + (R1 +
Ro)e(1-x)1e
+ e2(Z21 + (R2 + Ri)e0-x)/e - Loe x/e) + 0(e3) Y"(x,e) = Roe (1-)/e +
(Z'o + (R1 +
2&)e(1-x)/e)
+ e[Z'1 + (R2 + 2R1 + R'o)e-(1-x)/e + Loex/e] + 0(e2) Y'"(x,e)
=
R2e-(1-x)/e + E(R1 + 3Ro)e a-x)/e
+ (Z'o + (R2 + 3R1 + 3R'o)e (1-x)/e Y'v(x,e) =
R3
+
e
Loe-x/e)
+ 0(e)
e-(1-x)/e + I (R1 + 4Ro)e 0-x)/e [(R2 + 4R1 + 6R0")e (1-x)/e + Loex/e] + 0(1).
54
3. LINEAR BOUNDARY VALUE PROBLEMS
Thus, the differential equation becomes
t_Z'p+2R'e (1-x)le]+e[-Z1+ 2R' +
5R'o)e-(1-=)/e] + 0(e2)
= 0.
This implies that
and
Z'i=0 among other equations. Further, the boundary conditions become AO) = Zo(O) + eZi(O) + e2Z2(0) + e3 Z3(0) + L0(0)) + 0(e4)
y(l) = Zo(l) + e(Z1(l) + Ro(l))
+ 2(22(1) + R1(1))
+ e3 (Z3(1) + R2(1)) + 0(e4)
Y(1) _ (Zo(1) + Ro(1)) + e(Zi(1) + R1(1) + R0(l)) + 82(Z?(1) + R2(1) Y "(0)
'R, (1))
+ 0(e3)
L0(0)) + 0(e)
Thus, we have
and
Z
+ Ro(1) ¢ Al)
2. HIGHER-ORDER PROBLEMS
55
among other equations. As expected, then, Z" = 0 ZOO) = y(0),
Z0(1) = y(l)
or
Z0(x) = y(0) + (y(1) - y(O))x. F urth er,
&(X) = 0 R0(1) = Al) - Z0'(1) that
i mp lies
Ro(x) = y'(1) - y(l) + y(0).
Fi nally, Z'1' (x) = 0
Z,(0) = 0,
Z1(1) = -R0(1)
so that
Z, (x) = (Y'(1) + y(l) - Y(0)) X. Thus,
y(x, E) _ [y(0) + (y(1) - y(0))xl + E[(-Y,(,) + y(l) - Y(0)) (X - e ('-x)/e)] + o(E2) and
y'(x, E) _ [y(l) - y(0)1 - (Y'(1) + y(l) + O(E).
y(0))e-('-x)1e
56
3. LINEAR BOUNDARY VALUE PROBLEMS
In this example, then, y(x,e) converges to the solution Zo(x) of the reduced problem as a -- 0 throughout [0, 1], while y'(x, e) converges to
Z' (x) except near x = 1, where nonuniform convergence occurs [unless y(O) = y(1) - y'(1)]. Higher-order derivatives of Zo(x) are zero, and higher-order derivatives of y(x, e) converge to zero except at
x = 0 and x = 1, where they will generally be (algebraically) unbounded as a -- 0. A special example is illustrated in Fig. 10. Further examples may be found in the book by B. Friedman (1969).
PROOF OF THEOREM 2: Consider the nonexceptional case where we can write the general solution of the differential equation
x
The component parts Z (x, e), L(x, e)e-x/°, and R(x, FIGURE 10 a)e 0-x)/t of the solution y(x,a) = Z(x,a) + a3L(x,a)e-x/t + aR(x,e)e p`x)Je to e2y(4) - y(2) - 0, y(o) - 0, y,,,(0) = 1, y(l) = 1, Y(l) = 3.
57
2. HIGHER-ORDER PROBLEMS
(3.18) as
y(x,e) =
,
-1
Cj(e)Gj(x,e)exp[1 E
IL
m-n
+ eX'+', I c(e) Gj(x,e)exp[1 1
+
I
j=m-n+l
(1)
x
f
d] µj(t)dt]
Cj(e) Gj(x,e)
for undetermined coefficients Cj(e) [cf. Eq. (3.22)]. Using the differentiation formula (3.23), the boundary conditions y(Ai)(0) = 1, take the form 1i =
ja C m-n + I Cj (e) ex°-a,
0
e
-11
(asymptotically exponentially small terms)
ja+1
+I
+ Owl
M
j=m-n+1
for
C, (e) [Gj(",) (0, 0) + O(e)],
i = 1, 2, ... , r
while y(A)(1) = 1, imply a
1, _ Y. Cj(e) j=1
(asymptotically exponentially small terms) m-n
+ e14+,->,; I Cj(e)[(Lj(l))k' + O(e)] ja+1
+I M
j=m-n+l
Cj(e) [Gk')(1, 0) + 0(e)],
for
i=r+
since we can take Gj(xj, 0) = 1, j < m - n, with xj = 0 for j = 1, 2, .... a and xj = 1 for j = a + 1, ... , m - n. Considering these as m linear equations in the m unknowns Cj(e), we can obtain a unique solution if the determinant of coefficients is nonzero. This determinant, however, is a nonzero multiple of 0(e) = det
[OX
+ O(E),
J
3. LINEAR BOUNDARY VALUE PROBLEMS
58
where X is a matrix with bounded entries, 0 is given by (3.32), and 0
for
0 = ((#J(0))x'),
i,j = 1, ..., a
and
1,j= 1, ..,T.
51 =
Thus
0(0) _ (det 4))(det )(det ,). Note, first, that det ¢,
0 [cf. (3.32)] since the reduced problem has
a unique solution. Next, note that j (0), p2(0), . . ., µa(0) are those determinations of (-/3(0))'1(n'-n) in the left-half plane. Thus, they are nonzero constant multiples of ic, K2, ... , K°, where K = e(21101(m-n) This implies that det o = C det ((K"r)Y)
i, j = 1, 2, ... , a,
for some C
0.
Thus, det o = 0 if and only if a Vandermonde determinant is zero; i.e., if and only if K7. = (e(2n1)1(m-n))7- = (e(2n1)/(m-n))AB = KXd or
(e(2nl)/(m-n))Xe-7'g
=1
for integers a and /3 such that 1 < a < /3 < a. [For a discussion of Vandermonde determinants, see, e.g., Bellman (1970).] This equality is impossible here because Al , ... , X. are distinct modulo m - n. Thus 0. Similarly, i is nonsingular because A,+1 , A,+2, . . . , X.+T det Jo
59
3. GENERALIZATIONS OF THESE RESULTS
are distinct modulo m - n. Thus, 0(0) * 0 and 0 (e) will be nonzero for a sufficiently small. It is easy to show that the CC(e)'s will have power series expansions as a --> 0, which can be obtained by Cramer's rule. Alternatively, the expansion (3.34) could be obtained directly by an undetermined coefficients scheme [cf. O'Malley and Keller (1968) where a proof in the exceptional case is also given]. Finally, analogous
results would also follow provided 0(e) 0 for e > 0, even if 0(0) = 0. For example, it is simple to obtain the unique asymptotic solution of the form (3.34) to the problem e2y(4) - y - 7r2y = 0
A) = y(1) = 0,
y'(0) = y'(1) = 1
even though the solution of the corresponding reduced problem
Z"+7r2Z=0 Z(0) = Z(1) = 0 is not unique [and, therefore, 0(0) = 0].
3.
GENERALIZATIONS OF THESE RESULTS
Turrittin (1936), among others, shows how to construct asymptotic solutions for more general linear equations (without turning points). When the appropriate characteristic equation has multiple roots, the Puiseaux (Newton) polygon method must be used to obtain formal solutions. The proof of asymptotic validity uses integral equations. Analogously, Hukuhara (1937) and Turrittin (1952) show how to obtain asymptotic solutions for systems of linear differential equations whose order drops when e = 0. Much more complicated situations can be analyzed [cf. Stengle (1971)]. Constructing asymptotic solutions for such linear equations is analogous to obtaining asymptotic solutions to polynomial equations with coefficients depending on a
small parameter. Readers would find it instructive to study the polygon method for such problems [cf., e.g., Vainberg and Trenogin (1962) or Walker (1962)].
3. LINEAR BOUNDARY VALUE PROBLEMS
60
Knowing a fundamental set of asymptotic solutions, very general boundary value problems can be studied. Thus, O'Malley and Keller
(1968) consider boundary value problems when the order of the boundary conditions changes when e = 0; Harris (1960, 1973) considered boundary value problems for linear systems with coupled boundary conditions; Handelman et a!. (1968) and Hams (1961) considered eigenvalue problems; O'Malley (1969c) considered nonhomogeneous equations; and O'Malley and Mazaika (1971) considered multipoint problems with discontinuous coefficients. Using spectral representation, these results can be applied to boundary value prob-
lems in Hilbert space [cf. A. Friedman (1969) and Bobisud and Calvert (1970)]. Direct applications of these results occur in many fields [see, e.g., Boyce and Handelman (1961) or Nau and Simmonds
(1972) for a vibrations problem, Desoer and Shensa (1970) for a problem in electrical networks, and Keller (1973) for a nonlinear diffusion problem]. It is worth observing that cancellation laws can also be obtained for the frequently occurring case of equations with small coefficients multiplying the highest derivatives, but where explicit dependence on a small parameter is not known. Such results should be helpful in analyzing "stiff" differential equations [cf. Bjurel et a!. (1970)]. A numerical technique for boundary value problems is discussed in Abrahamsson et al. (to appear). A.
The Vibrating String Problem
As a simple example, we shall consider the eigenvalue problem which arises from the vibrations of a string with clamped endpoints [see Lord Rayleigh (1945)]. If the string has negligible stiffness, the appropriate eigenvalue problem is
y =-Ay,
0<x< 1
A) = y(1) = 0.
(3.36)
If stiffness effects are introduced, we must, instead, consider the higher-order problem e2y,° - y" = Ay,
0 < x < 1
A0) = y(0) = Al) = y(1) = 0,
(3.37)
3. GENERALIZATIONS OF THESE RESULTS
61
where e2 is proportional to the stiffness. As the stiffness (and the positive parameter e) tends to zero, we expect the eigenvalues and eigenfunctions of the full problem (3.37) to converge to those of the reduced problem (3.36), i.e., to n27r2 and sin n7rx, respectively, for n = 1, 2, .... The eigenfunctions, however, might be expected to feature nonuniform convergence as e - 0 near both endpoints. Four linearly independent asymptotic solutions of the differential equation y" + Ay = 0; namely, G1 (x, A, e) a z/e,
G3(x,A,e),
G2 (x, A, e)
e-0-z)/e,
and G4(x,A,e)
(3.38)
can be formally obtained, as in Section 2. Here, the G;'s have asymptotic series expansions in a with coefficients depending on x and A and, without loss of generality, G1 (0, A, e) = 1 = G2 (1, A, e). The general solution of the differential equation will be a linear combination of these four linearly independent solutions while the eigenvalues A are those values such that the boundary value problem (3.37) has nontrivial solutions. The boundary conditions, then, imply a system of four linear equations for the coefficients of the linear combination. The system will have a nontrivial solution with A as an eigenvalue if
and only if the determinant of coefficients 0(A) vanishes. For e sufficiently small, A(A) has the form 0(A)
=1 82
x det
+0
1
0
G3 (0, A, e)
0
1
G3(1,A,e)
G4(1,A,e)
G4 (0, A, e)
-1 +eG'(0,A,e)
0
eG3(O,A,e)
aG'(O,A,e)
0
1 +eG2(1,A,e)
G3(1,A,e)
eG'(1,A,e)
G
e-111
).
(3.39)
From this it follows that the eigenvalues A(e) as functions of a all have asymptotic series expansions in a with the leading terms A0 being roots
62
3. LINEAR BOUNDARY VALUE PROBLEMS
of the equation so(lo)
det
(G3(oxoo)
G4(0, AO, 0)
G3(1,Ao,O)
G4(1,A0,0)
= 0.
(3.40)
Since G3 (x, A, 0) and G4 (x, A, 0) satisfy the reduced differential equation, it follows that A0 is an eigenvalue of the reduced problem (3.36).
Note that if the eigenvalues A(e) were known asymptotically, the corresponding eigenfunctions y(x, e) could be determined (uniquely up to normalization).
For the problem (3.37), then, it is natural to seek asymptotic expansions for the eigenvalues A(e) and the corresponding eigenfunctions y(x, e) of the form X(e)
i0
Xj ej
(3.41)
and
y(x, e) = A(x, e) + B(x, e) a x' + C(x, e) e-(1-x)/e,
(3.42)
where A, B, and C have asymptotic series expansions in E. We shall determine these expansions termwise by an undetermined coefficients procedure. Substituting into the differential equation and boundary conditions of (3.37), we find that the "outer expansion" A(x, e) must satisfy the full equation
e2A'v-A" =AA while B(x, e) and C(x, e) must satisfy
e3B'v - 4e2B"' + e(5B" - AB) - 2B' = 0 e3 C'v + 4e2 C" + e(5 C" - AC) + 2C = 0 and
0 = y(0, e)
A (0, e) + B(0, e)
0 = y(l,e)
A(l,e) + C(l,e)
0 = ey'(O, e) ^- e(A'(O, e) + B'(O, e)) - B(O, e)
0 = ey'(l,e) ^- e(A'(l,e) + C'(l,e)) + C(l,e).
3. GENERALIZATIONS OF THESE RESULTS
63
Equating coefficients, then, we ask that a'i + A0 ai = ai-Z
-
t
i
XI ai-r for j = 0
{-Xiao + ai-
for j > 1.
i-1
2b, = 5bj-i - Y. Xlbi-i-l - 4bj2 + bj_'3 = 2/3iand i-1
2cj = -5cj_i + I Xrci-1-r - 4cj-2 - cj 3 = 2Yiro and
ai(0) = -bi(0) _ -d-i(0) - bj-i(0) ai(1) = -ci(1) = aj-i(1) + cj-i(1) for all integers j > 0 where coefficients with negative subscripts are defined to be identical zero.
For j = 0, then ao(x) + Xoao(x) = 0,
ao(0) = ao(1) = 0
bo(x) = 0,
bo(0) = 0
co(x) = 0,
co(1) = 0.
Thus, bo(x) = co(x) = 0, and there is a nontrivial solution for ao only
if X0=n27r2,n = 1,2,.... Then ao(x) = Aosin n7rx
for arbitrary A0 0. For a unique (up to sign) determination of ao, we (somewhat arbitrarily) normalize by asking that fo ao (s) ds = 1 and set A0 = .
64
3. LINEAR BOUNDARY VALUE PROBLEMS
For j = 1, we have
ai(0) = -bi(0)
a" (x) + Xoa, (x) = -X1ao(x),
a, (1) = -c1(1)
2b' (x) = 5bo(x) - Xobo(x),
bi(0) = ao(0)
2c' (x) = -5co(x) + Xoco(x),
cl (1) = -a0'(1).
n7r and cl (x) _ (-1)"V2 n7r and, from the differenThus, b, (x) = tial equation for ao, we have
(aoa, - aoa')' = X,ao. Integrating from 0 to 1, then A, = 4n27r2, while the boundary value problem for a, implies that a, (x) = AI sin n7rx -
n7r(1 + 2x)cos n7rx
for A, arbitrary. To specify a, uniquely, we (again somewhat arbitrarily) impose the orthogonality condition f ' ao (s) a, (s) ds = 0,
so
A, =
.
Continuing, suppose that the al(x), bl(x), cl(x), and A, are known for
1 < j, j > 1. Then the differential equation and boundary conditions imply that
a'i(x) + \Oaj(x) = -Aiao(x) + ai-i(x),
ai(0) = -bi(0)
ai(l) = -ci(l) bj(x)
= bi(0) +foz Qi-i(s)ds
and
ci (x) = ci (1) +
f
X
Yi-I (s) ds,
65
3. GENERALIZATIONS OF THESE RESULTS
where aj_1, 8j_,, yj_,, bj(0), and cj(0) are known in terms of the preceding coefficients. Using the differential equations for ao and aj, then
(apaj - aoaj) = Xjao - a0aj_1, so integration implies that Xj = bj(0)ao(0) - cj(l)ao(l) +JoI ao (s) aj- I (s) ds.
The differential equation for aj, then, yields aj(x) up to an arbitrary term Aj sin n7rx. The constant Aj can be uniquely specified by requiring
that fo ao(s)aj(s)ds = 0. Thus, all coefficients in the expansions for Jo(e) and y(x, e) can be obtained termwise by this scheme of undetermined coefficients. That the resulting expansions are asymptotically correct has been proved by Handelman et al. (1968). To summarize, we have shown that all eigenvalues of the vibrating string problem (3.37) with small stiffness are of the form X(e) = n2 7r2 + 4en2 ore + E2( . ),
n = 1, 2, ...
(3.43)
with corresponding (normalized) eigenfunctions
y(x, e) =
nor
sin n7rx
n
r sin n7rx +eL n7r
-(1 - 2x)cosn7rx + e_z/E + (-l)e-0-z>1t1 (3.44)
Note that the eigenvalues converge to n27r2 and the eigenfunctions to
\sinnnrx as e -p 0, as expected from the reduced problem (3.36). Convergence of the eigenfunctions is uniform throughout 0 < x < 1, while derivatives of the eigenfunctions will converge nonuniformly as
e -+ 0 at both endpoints x = 0 and x = 1. (This could have been anticipated, as in Theorem 2.)
66
3. LINEAR BOUNDARY VALUE PROBLEMS
Two-Parameter Problems
B.
A direct and useful extension of the preceding results is to problems
involving two parameters. Thus, we consider the boundary value problem consisting of the linear equation (x)y(m-,) + ... + am(x)yl
e[ yam) + a,
+ µ$(x) [y(") + R, + Y(x) [ y(P) + Y,
(x)y("- ,) + ... + on (x)yl
(x)y(P-,) + ... + YP(x)yl
=0
(3.45)
on the interval 0 < x < I and the boundary conditions y(k)(0) = i;,
i = 1, 2, ..., r
y(k)(1) = 1;,
i = r + 1, r + 2, ... , m,
(3.46)
where
m>X1 jX2>...>Xr>0 and
m>Xr+,>Ar+2>...>Xm> 0. Here, a and µ are small, positive parameters simultaneously approaching zero while
m>.n>p>0.
(3.47)
The limiting behavior of the asymptotic solutions to this differential equation will be completely different in the three cases where the parameters e and .t are interrelated such that (
(ii)
e
a
µ(m-P)/(" P)
or
(iii)
a
-* 0.
These cases are all studied by O'Malley (1967b).
3. GENERALIZATIONS OF THESE RESULTS
67
These sharp distinctions become already apparent in analyzing the simple constant coefficient equation
ey" +pay' +by= 0,
O*b,
a
where two linearly independent solutions are given by
[-
Y, (x)
exp
Y2(x) =
exp[-
4ea2
x 1 1+
(I -
)")I
and
When (i) a//2 N°x/e and
e
/
x1 1
/
-11-
a
)12)
0, note that the solutions behave approximately like e = p2, they behave like e-N°x/Ze X
exp[+(pax/2e)(1 - 4b/a2)h/2]; and (iii), when p2/e - 0, they are approximated by exp [+(-bx/e)'/2 ]. Putting it roughly, let us say that
the second-order equation degenerates to the reduced equation, by = 0, (as a and p tend to zero) where, in Case (i), the degeneration is through the intermediate equation pay' + by = 0; and, in Case (iii), the degeneration is direct (almost as if p = 0). In order to analyze the limiting behavior of y, and y2, then, it is essential to know the limit of e/p2; i.e., a and p must be interrelated in order to ascertain the limiting behavior of the asymptotic solutions. In this section, we shall only consider the first (but, perhaps, the most interesting) case appropriate for the mth-order equation (3.45), namely e
p(m-p)/(n p)
-' 0
as
p -' V.
(3.48)
Here it is convenient to introduce the new small positive parameters = p,/(n-p)
(3.49a)
and
C
p(m-p)/(n-p),
(3.49b)
68
3. LINEAR BOUNDARY VALUE PROBLEMS
in terms of which we have
e=
and
.m-P,nm-n
µ
= yn-P
We shall now proceed treating and q as independent small parameters. Note first that the order of the differential equation (3.45) drops from m whcn and q are both positive to p <, m when = q = 0. From our successful analysis of one-parameter problems, we might
expect (under appropriate hypotheses) that the limiting solution within (0, 1) as 3' and q both tend to zero will be the solution of a reduced problem consisting of the reduced equation [(3.45) with e = µ = 0] and p of the original m boundary conditions (3.46). A cancellation law will be needed to determine which m - p boundary conditions are omitted in defining this reduced problem. We shall assume that the coefficients in the differential equation (3.45) are infinitely differentiable throughout 0 < x < 1 and that
0 * Y(x)
/3(x)
(3.50)
there. Then we might seek m linearly independent asymptotic solutions 1 (x, ,,q) of the form y
fz
y
vj(t,J,')dt]
[generalizing the previous representation (3.22)]. Somewhat more efficiently, however, we shall find m - n asymptotic solutions of the form _
x yy
J
Jryi
_
Vj(s,'qP n )ds
,
j = 1, 2, ... , m - n,
xj
(3.51)
n - p asymptotic solutions of the form Gj (x,
,
1
7lm-n) exp
LX
dsj'
vj (s' 77m-n) [f j = m - n + 1, . . ., m - p,
(3.52)
69
3. GENERALIZATIONS OF THESE RESULTS
and p asymptotic solutions of the form G (x y n-P J m P q,-n),
j = m -p + 1, ... , m.
(3.53)
Here the Gj(x, a, $)'s are to have double asymptotic expansions r, cc
IS-0 Gjrs(x) ar/33.
Gj(x, a,#)
i.e., for every integer N > 0, Gj(x,
( arcs + a'#) =r>0,s>0 G Gjrs(X)
((
77
N))N].
r+s
Further, the exponents have asymptotic expansions in the indicated parameters -q- P or Formally substituting (3.51), then, into the differential equation (3.45), we have I--
\Gjvjm +
L
(mGvjm_I + M( M G(m))
X1)2 (...) +
+
\Gjvj + X11
1)GjV
+ ai
nG'vj-i
v; -n + (71)2 (...
+ n(n2 1)Gjvj-2vj)
L
[(GjvjP
+
pG'vr' +P(p2
V,
+
(J,q
)2(...)
+
(J,q)PYp(x)Gj = 0.
To eliminate the most singular terms, we ask that vj satisfy
vj + /3(x) vj + r"-°y(x) vj° = 0,
(3.54)
70
3. LINEAR BOUNDARY VALUE PROBLEMS
obtaining m - n distinct roots vj(x, q" p) which have series expansions in q' -P with distinct selections
=
vj(x, 0)
(-a(x))1/m-n,
j = 1, 2, ..., m - n.
(3.55)
The corresponding Gj's are then obtained by a regular perturbation procedure. In particular, equating the next-order terms in the differential equation to zero, we find that Gj00 must satisfy the nonsingular linear equation Gjoo(mvj'"-' + n/3(x)vj-') + Gj00
+
m(m - 1) 2
vjin-2vj
nn-2 1) /3(x) vr2 vj + a, (x) vj°-1 + /3(x)/3, (x) vj"-1 = 0.
Thus we formally obtain m - n asymptotic solutions (3.51). Analogously, for the n - p solutions of the form (3.52), the vj's must be distinct nonzero roots of the polynomial
/3(x)vj + y(x)vf + 7,m-"v = 0.
(3.56)
Thus
0) _ v (x,
Y(x)
\- Q(x) 1\
'
j =m - n+ 1,...,m-p
(3.57)
and the corresponding Gj's can be obtained termwise as solutions of nonsingular linear equations (by a regular perturbation procedure). Finally, the remaining p Gj's of (3.53) are regular perturbations of the p linearly independent solutions of the reduced equation y(P) + Y1(x)y(' ) + ... + Yp (x)y = 0.
The m formally obtained solutions so constructed can be shown to be asymptotically valid throughout 0 < x < 1 as and -* 0 by generalizing the procedure of Turrittin (1936).
3. GENERALIZATIONS OF THESE RESULTS
71
We note that the exponents in the first m - n asymptotic solutions (3.51) approximate the exponents which would result from the simpler
equation ey(m) + µ /3(x) y(n) = 0, while the exponents of the n - p asymptotic solutions (3.52) approximate those of µ /3(x) y() + -y(X)y(p) = 0. Further, the former exponents are more singular (as and q tend to zero) than the latter since Case (i) applies. In Case (i), then, we say that the mth-order equation (3.45) degenerates to an intermediate nth-
order equation and, then, to the reduced pth order equation as a and µ tend to zero. Neither the intermediate nor the reduced problem will be singular since /3(x) 0 -y(x) throughout 0 < x < 1. We shall call the differential equation (3.45) nonexceptional when all m - p determinations vj(x, 0) of (3.55) and (3.57) have nonzero real parts. Otherwise, (3.45) is exceptional. Since the limiting behavior is more difficult to analyze in the latter case, we shall not consider it here
[see O'Malley (1967b), however]. [We note that Eq. (3.45) will be nonexceptional if both m - n and n - p are odd.] Let us, then, order the roots vj(x, 0) so that their real parts satisfy Re vj(x, 0) < 0
for j = 1, 2, ... , al
(3.58a)
Revv(x,0)>0 Revv(x,0)<0
forj=a, + 1,...,a, +Tj = m - n
(3.58b)
for j=m-n+l,...,m-n+a2 (3.58c)
and
Re vj(x, 0) > 0
for j=m-n+a2+l,...,m-n+a2+T2 =m-p. (3.58d)
Thus, we state
Cancel a = a, + a2 boundary conditions at x = 0 and T = Tj + T2 boundary conditions at x = 1, starting from
THE CANCELLATION LAW:
those involving the highest derivatives.
Note that the cancellation law is well defined if and only if
a
and
T < m - r.
(3.59)
72
3. LINEAR BOUNDARY VALUE PROBLEMS
Slightly modifying Theorem 2 (in the nonexceptional case), then, we have: COROLLARY:
(i)
Suppose
the reduced problem
0<x<1
+ yp(x)z = 0,
Z(P) + y, (x)z(p-1) +
zQi)(0) = 11,
i = a + 1, ..., r
z(,\,) (1)=1;,
i = r+T+1,...,m
has a unique solution z(x) and (ii)
... , X., are distinct modulo m - n, , X. are distinct modulo n -p, Xo,+1) X01+2, kr+i) Xr+2, , kr+,, are distinct modulo m - n, and , A, ,. are distinct modulo n - p. Xr+1,+1, Xr+11 +2)
(a) (b) (c) (d)
Al , A2,
Then, in the nonexceptional case [where (3.58) holds], the boundary value problem (3.45)-(3.46) has a unique solution y(x, e, µ) for a and µ sufficiently small and with a/(µ(m-p)/(n-p)) -* 0 as µ -* 0. It has the form
y(x, e, to = G (x, ' 71) +
A.71 A°'
ji
Gj (x) ) 71)
expz°j
x
[i_f
J
(s
r
j1 O 11
I
f
z L71+j(s''7n-p)d11
02
+
>b
Om-n+j (x) ) 71)
j= I
z
x expr
dSJ
1
V.-n'(31
L ,/p
I
7'n'-n)
12
j
x exp [
6m-n+v2+j (x)
71)
I
1
f
fJ
z
Vm-n+o2+j (s, ',m-n) ds ,
(3.60)
73
3. GENERALIZATIONS OF THESE RESULTS
where the functions G and Gk have double asymptotic series expansions in and q throughout 0 < x < 1 which may be obtained successively by a scheme of undetermined coefficients. Further G(x, 0, 0) = z(x) and d'y(x, e, µ) -* z(j>(x) dx>
j = 0, 1, 2,
within 0 < x < 1 as a and µ tend to zero.
Remarks
1. Note that the expression (3.60) for y can be differentiated any number of times. Also note that since the exponents in (3.60) all tend is to -oo as e and µ tend to zero, only the outer solution asymptotically significant within (0, 1). Moreover, G is a linear combination of the p asymptotic solutions (3.53). Since ,1 << , we
also note that the first sums in (3.60) decay much faster than the last;
i.e., the n - p asymptotic solutions (3.52) are asymptotically more significant than the m - n asymptotic solutions (3.51). [Of course, all m asymptotic solutions are necessary to obtain the uniformly valid representation (3.60).] Finally, we observe that if a specific relationship between and q becomes known (e.g., _ q3), the expansion (3.60) can be considerably simplified.
2. As for the theorem, a list of examples can be given for which a hypothesis of the corollary is violated and no limiting solution exists as a and µ tend to zero. 3. Generalizations of these results to systems of differential equations involving many parameters have also been made [cf. Wasow (1964), Harris (1965), and O'Malley (1969b)].
4. Many-parameter problems can be expected to occur often in practical problems. Among many applications of two-parameter problems, the reader might refer to DiPrima (1968) for one in lubrication theory or to Vasil'eva (1963b) or Chen and O'Malley (1973) for ones in chemical reactor theory and dc motor analysis.
74
3. LINEAR BOUNDARY VALUE PROBLEMS
EXAMPLE:
Finally, consider the initial value problem
ey" + pay' + by = 0 Y(0) = 1,
y'(0) = 0,
where a and b are constants, x > 0, and the small parameters a and p are such that p and a/p2 both tend to zero. Its unique solution y(x, e, µ) 1
2(1 - 4eb/p2a2)1/2
X { (-1 +
(1-
b2\1/Z)exP[_(l+l_)
(µ a
tends to the trivial solution of the reduced equation, by = 0, in the limit provided both a and b are positive. As for the general problem (3.45)-(3.46), we could introduce = and 71 = e/p2 and then construct asymptotic solutions of the form
y1 = G1 (x, µ, 6) exp
[
E
j
x
vi (,
p
µ2) ds]
with v1(x, 0) = -b and rr
y2 =
f
X
G2(x,µ, Z)expvz(S, i) d ] µ
µ
with v2(x, 0) = -b/a. Both solutions will decay exponentially away
from x = 0 provided a and b are both positive and we will have 01 = 02 = 1, T1 = T2 = 0. (See Fig. 11.) Note that the "boundary layer thickness" of Y1 is much less then that of y2. Thus, the corollary
75
3. GENERALIZATIONS OF THESE RESULTS
Y
1
x FIGURE 11
The linearly independent solutions yi and y2 of ey" + pay'
+by =0,a>0,b>0,e/p2->0. implies that the unique solution y will have the form
exp[-f xvi1s,112 ) ds]
y(x,e,µ) - µ2G1 (x,
/
rr
(+ G2(x,A, 2
µ
/'x
expLlJ v2s, 2) 0
µ
ds],
where the terms in the expansions for G1 and G2 can be obtained by an undetermined coefficients scheme. Note that the outer solution G(x, p, e/p2) of (3.60) does not appear here because the two linearly independent solutions of this second-order differential equation are accounted for by yl and y2; i.e., G is asymptotically zero. Note, too, that this representation agrees with the known exact solution. This and other boundary value problems for variable coefficient secondorder equations are discussed by O'Malley (1967a).
CHAPTER 'T
NONLINEAR INITIAL VALUE PROBLEMS
1. THE BASIC PROBLEM We will consider the nonlinear system
_ dt =
dx
dy E
u(x,Y, t, E)
= u(x,Y, t, E)
of scalar equations for t > 0 subject to the initial conditions x(0, E) = x°(E) Y(0, E) = Y°(E) 76
77
1. THE BASIC PROBLEM
Here e is a small positive parameter, u, v, x°, and y° have the asymptotic expansions 00
u(x,Y, t, e) - 2 ui (x,Y, t)e; ;=o
v (x,Y, t, e) -
;o v; (x,Y, t)ei 00
x°(e) .,, 2 xj eJ ;=o 00
yo(e) - 2 yoei
ase->0,andtheu;'s and v;'s are infinitely differentiable functions of x, y, and t. The reduced problem is
dt =
u0(x,Y,t)
t>0
0 = v0(x, y, t),
x(0) = x$. We shall make the two assumptions (Hi)
that there is a continuously differentiable function 4(X, t) such that vo (X,
o(X, t), t)
=0
and that the resulting nonlinear initial value problem dx
_
dt - u0 (x, p(x, t), t)
= ox' t)
x(0) = x$ has a (unique) solution X0(t) on some closed bounded interval, say
0 < t < 1, such that vor(Xo(t), Yo (t), t)
-K
78
4. NONLINEAR INITIAL VALUE PROBLEMS
on 0 < t < 1 for some constant K > 0 and for Y0(t) = 4.(Xo(t), t)
and (Hii)
that for the same K > 0,
v,(Xo(0),A,0) < -K for all values A between Yo(0) and y8.
Under these hypotheses, we shall be able to construct an asymptotic solution of the initial value problem (4.1)-(4.2) which has (X0(t), YO Q)) as its limiting solution for t > 0 as e - 0. Since X0(0) = and Y (0) 0 yo, in general, x(t, e) will converge uniformly in [0, 1] as a -> 0 but convergence of y(t, e) will usually be nonuniform at t = 0.
Note that the pair (X0 (t), Yo (t))
satisfies the reduced problem (4.3) and, by (Hi) and the implicit function theorem [see, e.g., Hale (1969)], the reduced problem has no other nearby solution. Note, too, that if the reduced problem (4.3) has two (or more) distinct solutions (X0(t), Y(t)) corresponding to different choices for (p(X, t), condition (4.8) can only hold for one of them. Finally, observe that (4.8) follows by continuity from (4.6) for small values of the "boundary layer jump" J = y8 - YO(0). It also follows from (4.6) when u0 is linear in y. In general, however, (4.6) and (4.8) are independent hypotheses.
To illustrate the hypotheses and to predict the form of solution obtained, we consider the linear problem ex" + a(t)x' + b(t)x = 0,
0
x(0) = a,
x'(0) = y,
1. THE BASIC PROBLEM
79
where a and b are infinitely differentiable and a(t) > 0. We put this into the system form as dx dt
y
0<<1
E d1 = -b(t)x - a(t)y, x(O, e) = a,
y(O, e) = y.
As we showed in Chapter 2, this problem has a unique asymptotic solution
r
r
x(t, e) = A(t, e) + e (t, e) exp L- E
J0
a(s) ds]
y(t, E) = A'(t, E) + [-a(t) E(t, E) + EA'(t, E)] X exp C
E
f
t
a(s) ds]
where A and A have asymptotic series expansions as e - 0 with infinitely differentiable coefficients and
A(t, 0) = a exp L- f x b(s) ds1. 0
a(s)
Introducing the stretched variable T = t/E,
note that e--(O)r decays exponentially to zero as e - 0 for any fixed t > 0. Further,
\
r
exp C
E
(a(s)
- a(O)) ds]
0
can be expanded as a power series in e with coefficients which are polynomials in T. Expanding the relevant functions of t = eT as
80
4. NONLINEAR INITIAL VALUE PROBLEMS
functions of r, we can rewrite the solution of the initial value problem in the form X((t, e) = A((/t, e) + Epl(T, E) e -(01
At, e) = '4'(t, e) + p2 (T, 6) e401,
where 00
p;(r,e) - 2 py(r)ei, i = 1, 2,
as
J=o
a,0
and each p, is a polynomial in T. Note that, when t > S > 0, r --> oc and
p;(r,e)e(oh = O(em)
as
e-0
for any M. Away from t = 0, then, (x, y) will converge to the outer solution (A, A') as e -* 0. Moreover, the solution is expressed as the sum of an outer solution (a function of t having an asymptotic series expansion as e -* 0) plus a boundary layer correction (a function of T having an asymptotic expansion as e --> 0 with terms tending to zero as r - oo). We note that the reduced problem for this linear system dx It = y,
X(0) = a
0 = -b(t)x - a(t)y. Further, for this system, 4 is uniquely given by ,A(x,
t)
b(t) x
while
v0, (x, y, t) = -a(t) < 0
81
1. THE BASIC PROBLEM
for 0 < t < 1 and the solution of the reduced problem is given by (A(t, 0), A'(t, 0))
_
(aexp[_fl, a(s)
r -a b(t) exp L-
X
a(s) ds
J
.
As in the linear example, the asymptotic solution of (4.1)-(4.2) will be an additive function of the variable t and of the stretched variable (4.9)
T = tle,
which tends to infinity as e -* 0 whenever t > S > 0, S arbitrary but fixed. Use of this stretched variable will allow us to describe the nonuniform convergence at t = 0. We shall seek a solution of the form x(t, E) = X (t, E) + Em(T, E) (4.10)
y(t, E) = Y(t, E) + n(T, E),
where X, Y, m, and n all have asymptotic series expansions as e -> 0; i.e., 00
X (t, e) -
X (t)e1, j=0
Y(t, E) ~ 2 Y(t)eJ j=0 00
m(T, E)
mj (T)E1, j=0
n(T, E) ~ 2 nj (T)Ei . =0
We shall ask that all terms in the expansions for m and n tend to zero as T tends to infinity. (This condition replaces the familiar matching (patching) conditions used elsewhere.) Away from t = 0, then, (x,y)
will converge to the outer solution (X, Y) as e - 0, while the boundary layer correction (em, n) will be significant only near t = 0. Our expansion procedure will consist of obtaining, first, the outer expansion (i.e., the expansion of the outer solution) and, second, the complete expansion.
4. NONLINEAR INITIAL VALUE PROBLEMS
82
Since the solution is asymptotically given by the outer solution for t > 0, (X (t, e), Y(t, e)) must formally satisfy the system (4.1). Substi-
tuting into (4.1), we have equality when e = 0 since (X0, Y) is a solution of the reduced problem. Using Taylor series expansions of u and v about (X0, Y,t,0) and equating coefficients of ei for j > 0, we must have dX
uo.(Xo,Yo,t)X + uoy(Xo,Y,t)Y+ ai_i(t)
Wt- =
0 = v0 (X0,Yo,t)X +voy(X0,Y,t)Y+fli_1(t), where ao (t) = ul (Xo, Yo, t), Qo (t) = v1(X0, Yo, t) - d Yo/dt, and, gener-
ally, aj_1 and Ni_1 are known successively in terms of the Xi's and rs with 1 < j. By assumption (Hi), $X(Xo, t) _
-vo((X0, Y, t) voy Xo, Yo, t)
is defined throughout 0 < t < 1, so we have Y(t) _ $X(X0(t),t)X(t) + Qi-1(t) dX (t) dt
uX ( X0
(t), t) X (t) + &i_ 1(t),
where &i_1 and Pi-1 are known successively and u is defined by (4.5). We will obtain X (and thereby Y) uniquely as the solution of its linear equation once the initial value X (0) = x° - mi_1(0)
is determined. Thus, the outer expansion (X, Y) can be completely determined termwise provided the constant mi_1(0) is known at each step. We wish to emphasize that, to obtain the outer solution, it is not sufficient to know the prescribed initial value x°(e). One also needs the initial value m(0, e) of the boundary layer correction. The complete expansion (4.10) must also satisfy the system (4.1). Substituting (4.10) into (4.1), we find that the boundary layer correc-
1. THE BASIC PROBLEM
83
tion (em, n) must satisfy the nonlinear system dm
dx
dTdt
_
dX
dt
U(X (eT, E) + Em(T, E), Y(ET, E) + n(T, E), ET, E)
- U(X (ET, E), Y(ET, E),
ET, E)
(4.12)
do_E(dy_dY) aT
dt
dt J
V (X (ET, E) + Em(T, E), Y(er, e) + n(T, e), ET, E)
- v (X (ET, E), Y(ET, E), ET, E). Likewise, (4.2) implies the initial condition n(0, e) = Y, (E) - Y(0, E).
We shall obtain the expansions for m and n by expanding both sides of (4.12) and equating coefficients of like powers of e for finite values of T.
At e = 0, we have the nonlinear system dd0°
= uo(Xo(0), Y(0) + no(T),0) - Uo(Xo(0), Y(O),O) ° no (T) U (no (T))
d
dT° = vo(Xo(0); Y(0) + no(T),0)
- vo(Xo(0), Yo(0),0)
(4.13)
° no(T)V(no(T)), where U and V, by the mean value theorem, are appropriate partial
derivatives. Note that the initial condition implies that no(0) = J - y$ - Y (0), the boundary layer jump, and V(J) < -K by assumption (Hii). By the differential equation, then, Ino(T)I is initially decreasing. By assumption (Hii), moreover, V(no(T)) will remain negative and the differential equation implies that Ino(T)I will decrease monotonical-
4. NONLINEAR INITIAL VALUE PROBLEMS
84
ly to zero as T --> oo such that
Ino(T)I <_ Ino(0)Ie '.
Direct solution of the nonlinear differential equation for no is seldom possible, but the integral equation no(T) = no(0) + l T no(s)
V(no(s)) ds
(4.14)
on the semi-infinite interval T > 0 can be uniquely solved by successive approximations [cf. Erdelyi (1964)]. (We note that it is essential here that K be positive.) Knowing no, we have 00
rno(T) _ - f no(s) U(no(s)) ds = O(e- t)
(4.15)
since mo - 0 as r - oo. Then, X1(0) = x° - mo(0) is known, so the terms X, (t) and Y(t) in the outer expansion are uniquely determined. Expanding the right sides of (4.12) about (x,y, t, e) _ (X0(0), YO(0) + n0(T), 0, 0), and equating coefficients of ej for each j > 0, we ask that ml and nj satisfy the linear system dmj dT
= u0r( Xof0), YO(O) +n oOT,0) n,OT +A_1T()
dn dT
v
'
(4.16)
n, T O +Bi_ I (T), ( Xo (O), Yo O O + no O T,0)
where Aj_1 and Bj_1 are known successively and, by induction, satisfy Ai-1(T)
=
O(e K(1-a)r)
=
Bj-1(T)
as
T -> 00
for any S > 0. Since ni (0) = y° - Y(0) is known successively, we can integrate the linear equation to uniquely obtain nj(T). Then dmm/dT is determined and we have (T)_-
m,
f T
dmj
(s)ds.
1. THE BASIC PROBLEM
85
Moreover, since uq,(Xo(0), Yo(0) + no(T), 0) < -K < 0,
nj(T) = O(e-(1-0h) = mj(T)
as
T 1 00;
i.e., both mj and nj are "functions of boundary layer type" in the terminology of Vishik and Lyusternik (1957). Finally, since
X +1(0) = x° i - m1(0),
the (j + 2)nd terms in the outer expansion are now completely specified. Thus, we are able to recursively determine the complete expansion termwise. Summarizing, we have THEOREM 3:
For each integer N > 0, the initial value problem
(4.1){4.2) under the hypotheses (Hi)-(Hii) has a unique solution for e sufficiently small which is such that N
x(t, e) = Xo(t) + 2 (X (t) + mJ-1(t/e)lej + eN+1 R(t, e) N
y(t, e) _
j=0
(Y (t) + nj(t/e)1 ej + e N+1 S(t, e),
where R(t, e) and S(t, e) are uniformly bounded throughout 0 < t < 1. Remarks
The expansion technique developed here follows that of O'Malley (1971a). With no serious complications, it extends to problems where
x and y are vectors. Likewise, by strengthening the hypotheses to achieve asymptotic stability of the solution to the reduced problem, the results are valid on the semi-infinite interval t > 0 [cf. Hoppensteadt (1966)]. Similarly, Cauchy problems for ordinary differential equations in any Banach space can be studied and used to obtain solutions to initial boundary value problems for partial differential equations [cf. Trenogin (1963, 1970), Hoppensteadt (1970, 1971), and Krein (1971)]. Likewise, Miranker (1973) has used similar methods to
4. NONLINEAR INITIAL VALUE PROBLEMS
86
develop a numerical scheme for integrating stiff differential equations, and Murphy (1967) gave techniques for boundary layer integrations. As a direct application of Theorem 3, the reader should note the paper by Heinekin et al. (1967) which presents a problem in enzyme kinetics and the paper by Hoppensteadt (1974) which discusses an interesting genetics model. If v0, were positive, the solution of the initial value problem would
become unbounded as the positive parameter e -> 0 for t > 0. The corresponding terminal value problem, however, would be well behaved for e -> 0. ExAMPLE: We will now consider the nonlinear problem dx
dt =xy dy = y3 +y with x(0) = x°(e) and y(O) = y°(e) prescribed. Here the reduced problem
dXXY' dt
-Y3+Y=0
X(0) = x° has the three solutions:
For y$ > \/3
X1l)(t) = x$e`,
Y0111(t) = 1
X" )(t) = x$ e-`,
Y01'1(t)
X131(t)
Y(3)(t) = 0.
= xp,
_ -1
, we can construct a unique asymptotic solution
(x(1)(t, e),y(1)(t, e)) of the form (4.17) converging to (XW) (t),
0 < t < 1 since (HI)
v°y (X W) (t),
(t), t) = -2 < 0
(t)) for
87
1. THE BASIC PROBLEM
and
(Hii)
v0,(x$,y,0) = -3y2 + 1 < 0
for y between yg and
1.
Similarly, for y$ < --/3, there is a unique asymptotic solution of the form (4.17) converging to ft)(t), YW)(t)) for 0 < t < 1. Since voy
(X 13) (t), Yi3i(t), t)
= 1 > 0,
we would expect any solution to converge to (XW)(t), YW)(t)) as e , 0 unless y°(e) = 0.
To be more explicit, we will obtain the first few terms of the asymptotic solution (x(1)(t,e),y(1)(t,e)). For notational simplicity, we will drop the superscripts. The appropriate outer solution will have an expansion X (t, e) = xp e' + eX1(t) + e2 X2 (t) +
Y(t, e) = 1 + E y (t) + e2 Y (t) +
which satisfies the original system for t > 0. Equating coefficients of ej in the differential equations, for each j > 0, we have
dX1=x$e`Y+XI
0=-2Y, dX2=x$e`Y+X2+X1Y
0=-2Y-3Y2- dY, etc. Integration then implies
Y(t)=0 Xl (t) = X1(0)e`
88
4. NONLINEAR INITIAL VALUE PROBLEMS
and
Y (t) = 0 Met
X2(t) = X2
with X1(0) and X2(0) still undetermined. Unless, by chance, y0(e) = 1 + O(e3), then, the outer solution cannot be a valid representation of the solution near t = 0. To account for the nonuniform convergence there, we shall represent the solution in the form x(t, e) = X (t, e) + em(T, e)
y(t, e) = Y(t, e) + n(T, e),
where T is the stretched variable T = t/e and m and n tend to zero as T tends to infinity. Substituting into the original system [which is also satisfied by the outer solution (X, Y)], we find that the boundary layer corrections m and n must satisfy the nonlinear equations dm
_
dT = X (ET, e)n(T, e) + e( Y(eT, e) + n(T, e)) m(T, e) do = (1 - 3 Y2 (eT, e)) n(T, e) - 3 Y(eT, e) n2 (T, e) - n3 (T, e). WT
Likewise, by the initial conditions, x° (e) = X (0, e) + em(0, e)
y°(e) = Y(0, e) + n(0, e).
Substituting m(T, e) = m0 (T) + em1(T) + n(T, e) = no (T) + en1(T) +
into the system, then, we successively obtain equations for the
1. THE BASIC PROBLEM
89
coefficients (mj, nj). In particular, since Yo(t) = 1,
d = Xo(0)no(T) d dTO
no(T)(2 + 3no(T) + np(T))
and
no(0) = yo - 1
X(0)=x10 -mo(0). Further, since no(0)(y0 +yo02),
dT 1-0
Ino(T)I is initially decreasing provided y$ > 0, and, since 2 + 3n + n2
> K + K2 > K for n > -1 + K > -1, for all T > 0,
1no(T)I < Jno(0)Ie-'
provided y$ > K > 0. This assures us that we can obtain no(T) by solving the nonlinear integral equation no(T) = (yo - 1) -1 T no(s) (2 + 3no(s) + no (s)) is,
T>0
by successive approximations. [For this particular problem, we can actually integrate the differential equation for y2 as a Riccati equation from which it follows that
no(T) = (I -
(I -
= O(e-2r)
y'2)e-2T)-1/2 - 1
as T - oo. ]
Knowing no and asking that mo -> 0 as T - oo, we have
mo(T) = -xo r no(s)ds = O(e-2T).
90
4. NONLINEAR INITIAL VALUE PROBLEMS
In particular, note that this provides the initial value
X1(0) = x° + x$ f no(s) ds needed for the outer expansion. Continuing, the /equations for m1 and n1 form the linear system
m1 = xonl(T) + (X1(0) + Txo) no(T) + (1 + dTl
-(2 + 6no(T) + 3no(T))
no (T)) mo(T)
n1 (T)
n1(0) = y°. Integrating, then, n1 (T) = y° a -27 exp
[-3 1
T
(2no (s) + no (s)) ds
and, since m1 -> 0 as T - oo, M, (T) _ - f Ixon1(s) + (Xi(0) + Sxo)
no (S)
+ (1 + no(s)) mo(s)]ds
is also exponentially decaying as T -* oo. Taking X2 (0) = x2 - m1(0), we completely determine the third-order terms in the outer expansion. From only partial results, then, we have obtained x(t, e) = x$ e` + e[ (x° + x$ f - no(s) ds)e` - xo f / no (s) d s] + O(e2)
y(t, e) = 1 + no
/
+ e1
(t) e
yoe-zr/texp [-3 f t' (2no(s) + no(s))
]) + O(e2)
on any bounded interval 0 < t < T provided yo > 0. Here no (T) = r1
02)e)-1/2
-
(_yO
o
- 1.
1. THE BASIC PROBLEM
91
Note that Theorem 3 guaranteed the existence of the asymptotic while the constructed solution is valid solution only for yg >
for y00 > 0. Similarly, we could obtain a solution converging to (x$a `,-1) for all t > 0 provided y00 < 0, while the theorem implies x (t)
Y (t) 1
L 0
.5
I 1
t
-1
-2
FIGURES 12 AND 13 The solution x(t), y(t) for dx/dt = xy, x(O) = 1 and dy/dt = -y3 + y, y(O) prescribed in the two cases y(O) > 0 and y(O)
< 0.
4. NONLINEAR INITIAL VALUE PROBLEMS
92
this result only on bounded intervals for y00 < -'/3. Thus, it is likely that the expansions of Theorem 3 are valid under assumption (Hi) and a condition somewhat weaker than (Hii). (See Figs. 12 and 13.)
PROOF OF THEOREM 3:
Let us define
N
N
YN(t, E) = I y (t)ei
XN(t,e) = F, X(t)ei, j-0
j=0
N-1
N
((
nN(T,E) = F, nj(T)Ei.
mN(T,E) = I mj(T)EJ,
j-0
j-0
By the definitions of the X's, y's, mj's, and nj's, we have dXN
dt dYN
dt ,j
N
dT
= u(XN, YN, t, e) + O(eN+1)
= V(XN, YN,t,e) + O(eN+l)
= u(X" + em", t" + n",t,e) - U(XN, YN9 t, E) + ` (ENe 1-01)
do N
dT
V(XN + EM N, yN + nN, t, E)
- V(XN yN t E) +
O(EN+1e-K(1-S)r)
where the 0 symbols hold for all t in [0, 1]. Substituting the solution (4.17) into the system (4.1) and the initial conditions (4.2), we obtain eN+1 d R = U(XN + EmN + EN+1R, yN + nN + EN+1 S, t, E)
- U(XN + em', YN + n N, t, E) + O(E N+1) + O(E N e K(1-S)r) N+2 E
_ = V (X N + EmN + dS dt
E
N+1 R, yN + n N + E N+1 S, t, E)
- V(XN + EmN, yN + nN,t,E) + O(EN+1),
93
1. THE BASIC PROBLEM
where the 0 terms are independent of R and S. Further, R(O,e) = S(O,e) = 0(1). Integrating, then, R and S will satisfy a pair of integral equations of the form R(t, e) = R°(t, e) + l t U(R(p, e), S(p, e), p, e) dp S (t, e) = S ° (t, e) + E fo V (R(P, e), S (p, e), p, e)
l
r XexpLE
dp,
P
where N+1lU(XN+EmN+EN+IR,YN+nN+EN+1S,t,E)
U(R,S,t,E) =
- U((XN + EmN, yN + n N, t' e)],
V(R,S,t,E) =
N+llv(XN+EmN+EN+IR,YN+nN+EN+1 S,t,e) E
- v(XN + EmN, yN + nN, t, e) - EN++1Sv,(X Nr+ EmN, yN + nN,t,E)J,7
R°(t, E) = 0(1)
C1 +
J
t
l l
1 1 + E exp[-ic(l - 8)s/E]) d] = 0(1),
and
S°(t,E) = 0(1) + 01
/'t
XJ
t
expL1E
J
vy(XN+EmN,YN+nN,s,E)ds]
By the construction and assumption (Hii), however,
1y(XN+EmN,YN+nN,t,e)
-2 <0
A
94
4. NONLINEAR INITIAL VALUE PROBLEMS
provided E is sufficiently small. Thus, S°(t,e) is bounded throughout [0, 11 and R(t, E) and S(t, E) can be uniquely determined by successive approximation for E sufficiently small [cf. Erdelyi (1964) and Willett (1964)]. Thus, there is one solution of (4.1)-(4.2) of the form (4.17).
2. TWO-PARAMETER PROBLEMS
Physical problems frequently involve the solution of boundary value problems with many small parameters. Thus, we are naturally led to consider boundary value problems for nonlinear systems of first-order equations with small parameters multiplying the derivatives.
Before examining initial value problems for such systems, we will first reconsider the linear constant coefficient problem Ey" + pay' + by = 0
y(0) = 1,
y'(0) = 0
on the interval 0 < t < 1 where the coefficients a and b are both positive and the small positive parameters E and p are such that p and
E/µ2 both tend to zero. Recall that we previously showed that the solution converges nonuniformly at t = 0 as the parameters tend toward zero and that the limiting solution elsewhere is the trivial solution of the reduced problem. Introducing the small parameters
and
E2 = E/p2 ,
we rewrite the second-order equation as dy El
dt=Z
El E2 T = -by - az,
0 < t < 1,
95
2. TWO-PARAMETER PROBLEMS
where y(O) = 1 and z(O) = 0, noting that the reduced system has only the trivial solution Ye(t) = Z00(t) = 0. Introducing the stretched variables Tl =
-t
and
El
T2 =
-t , El E2
then, note that both Tl and T2 - oo for t > 0 as El and E2 -* 0, but that T2 is then much larger than Tl. In terms of these variables, we have yP, (t) =
I
2(1 - r4E2 b/a2)1/z
X exp L-
+
2E2
{(l
+ (1 - 4E2 b/a2 )1 /2
/
(1 - alb - (1 - 462 b/a2 )1/2 / J e-nT,/o
(-1 + (1 - 4E2b/a2)1/2)
X exp [- a2 (-1 + (1 - 462 b/a2 )1/2
/J
e -a2
}.
More simply, we note that the solution has the form y(t;El,E2) = n(T1 ;E2)e
6'`i/a+E2g(T2;E2)e-aT2
z(t;El,E2) = p(TI;E2)e brI/a + h(T2;E2)ea2,
where n(T; E), p(T; 6), g(T; E), and h(T; E) all have asymptotic expansions
of the form 00
q(T; E) ^' I qj (T)EJ, j=0
as
E -* 0
where each qj is a polynomial in T of degree < j. Here, then, the outer expansion (valid away from t = 0) is asymptotically zero to all orders E E22. We note that it is essential in this example that the coefficients a and b/a be positive in order to have convergent behavior for t > 0 as El and E2 both tend to zero.
4. NONLINEAR INITIAL VALUE PROBLEMS
96
Now we will seek the asymptotic solution of the more general nonlinear system dx
dt = El
U(X, y, z, t; El , E2)
d = v (x,Y, Z, t; El, E2) z
El E2 T
(4.18)
= w(x, Y, Z, t; El, E2)
of three scalar equations on the interval 0 < t < 1 (or any bounded interval) subject to the initial conditions x(O;E1,E2) = x°(El,E2) (4.19)
Y(O; El, ED = y°(El,E2) z(0; El, E2) = Zo (El, ED
as the small parameters El and E2 both tend to zero. We will assume that u, v, w, x0, y0, and z° all have double asymptotic expansions in El and E2. For example, U(x,Y, z, t; El , E2)
00
JI.j2 -0
Ujij2 (x' y' z' t)El E22 .
That is, for any integer J > 0, U(X, Y, Z, t; El, E2)
_
Uj1j2 (x,Y, Z, t) El
E22
O(J+l)
Ui.j2)>(0.0)
jl+j2 <J
as
A = max(El, E2) -* 0.
Further, we shall assume that all coefficients in the expansions for u, v, and w are infinitely differentiable. It might seem somewhat more natural to instead consider systems
d = u(x, y, z, t; µ, K) dy dt = v (x,Y, z, t; p, K) dz K = w(x,Y, Z, t; µ, K), dt
97
2. TWO-PARAMETER PROBLEMS
where µ and K are both small parameters. As we shall see, however,
the asymptotic solutions can be completely different in the two extremes when K/µ -- 0 as µ -* 0 or µ/K -+ 0 as K -+ 0. In the first case, we have a special system of the form (4.18) with E, = µ and E2 = K/µ. In the second, we must interchange the roles of the y and z coordinates in order to obtain a system of the form (4.18). The case when µ = cK for some nonzero constant c leads to a one-parameter problem of the form (4.1) for a vector y. Analogous considerations apply for the infinite interval problem which describes the "shock layer" limiting behavior of the one-dimensional flow of a fluid when the viscosity and heat conductivity both tend to zero [cf. Gilbarg (1951)].
The reduced problem corresponding to (4.18) is dx dr= uao (x, y, z, t)
0 = vOO (x, y, z, t)
(4.20)
0 = WOO (x, y, z, t)
x(0) = 4. We shall suppose that continuously differentiable functions 4(X, Y, t) and p(X, t) exist such that woo (X, Y, 0, t) = 0
(4.21)
v (X, p, t) = v00 (X, p, q(X, p, t), t) = 0
(4.22)
and
and that the (unique) solution X00(t) of the initial value problem dX dt
= a(X' t)
X(0) = A exists for 0 < t < 1, where a(X, t) = u00(X, P(x, t), q(X,,p(X, t), t), t).
(4.23)
4. NONLINEAR INITIAL VALUE PROBLEMS
98
Then (Xoo (t), Yoo (t), zoo (t))
will be a solution of the reduced problem (4.20) where we define Yoo(t) = (Xoo(t), t)
(4.24)
and
zo0(t)
=
0(X00(1), YOO(t), t).
(4.25)
In order to prove that the original problem (4.18)-(4.19) has a solution converging to (X00, Y00, Zoo) away from t = 0 as El, E2 -* 0, we shall also assume that for some K > 0, woo-(X00(t), Y00 (t), zoo (t), t) < -K
(4.26)
6y(X00(t), Y00(t),t) < -K
throughout 0 < t < 1 and that woo,(xooYoo,IL,0) < -K
(4.27)
17y(x00, X, 0) < -K
for all µ between O(x$O,y0 , 0) and and and for all A between Y00(0) and y00. Note that these assumptions are directly analogous to hypotheses
(Hi) and (Hii) used for the one-parameter problem, except that Z00(0)
0(x000, y000, 0) in general.
Under these conditions, we shall obtain an asymptotic solution of (4.18)-(4.19) of the form x(t; E1 , E2) = X (t; El, E2) + E1 m(71 ; El, E2) + El Elf (T2 ; El, E2)
Y(t;El,E2) = Y(t;El,E2)+n(T1;El,E2)+E2g(T2,El,E2) Z(t, El, E2) = Z Q; El, E2) + P(T1 ; EI , E2) + h(T2 a El , E2 ),
(4.28)
2. TWO-PARAMETER PROBLEMS
99
where Tl and T2 are the stretched variables Tl = t/El
and
(4.29)
T2 = t/El E2
and X, Y, Z, m, n, p, f, g, and h all have double series expansions in El and E2. Further, m, n, and p tend to zero as Tl - oo while f, g, and h tend to zero as T2 - 00.
The formal expansion can be obtained in three stages by directly generalizing the procedure used for one-parameter problems. First, we determine the outer expansion (X (t; El, E2), Y(t; El, E2), Z(t; El, E2))
(4.30)
as a formal solution of the system (4.18) with first term (X00(t), Y00(t),
Z00(t)). Using (4.21), (4.22), and (4.26), higher-order terms will recursively satisfy linear systems with only X(0; El, E2) unspecified. Second, we ask that the intermediate expansion (X(EI TI ; El, E2) + El m(Tl ; El , E2 ), Y (El Tl ; El, E2)
+ n(TI;El,E2),Z(ElTl;El,E2) +p(Tl;El,E2))
(4.31)
satisfy the system (4.18) as a function of the lesser stretched variable TI. Equating coefficients when El = E2 = 0, we obtain the nonlinear system dmoo
A
= u o0 (Y--(n) Y--(n) + n00 () Z (0) + Poo (T) 0) , 00 1
1
- u00 (X00(0), Y00(0), ZOO (0), 0)
dd = v00(X00(0), Yoo(0) + noj(Tl ), ZOO(0) + poo(Tl ), 0) - vOO (X00(0), Yoo(0), Zoo (0), 0)
0 = woo(Xoo(0), YOO(0) + noo(Tl),ZOO(0) +poo(Tl),0)
- woo (X00(0), Y00(0), ZOO (0), 0)
=
woo(Xoo(0), Yoo(0) + n o(Tl ), ZOO(0) + poo(Tl ), 0).
100
4. NONLINEAR INITIAL VALUE PROBLEMS
Using the definition of 0, we take Z00(0) +poo(7l) = O(Xoo(0), YOO(0) + noo(rl),0) or
pOO(7l) = O(Xoo(0), YOO(0) + noo(rl),0) - O(X00(0), Y00(0),0).
Then, by (4.22), noo satisfies the nonlinear equation dT00
=
6
(Xoo(0),Yoo(0) + noo(rl),0) - U (X00(0), Y00(0),0)
noo(Ti)17 (noo(Ti ))
and, by (4.28), nw(0) = A - YOO(0). By hypothesis (4.27), then, a unique noo can be obtained by use of successive approximations in the nonlinear integral equation
noo(7l) = noo(0) +noo(s)V(noo(s)) ds,
7l > 0.
Further, we will have Inoo(7i)I < Inoo(0)Ie-"
for 7l > 0.
Knowing noo, we also have poo (7l) and moo (Ti) _ -
J
dd 00 (s) ds.
Moreover, poo (7l) = O(e ") = moo (7l) as -r, -* oo. Higher-order coefficients mj, nj, and pj will satisfy linear systems and can be completely determined recursively up to specification of n(0; El, E2). They will also decay exponentially as 7l -* oo. Third, we ask that the complete expansion (x(ElE2T2;E1,E2),y(ElE2T2;El,E2),Z(E1E2T2,El,E2))
(4.28)
2. TWO-PARAMETER PROBLEMS
101
satisfy the system (4.18) as a function of the greater stretched variable z2. When E1 = E2 = 0, this implies that doo = uoo(xo0,A, Zoo(0) + poo(0) + hoo(r2), 0)
- uoo(x$o,yo°o, zoo (0) + Poo(0), 0) dgoo
= v00(xo0,yA0, Zoo(0) + poo(0) + hoo(r2), 0)
- voo(x$o,yoo, zoo(O) + poo(0), 0) d oo = woo(xo0,yo0, Zoo(0) + poo(0) + hoo(72), 0)
- woo(xoo,A, Zoo (0) + poo(0), 0) hoo(72) n'(hoo(72)).
Note that hoo(0) = zoo - Zoo(0) - poo(0) = zoo - $(xoo,yoo, 0) and the condition (4.27) and the differential equation imply that Ihoo(72)I will decay to zero monotonically as z2 increases. Moreover, foo, goo,
and hoo will all be uniquely determined as exponentially decaying terms as z2 - oo. Higher-order terms of f, g, and h will likewise be recursively determined as exponentially decaying terms up to specification of h(0; E1 , E2). Applying the initial conditions for x, y, and z termwise successively yields the unknown initial conditions for X, n, and h. Thus, the expansion (4.28) can be formally obtained. Details of
the proof and expansion procedure are given by O'Malley (1971a) (with a reversal in subscript notation). To summarize, we observe that the asymptotic solution obtained is a sum of functions of t, of Tj = -r/E,, and of -r2 = tIE1 E2. The expansion
process first finds the outer expansion depending on t, then constructs
the boundary layer correction depending on the least singular stretched variable TI, and, finally, adds the contribution due to the most singular stretched variable z2. (One often refers to such boundary layer corrections as being of thickness E, and E, E2, respectively. The thicker correction is added first.) Many-parameter problems should be
4. NONLINEAR INITIAL VALUE PROBLEMS
102
treated in analogous fashion. An application of this technique to obtain asymptotic solutions to certain Cauchy problems in a Banach space is given by Gordon (1974), while Chen and O'Malley (1974) treat a problem in chemical flow reactor theory.
DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
3.
In this section, we wish to consider the initial value problem consisting of the nonlinear differential-difference equation
z(t) = f (t, x(t), x(t - 10' At - µ))
for
t>0
(4.32)
and the initial condition
x(t) = 0(t)
for
-µ < t < 0
(4.33)
as the positive delay parameter µ tends to zero. We shall assume that (Hi) f (t, x, y, u) and 0(t) are infinitely dtfferentiable in all arguments and that they are independent of µ, (Hii)
the nonlinear reduced problem X0 (t) = f (t, Xo (t), Xo (t), X0 (t)) (4.34)
X0(0) = 4(O)
has a unique continuously differentiable solution X0(t) on some interval
0
(Hiii) for some K > 0, I f (t, X o (t), X o (t), X o (t))I < e ' < 1
(4.35a)
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
103
for 0 < t < T, and Ifu(0,$(0),$(0),u)I < e-' < 1
(4.35b)
for all u such that I u 1 < IXo(0)I + IXo(0) - 4(0)I.
[By the implicit function theorem, note that the last assumption implies (among other things) that Xo (t) can be uniquely obtained as a function of X0 and t.] Under these conditions, the solution x(t, µ) will
converge to X0(t) as µ -* 0 on the closed interval 0 < t < T. Derivatives of the solution, however, will converge nonuniformly near t = 0.
Before proceeding, we note that the problem could be solved by a stepwise integration scheme for moderate values of µ. We would define
x(t) = x0(t) = c(t)
for
-µ < t < 0
and then, for each integer j with 1 < j < 1 + T/µ, we would set
xj(t) =f(t,xi(t),xj-I(t - µ),xj-I(t - µ))
xi((j -
1)µ) = xj_I ((j - 1)1k)
for (j - 1)µ < t < jµ, so that x(t) will generally be discontinuous at the values t = kµ, k > 0. Note that this method is unsatisfactory when the stepsize µ is too small. Instead, asymptotic methods are then appropriate. They have been given by Vasil'eva (1962) and O'Malley (1971b). A different type of singular perturbation problem for differential-difference equations is discussed in Cooke and Meyer (1966). A heuristic connection with familiar singular perturbation problems results if we consider the example
x(t) = ax(t - µ), x(t) = o(t).
Ial < I
104
4. NONLINEAR INITIAL VALUE PROBLEMS
Expanding . (t - µ) as a power series in µ, we have
z(t) + ... ).
z(t) = a(.z(t) - µx(t) + 2
Terminating this series after two terms, note that the "truncated problem"
pa.X(t) + (1 - a) z(t) = 0
z(0) = 0) X(0) _ 40), and the full problem will both have solutions converging to the constant solution X0(t) = q(0) of to (t) = a lo(t) Xo(0) = $(0) provided a > 0. Conclusions in certain applied literature involving differential-difference equations have sometimes been based on such analogies. The analogy can be misleading, however, and such conclusions should be considered dubious. For singularly perturbed ordinary differential equations, the number of initial conditions required
for the full problem and the reduced problem differ by a finite number. For the differential-difference equation (4.32), however, an infinite number of initial derivatives are prescribed by (4.33) while the limiting differential equation for X0(t) requires only an initial value. We shall seek a solution x(t, µ) of (4.32)-(4.33) of the form
x(t, µ) = X (t, µ) + µm(9,µ)
(4.36)
for t > 0 where the outer solution 00
X (t, µ) - l7, X (t) µJ
satisfies the difference-differential equation for t > 0 and each term mj of the boundary layer correction 00
m(9,µ) ' l7, mi (0)
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
105
tends to zero as the boundary layer coordinate tt
µ
tends to infinity. Away from t = 0, then, the asymptotic solution will be determined by the outer expansion. Since the initial function (4.33) is independent of µ, (4.36) implies that
X(0) = -mj_,(0)
for each j > 1.
(4.37)
Thus, to calculate the outer expansion it is necessary to know the initial value m(0, µ) of the boundary layer correction. Note that the definition of X0(t) implies that the outer expansion X (t, µ) satisfies (4.32) when µ = 0. Equating coefficients of µ, X, must satisfy the linear differential equation A (t) = fx(t,Xo,X0,Xo)X1 + f (t,X0,Xo,Xo)(X1 - Xo)
+f(tIXo,Xo,Xo)(Xi - Xo) or
Xi (t) = A(t) Xi (t) + B0(t),
where
A(t) = (1 x
Xo, Xo, Xo))^' (f(t,X0,Xo,Xo) + f (t,Xo,Xo,Xo))
and Bo(t) is determined from X0(t). [Note that 1 - f is invertible for 0 < t < T by (4.35).] In general, each coefficient X(t) for j > 1 will be determined as a solution of a linear differential equation of the form
!j(t) = A(t) X (t) + Bj_, (t),
(4.38)
where Bj_1 is a smooth, successively known function. Thus, the X's
can be determined recursively on 0 < t < T up to specification of their initial values [by (4.37)].
106
4. NONLINEAR INITIAL VALUE PROBLEMS
The boundary layer correction terms mj(9) are determined successively by stepwise integration on the intervals p < 9 < p + 1, p > 0. Since the complete expansion (4.36) and the outer expansion X(t,E) both satisfy equation (4.32), the boundary layer correction m(9,µ) must satisfy m9(9,µ) = f (µe, X (µe,µ) + µ m(9,µ), $(µ(e - 1)), $(149 - 1)))
- f (0, X (µe, µ), X (µ(e - 1), µ), X(49 - 1), µ)) for 0 < 9 < 1
(4.39a)
M0 (9, µ) = f (9,X(µ9,µ) + µ m(9, µ), X (1(9 - 1), µ) + µm(9 - 1, µ),
1(µ(9 - 1),µ) + m9 (9 - 1, µ))
- f (µe, X W, µ), X (µ(9 - 1), µ), X(49 - 1), µ)) for 9 > 1.
(4.39b)
For µ = 0, then, m°(9) must be the continuous solution of moo (0)
=f(0,X0(0),0(0), (0)) -f(0,X°(0),X°(0),9°(0)) for
0<9<1
(4.40a)
and
moo (0) = f(0,Xo(0),Xo(0),X°(0) + moo (0 - 1))
- f(o,X°(o),X°(O),X°(o))
for
9 > 1.
(4.40b)
Thus, mo9 is stepwise constant. Setting
m09(9)=G°
for p <9
p>_0,
we have m°(9) = m°(0) + fo mo9(s) ds or P-1
m°(9)=m°(0)+7, GO +(9-p)G° r-o
for p<9
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
107
Since we asked that mo -+ 0 as 0 -+ oo, we select 00
X1 (0) = -mo(0) = I G°,
(4.42)
r=0
assuming this limit exists. To clarify this issue, we introduce the mapping F such that
Fu = F0u = f (0, $(0), $(0), u) and
for each j > 1.
Flu = f (0, 0(0), q(0), Ff-' u)
Note that (4.34) and (4.35) imply that F is a contraction mapping with
10(0) as its (unique) fixed point [see, e.g., Hale (1969) for the definition of a fixed point and a statement of the contraction mapping principle]. Hence, 10(0) = liim00 FP$(0) and since G0 = FP+'$(0) - X0(0) (4.42) becomes
X1(0) = lim rrt (Fr+ 4(0) ) - (P + 1)Xo(0)1 and the limit is finite [cf. Vasil'eva (1962)]. Note that (4.40) and (4.35) imply that Imoe(0)I <_ a KImoe(0 - 1)1
for 0 > 1
so that (estimating freely) Imoo(0)I S I1o(0) - p(o)le
9e
and, by integration,
mo(0) = 0(e-40)
as
0 -+ oo.
108
4. NONLINEAR INITIAL VALUE PROBLEMS
Equating coefficients of µ/ in (4.39) successively for each j > 1 implies the linear differential-difference equations mje(9) = Y-1(9) + W(9)mio(0 - 1)
for 9 > 0, where Y_j is a linear combination of m!(9), m!(9 - 1), and m1B(9 - 1) for 1 < j and
for 0<9
0
W(9) = fu (o,-0(0),.0(0),go (0) + moo (0
-
1))
for 9 > 1.
Thus, we have
for p<9
mje(9)=Gp(9)
and the Gp's are determined, in turn, for p = 0, 1, 2, .... Integrating stepwise, we have
mi (9) = mi (0) +
I
1f
1+1
Gi (s) ds)
!=0 \
+f Pe Gp (s) ds
for p<9
(4.43)
Note that the derivatives of the ml's will generally be discontinuous at integer values of 9. Since mj -* 0 as 9 - oo, we must select 00
Xi+1(0) = -mi(0) = I
(f 1+1
G/(s)ds).
(4.44)
Further, since (by induction) V_1(9) = O(e-x(1-8)')
as
9
00,
integration implies that mj(9) is also exponentially decaying as 9 -* 00 and that the sum in (4.44) is finite. Summarizing, we have
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
THEOREM 4:
109
Under the assumptions (Hi)-(Hiii), the initial value
problem (4.32)-(4.33) has a unique solution x(t, µ) for µ sufficiently small which is such that, for each integer N > 0,
x(t, µ) = X0(t) + E (X (t) + mj-1(t/µ))µ' + µN+1 R(t, µ),
j=
1
where the mj's -+ 0 as t/µ - oo and R(t, µ) is uniformly bounded throughout 0 < t < T. The theorem can be proved as has been done by O'Malley (1971b). No further complications arise if the functions f and 0 have asymptotic expansions as µ -* 0. We note that the steplike construction of the boundary layer correction (cf. Fig. 14) is quite novel compared to
the preceding, but that the result is still exponentially decaying as
0 - co. M(6, E)
I
1
I
I
1
2
3
4
e= c/v
FIGURE 14 The steplike nature of the boundary layer correction for the differential-difference problem (4.32)-(4.33).
When f = 0, Eq. (4.32) is called an equation with retarded argument. For such problems, calculation of the boundary layer correction terms becomes considerably simplified. In particular, one
4. NONLINEAR INITIAL VALUE PROBLEMS
1 10
has mj (9) = 0
for
9 > j;
i.e., these terms ultimately become zero. Here mje = Y_j is known successively, and we have
f' Y_, (s) ds e
mi (e) = 0
for
9<j
for
9 > j.
This implies the initial value
Xi(0) = fo
Y-2(S)d
needed for the outer expansion. For example,
mo(0)=0
for 9>0
and
m, (9)
I
f (0, 0(0), 0(0), 0) (X0(0) - (0)) f' (s - 1) ds
for 0 < 9 < 1
0
for 9> 1
so
X1(0)=0 and
X2 (0) = If (0, 0(o), 0(o), 0) (Xo (o)
Since mj(9) = 0 for 9 > j, Theorem 4 implies
- o))
-
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
COROLLARY:
111
Consider the initial value problem (4.32)-(4.33) when f is
independent of u and assumptions (Hi) and (Hii) hold. Then, for µ sufficiently small, there is a unique solution x(t, µ) for 0 < t < T which is such that, for each integer N > 1, N
x(t, µ) = 7, X(t)µj + O(µN+I) j =0
for µ(N - 1) < t < T.
We note that equations with small retarded arguments occur in many applications. For example, they have often been used in population models to improve on the classical Volterra-Lotka models which involve no delay. There, the biologist Hutchinson (1948) states "there is a tendency for the time lag to be reduced as much as possible by natural selection." Thus arguments for small delay problems are
found throughout the literature on epidemics and population. An interesting application of these methods to an optimal control problem is given by Sannuti and Reddy (1973). Let us examine in detail a special problem where we have a linear difference equation in z(t), i.e., EXAMPLE:
t>0
z(t) = ax(t - µ), x(t) = 0(t),
-µ < t < 0
for jal < 1. Here the reduced problem
z(t) = a. (t) x(0) _ 0(0) has the unique solution X0(t) = q(0). Further, if we seek an outer expansion
X(t,µ)
$(0) + pX1(t) + µ2 X 2 (t) + µs(
)
112
4. NONLINEAR INITIAL VALUE PROBLEMS
for t > 0, we formally have I (t) + µ212(t) + µ3(...
)
and
x(t - µ, µ) - All (t) + µZ (Xz (t) - XI (t)) + A'( ... ). Equating coefficients, then, in the difference equation
X(t, µ) = aX(t - µ, µ) implies
XI (t) = all (t) Xz (t) = a(X2 (t) - XI (t)),
etc. Thus, each X(t) is constant and the values X(O) will be determined through the boundary layer correction terms. Moreover, since the outer solution is a constant function of t, a boundary layer correction at t = 0 will be clearly necessary unless 0(t) = .0(0). Representing the solution in the form x(t, µ) = X(0,µ) + µm(0,µ)
for t > 0 and 0 = t/µ [since X (t, µ) = X(0, µ)], the boundary layer correction m(9,µ) must satisfy the difference equation
a(µ(9 m9 (0, µ)
-
am9(0 -
1)), 1, µ),
0<0<1 0 > 1.
Thus,
m9(0,µ) = 0+I4(o - p - 1)),
p < 0 < p + 1, p > 0.
3. DIFFERENTIAL-DIFFERENCE EQUATIONS WITH SMALL DELAY
113
Integrating, then,
m(9,µ) =m(0,µ) +
(fk+1 P
k1
me(S,µ)d) + f 0 me(S,µ)ds
for p<9
I ak+I) ( 0 m(0,µ) _ -lim1 Jl P''00 k=0
S)d
1
(17 (0(_1k) a a) µ
- 0(0)).
Thus,
m(e,µ) = aµ [-ia° ) + $(µ(e - p - 0) ap+l µ(1
- a) [a $(-µ) + (1 - a) $(µ(9 - p - 1)) - $(0 )]
for p<90. In particular, it is important to note that m decays like a-90 as 9 -+ oo for K = -In Ial. Finally, the initial condition implies q(0) = X(0,µ) + µm(0,µ), so we have obtained the,outer expansion as
0,10 = X0,10 =
1
1 a [a 0(-µ) + (1 - 2a) 0(0)]
Thus, the asymptotic solution for t > 0 is given by the outer expansion with
X (t) = (1 a a)(-1)'
i
J(0),
1 > 0.
114
4. NONLINEAR INITIAL VALUE PROBLEMS
Near t = 0, however, the boundary layer correction is important. It is given by µm(t/µ, µ) ^-- µ J o m;(t/µ)µi, where aP+1
mj (9) =
$0+1)(0)(a(-1)' + (1 - a)(B - p - 1)'+'),
1-a(j+1)!
p0. Alternatively, using the greatest integer function uniformly valid expansion for t > 0 given by
x(t, µ) - $(0) +
x (I
a
\1
+ a[IIA]
a
/
I
we have a
j
{-a + (1 - a)( l
µ +
[t])J+'})
CHAPTER .5
NONLINEAR BOUNDARY VALUE PROBLEMS
1. SOME SECOND-ORDER SCALAR PROBLEMS We would now like to consider three examples of boundary value problems of the form
d 2
e
t2
+1 dt,t,e),
x(0, e) = a(e),
0
x(l, e) = /3(e)
which illustrate the varied phenomena possible in the limit as e --j 0. In Section 2, we will obtain asymptotic solutions for a restricted class of problems. 115
116
5. NONLINEAR BOUNDARY VALUE PROBLEMS
EXAMPLE 1:
Coddington and Levinson (1952) introduced the exam-
ple
d'x
¢x
dtz + dt +
x(0) = a,
dx 13 JJ = 0 dt
x(1) = P.
Here the differential equation can be explicitly integrated (since it is a Bernoulli equation in dx/dt). Applying the boundary conditions, however, one finds that the problem has no solution for a sufficiently small when the constants a and /3 are unequal. As this example would suggest, progress toward obtaining asymptotic solutions to problems like (5.1), where F(x, dx/dt, t, 0) is nonlinear in dx/dt, has been limited. Vishik and Lyusternik (1958, 1960) have discussed the general case where F(x, dx/dt, t, 0) = O((dx/dt)'), 1 > 0, as I dx/dt I --> oo. They conclude, under appropriate assump-
tions, that x'(0) will remain bounded as e - 0 and nonuniform convergence as e - 0 cannot occur at t = 0 if 1 > 2. (Note that 1 = 3 in Coddington and Levinson's example.) However, it is possible to have x'(0) = O(e hI(2 1))
if
0<1<2
and
x'(0) = O(ec%)
for some C > 0 if 1 = 2.
The same conclusions hold at the endpoint t = 1. We note that these estimates on endpoint behavior could be helpful in selecting appropriate stretched variables (or boundary layer coordinates). Vishik and Lyusternik also give complicated expansion techniques for such problems which have been generalized to systems by Kasymov (1968).
Similar problems are also discussed by Yarmish (1972). Cohen (1973b) obtains the limiting behavior for a restricted class of nonlinear problems.
117
1. SOME SECOND-ORDER SCALAR PROBLEMS
EXAMPLE 2:
The nonlinear problem dZx
dx
dx
Z dtZ+dt-wt=0
x(0) = 1,
x(1) = 0
has the solution x(t, e) = -e In [ 1 + e- '/e - e I/e ]
so x -* 0 as e - 0 away from t = 0. Further dx dt
= _e-'le[l + e-'le - e Ile]'
so x'(0) = -e'le, as Vishik and Lyusternik's results anticipate.
Other problems produce nonuniformities in the interior of the intervzl [0, 1]. Thus, consider EXAMPLE 3:
(dx)3 =0 dtZdx ddt d2x
x(0)=0,
x(1)=I.
We note that this equation differs from that of Coddington and Levinson only in the sign of the "damping" term. The solution here, however, is defined as e -- 0 and satisfies x(t, e) = e log
[Je`'-1/2)/e + [ 1 +
aeZ('-1/Z)/e
+ O(e '/Ze)]'/Z] + O(ee 'IZe).
Thus, the limiting solution as e - 0 (cf. Fig. 15) is Xo(t) =
(0 Sl
t - 1/2
for for
0 < t < 1/2 %
1/2
.
5. NONLINEAR BOUNDARY VALUE PROBLEMS
118
x0 (t)
1
5
I
0
FIGURE 15
1
.5
0t
The limiting solution for ex - x + (z)3 = 0, x(0) = 0,
x(1)=},e-0. Note that this limiting solution satisfies the reduced equation through-
out [0, 1] but has a discontinuous derivative at t = }. We note that other such "solutions" of the reduced equation satisfying the boundary conditions are given by
0
t,
{I,
#
{a+r,
0
4
and
fi(t)- ft,
-r,
0
These, however, do not provide limiting solutions for the original problem as e - 0.
2. SECOND-ORDER QUASI-LINEAR EQUATIONS
119
Such "angular" limiting solutions result for a class of nonlinear equations [cf. Haber and Levinson (1955)]. Further, asymptotic expansions of such solutions can be obtained by the same techniques used in Chapter 4 for initial value problems [cf. O'Malley (1970b)]. Another type of nonlinear problem with angular solutions is typified by Example 4 of Section 1.1, while the "transition layer" problems of Fife (1974) represent another type of interior nonuniformity.
SECOND-ORDER QUASI-LINEAR EQUATIONS
2.
In this section, we shall consider the quasi-linear problem
ex" + f(t,x)x' + g(t,x) = 0 x(0,e) = a,
(5.5)
x(l,e)
for prescribed constants a and 8 under the hypotheses that (Hi)
the terminal value problem f (t, X) -X' + g(t, x) = 0 (5.6)
x(1)=1B
has a solution X0(t) for 0 < t < 1 such that, for some K > 0,
f(t,Xo(t)) >
K
(5.7)
there, and that
f(O,X) > K for all X between a and X0(0).
(5.8)
120
5. NONLINEAR BOUNDARY VALUE PROBLEMS
We will also suppose that f (t, x) and g(t, x) are infinitely differentia-
ble in the indicated domains. We note that this problem has been discussed by several authors, including Coddington and Levinson (1952), Wasow (1956), Willett (1966), Erdblyi (1968a), and O'Malley
(1968a). Recall that the linear problem where f (t, x) = a(t) and g(t, x) = b(t)x was solved (by methods quite different than below) in Section 3.1. Under the hypotheses (Hi) and (Hii), we shall obtain an asymptotic
solution to (5.5) of the form x(t, e) = X (t, e) + (T, e),
(5.9)
where 00
X (t, e) --- I X (t)ej j0 0 S(T, e)
I j(T)ej -0
as e - 0 and T is the stretched variable T = t/e.
Here the terms Ej and their derivatives dEj/dT should tend to zero as
T - oo. Away from t = 0, then, we will have x(t, e) -+ X (t, e) as e
0.
Proceeding formally, we must ask that the asymptotic expansion of the outer solution X(t,e) satisfy the differential equation of (5.5) for
t > 0. Since the first term X0(t) satisfies the nonlinear reduced problem (5.6), the second term X, must satisfy the linear equation
f(t,X0)Xi + (ff(t,X0)Xo + gx(t,Xo))Xi = -X'o, and higher terms X will satisfy analogous differential equations with only a different, but successively known right-hand side. Since the boundary layer correction (T, e) is asymptotically negligible at t = 1,
2. SECOND-ORDER QUASI-LINEAR EQUATIONS
121
we must have X(l,e) = X13. Hence X;(l) = 0 for each j > 1 and further terms in the outer expansion can be uniquely determined successively from their linear differential equations. Thus, the asymp-
totic solution away from t = 0 can be formally determined without use of a boundary layer correction. Note that this contrasts with the situation for initial value problems where the outer expansion cannot be obtained without determining the initial values of the boundary layer correction. Since the outer solution satisfies (5.5), the boundary layer correction (T, e) must satisfy the nonlinear equation zr
d d22
+ f 1 ET, X (ET, e) + (T, e))
dT
+e [ (f (ET, X (ET, e) + (T, e)) - f (ET, X (ET, E)))
it-
+g (ET, X (ET, E) + (T, E)) - g(ET, X(ET, e)) ] = 0
(5.10)
and the initial condition (0, e) = a - X (0, e).
When e = 0, then, we have l_o
dT2 +f(0) XO(O) + SO(T))
= 0.
Since both 0 and duo/dT - 0 as T - oo, 0 must satisfy the nonlinear initial value problem
-JOf0f(O,Xo(O)+r)dr (5.11)
to(0)=a-X0(0)
122
5. NONLINEAR BOUNDARY VALUE PROBLEMS
on the interval T >0. Because f remains positive, decreases monotonically as T increases. Thus, 0 exists for all T> 0 and, by (5.8), satisfies I6o(T)I S I6o(0)Ie "T
for T > 0.
In general, of course, we cannot obtain an explicit solution of (5.11), but we can obtain the unique solution through successive approximations.
Further terms of the boundary layer correction, however, satisfy linear equations z dTj
+f(0,X0(0) +o (T))
+ fz(0, XO(0) +
J_i
to (T))
fj=
Cj-1(T),
where the Cj_I's are known successively and are O(e-K('-s)7) as T -> 00
for any 8 > 0. Direct integration then implies that tj(T) =
-X(0)exp[-f7 f(0,x0(0) + to - J0 exp
(s)]
[- I f(0,X0(0) + to (S)) ds]
X fP , Cj-. (r) dr dp
(5.12)
and, since f (0, X0(0) + 60(T)) > K for all T > 0, 6j(T) = O(e (I-s)+)
as
T -> 00.
Formally, then, we have uniquely obtained an asymptotic solution x(t, e) of the boundary value problem (5.5). For any integer N > 0, let x(t, e) =
I (x (t)/ + tj(t/e))ej + eN+I R(t, N
e)
(5.13)
2. SECOND-ORDER QUASI-LINEAR EQUATIONS
123
Then it can be shown [cf. O'Malley (1969a)] that, for e sufficiently small, the boundary value problem has a unique solution which is of the form (5.13) with R(t, e) uniformly bounded throughout 0 < t < 1. If f (t, x) were negative, we would instead have nonuniform convergence at t = 1 with the appropriately altered hypotheses. Note that hypothesis (Hii) might be weakened to the condition that I
f (0, X0 (0) + r) dr >
K
for all values between a - X0(0) and 0 [cf. Fife (1973)]. As an example, consider the highly nonlinear problem
x(0) = a,
x(1) = 0.
The reduced problem
e"x' - (sin--)e2x = 0,
x(1) = 0
has the solution
X0(t) = -In (1 + cos
0
The first term 0 of the boundary layer correction must satisfy
dT0
exo(o)e'dr, - Jb o
X0(0) = a + In 2.
Thus,
60(T) = -ln[(l - eT-/2) +
je-1e T/2] = 0( '/Z),
124
5. NONLINEAR BOUNDARY VALUE PROBLEMS
t
-1.
FIGURE 16
The limiting solution of ex" + e'x' - (ir/2)sinirt/2 = 0,
x(0) = x(1) = 0.
so
e2-a
x(t, e) = -In 1 1 + cos 2 )1 1 - e-1/ +
\
\
e_hI
J
+ O(e)
for 0
(cf. Fig. 16).
Note that this method can also be used to obtain the asymptotic solution of Example 4, Section 1.1, in Cases 1 and 2. It does not apply, however, in Cases 3-6. Before proceeding, we wish to observe that the expansion procedure
developed here for ex" + f (t, x)x' + g(t, x) = 0 will not work when f (t, x) has a zero. In the special case, however, when f (t, x) = 0, i.e., ex" + g(t, x) = 0 x(0) = a,
x(1) = Q,
125
3. QUASI-LINEAR SYSTEMS
a limiting solution within (0, 1) will be given by any X0(t) such that g(t, xo(t)) = 0
provided gz(t,X0(t)) < 0 and gz(0,x) and gz(1,x) are appropriately restricted [cf. Vasil'eva and Tupciev (1960), Yarmish (1972), and Fife (1973)]. At t = 0, nonuniform convergence will generally require a
boundary layer correction in the stretched variable To = t/f , while at t = 1 there twill be nonuniform convergence involving the variable
Tj = (1 - t)IV°. 3. QUASI-LINEAR SYSTEMS
Generalizing the boundary value problem (5.5), let us consider the quasi-linear system dx dt = e
f(x,Y,t,E) = fi(x,t,E) + f2(x,t,E)Y
(5.14)
dt = g(x,Y, t, E)
g1 (x, t, E) + g2(x, t, e)Y
of two scalar equations on the interval 0 < t < 1 with the boundary conditions a, (e) x(0, e) + eat W AO, E) = a(e) //
(5.15 )
bl(e)x(l,e) + b2(E)Y(l,E) = N(e),
where the f, g,, a;, b,, a, and ft all have asymptotic series expansions as e -- 0 such that the coefficients in the expansions for f; and g; are infinitely differentiable functions of x and t. (A reason for introducing the e multiple of a2 will be given below.)
We note that the scalar problem (5.5) can be put in the form (5.14)-(5.15) as follows: dx
dt =Y e dt = -g(t, x) - f (t, x)Y x(O,E) = a,
x(l,E) = Q.
(5.16)
126
5. NONLINEAR BOUNDARY VALUE PROBLEMS
For f positive, recall that the limiting solution as e -+ 0 features nonuniform convergence at t = 0 such that x is bounded and y is unbounded there. By analogy, we shall seek conditions which imply convergence of
the solution of the problem (5.14)-(5.15) away from t = 0 to a solution of the reduced problem dx dt =fi(x,t,0) +f2(x,t,0)y 0 = g1 (x, t, 0) + g2(x, t, 0)y
b,(0)x(l) + b2(0)y(1) _ /0(0) as e - 0. To satisfy this problem, we would need y
_
g,(x,t,0) g2(x,t,0)
and
dx
(x, t, dt = fi (x, t, 0) - f2(x, t, 0) gl $2(x, t, 0)
0),
where the terminal value x(l) is a root y of the nonlinear equation
P(y) = b, (0)-y - b2(0) gl(Y,1,0) 92(7,1,0)
- /B(0) = 0-
Selecting some root y, define X0(t) and Yo(t) by asking that X0 satisfy the terminal value problem dXo
gl (Xo, t, O)
0) dt = f (Xo, t, 0) - A (Xo, t, g2(Xo, t, 0)
Xo(1) = with
Y(t) _ _g1(X0,t,0) 92 (Xo, t, 0)
Then (X0, Y) will satisfy the reduced problem.
(5.17)
3. QUASI-LINEAR SYSTEMS
127
Let us assume that (Hi) a solution (X0(t), Y(t)) of (5.17) is defined on 0 < t < 1 and is such that
g2(Xo(t), t, O) < -K
(5.18)
there (for some K > 0) and that gz(Xo(1), Y0(1),1,0) 0.
bi(0) - b2(0)
$2(Xo(1),1,0)
(5.19)
Note that such a solution of the reduced problem will be uniquely determined once X0(1) is selected and that (5.19) implies that X0(1) is a simple root of P(y) = 0, i.e., P'(y) 0. [Recall that for the problem
(5.16), (5.18) requires the reduced problem to have a solution on 0 < t < 1 such that the positivity condition (5.7) holds. For (5.16), (5.19) is always satisfied.]
We shall now seek an asymptotic solution of (5.14)-(5.15) of the form X(t, E) = X (t, e) + (T, E) (5.20)
At' E) = Y(t, E) +
'n(T, e),
where X(t,E)
-
00
Xj(t)ej, rr
,
OT, E) - I 00 Sj(T)Cj,
j-o
Y(t,E) ,., I y(t)ej j-0 00
'n(T,e) - 17Ij(T)Ej
j-o
as e -- 0. Here (X0, YO) is the solution of the reduced problem under consideration and the terms j and q1j all tend to zero as the stretched
128
5. NONLINEAR BOUNDARY VALUE PROBLEMS
variable T = t/e tends to infinity. Away from t = 0, then, (x,y) will be asymptotically represented by the outer solution (X(t,e), Y(t,e)). The boundary layer correction ( (T, e), q(T, e)/e) is needed to obtain the nonuniform convergence at t = 0. Note, in particular, that we
anticipate that y will generally be unbounded at t = 0. We shall proceed to formally generate the expansions, giving the additional hypothesis (Hii) when needed to construct the boundary layer correction.
The outer solution (X (t, e), Y(t, e)) must satisfy the system (5.14) and the terminal boundary condition. When e = 0, these equations are satisfied by the solution (X0(t), Yo(t)) of the reduced problem. Higher-order coefficients in the expansion must therefore satisfy linear systems of the form
dX
/
dt = ff1 xo(t), NOW, t,o\ lx,. + f2 (Xo(t),t,o)Y + P_1(t)
0 = gX(Xo(t),NOW, t,0)X +g22(xo(t),t,0)Y+ Q;_, (t)
on 0 < t < 1 plus the boundary condition bl(0)X;(l) + b2(0) Y(1) = 4_1, where P-,, Q;_j, and j_j are known successively in terms of preceding coefficients. By assumption (5.18), Y(t) is determined as a linear function of X,(t); i.e.,
Y(t) _
-[gx(Xo, Y , t) X (t) + Qj-1(t)] 92(Xo, t, 0)
Thus
bj(0) -
b2(0)gX(xo(l), Yom, 1,0)
x(1)
82(xo(1), 1,0)
is known and assumption (5.19) implies that the terminal value x(1) is uniquely determined. Finally, integrating the resulting linear termi-
nal value problem for X(t) determines both X and Y uniquely
129
3. QUASI-LINEAR SYSTEMS
throughout 0 < t < I. Assumption (Hi), then, suffices to formally obtain the outer expansion (X, Y) uniquely. Note that no knowledge of the boundary layer correction terms is required to determine this outer expansion, which hopefully provides an asymptotic solution away from t = 0. Since the outer solution satisfies the system (5.14), (5.20) implies that the boundary layer correction must satisfy the nonlinear system
d, = ii(T,e)f2(X(eT,e) + S(T,E),ET,E) + elf, (X (ET, E) + (T, E), ET, E) - f, (X (ET, E), ET, E)
+ Y(ET,E)(f2(X(ET,E) + (T,E),ET,E) (5.21 a)
-f2(X(ET,E),ET,E))J d,q WT_
= 1j(T, e)g2 (X (ET, E) + (T, E),
ET, E)
+ E[gl (X (ET, E) + S (T, E), ET, E)
gl (X (ET, E), ET, E)
+ Y(ET, E) ($2(X(ET, e) + (T, E), ET, E) - 92W ET, e), ET, E))
J
(5.21b)
as well as the initial condition al
e) + a2 (00, E) = a(e) - al (e) X (O, e) - Eat (E) Y(0, e). (5.22)
At e = 0, then, we ask that rr ='q0(T)f2(XO(0) + SO( T),0,0)
(5.23) d,qo
dT
=
rI0(T)g2(XO(0) + SO(T),0,0)
for T > 0 and al(0)t0(0) + a2(0)r10(0) = a(0) - al(0)XO(0).
(5.24)
130
5. NONLINEAR BOUNDARY VALUE PROBLEMS
Proceeding freely, we have drlo -_ dT
g2(X0(0) + 0(T),0,0) d6o dT
fz (X0(0) + o(T), 0, 0)
and integrating
PE()$2(X0(0)+r,0,0) %(T)
=I
f2(Xo(0) + r,0,0)
Jo
dr (5.25)
since both 0 and Jo -* 0 as T -* oo. Thus, the initial condition implies that to(0) must be a root A of the nonlinear equation gz X0(0) + r, 0, 0)
Q(A) = a, (0)A + a2(0)
X
fo
dr - a(0)
f2 (XO (0) + r, 0, 0)
+ a, (0)X0(0)
= 0.
Picking some root A, 60 will satisfy the nonlinear initial value problem dt0 dT
_
to
fz `X°(0) + o' 0' 0)
J°
$z (X0(0)
+ r, 0, 0 dr
f2 (X0(0) + r, 0, 0)
(5.26)
z;0 (0) = A.
In particular, to(T) = Jo(T) = 0 if A = 0. Otherwise, additional hypotheses are necessary. Thus, we will assume (Hii)
f
The equation
$2(0(0) + r,0,0)
A
a2(0)
f2(Xo(0) + r,0,0) J
dr = a(0) (5.27)
has a solution A such that
a1(0) + a2(0)
g2(Xo(0) + A,0,0) 0.
f2(X0(0) + A,0,0)
(5.28)
3. QUASI-LINEAR SYSTEMS
131
Further, for some K > 0, suppose
$2(R,0,0) < -K
(5.29)
1 f2(R, 0, 0)I > K
(5.30)
and
for all values of R between X0(0) and X0(0) + X.
We note that (5.28', implies that A is a simple root of (5.27), i.e., Q'(A) 0. Further for the problem (5.16) hypothesis (Hii) simply requires that (5.8) hold and there is then only one determination of X.
In general, (5.26), (5.27), and (5.30) imply that
is
negative, so I60(T)I decreases monotonically as T increases. Thus, 60 will exist for T > 0 and we will have IAIe-'
for T > 0,
where a is such that f2 (R, 0, 0)
92 (S, 0, 0)
< _CF < 0
A(S9090) for all values R and S between X0(0) and X0(0) + X. The estimate for 0 implies that I'qo(T)I also decreases monotonically such that
%(T) = O(e °')
as
T -* 00.
In general, of course, it is necessary to solve (5.26) by successive approximations because the differential equation cannot be explicitly integrated. Knowing 0, however, rl0 is given by (5.25). From higher-order terms in (5.21), we obtain the linear variable coefficient system dT
Jj(T)J2("0(0) + 60 (T), 0, 0)
+ Sj(T)f2x(XO(0) + SO(T), 0, 0) ' O(T) +
-I (T) (5.31)
d,qj dT
rij(T) g2 (XO(0) + 60(T), 0, 0) + 6j(T)g2z(XO(0) -I- SO(T), 0, 0) Jo(T) + Yj-I (T),
5. NONLINEAR BOUNDARY VALUE PROBLEMS
132
Qj_1 are known successively and satisfy
where
j-1(T) = O(e-51) = Qj-1(T)
as
T -+ 00
for any a such that 0 < & < min (a, K). Further, the initial condition (5.22) implies that (5.32)
a2(0) nj(0) = jlj-1,
where &j- 1 is also known successively. Rearranging (5.31), we have dry j
_
d (jg2(xo(o) + 0(T), 0, 0)
dT
tT),0,0)
dT
Qj-1(T),
+
(2(X0(0) + SO(
where Qj_1 is known and exponentially decaying. Since both Ej and
,qj -*OasT -- 00, rr //
jlT) = Sj\T)
92 (XO(0)
+ SO(T), 0, 0)
f2 (X0(0) + SO(T), 0, 0)
_f00
(5.33)
Qj-1 (s)
and, using (5.31) again, Ej must satisfy
/
d dT
= tj 92(Xo(0) + 0 (T),0,0)
f2 (X0(0) + o(T),0,0)
6 f2(X0(0) + 0(T),0,0) + Pi- I (T),
where f_1 is known and exponentially decaying. Thus, Sj(T) = f2(XO(O) + O(T),0,0)
x
j(0)exp[J0 g2(Xo(O) + 60(s),0,0)ds]
+ Jo exp [
f2(X0(0)+X,0,0)
f
$2 (X0 (0) + s (s), 0, 0) ds] J_ 1(r) dr
.
(5.34)
3. QUASI-LINEAR SYSTEMS
133
Using condition (5.28), then, j(0) is uniquely determined and so are j(T) and J (T). Finally, since g2(Xo(0) + 60(T),0,0) < -K < 0, it follows that
j(T) = O(e-°') = qj(T)
as T -* 00,
where a > 0. Thus, we have THEOREM 5:
Under hypotheses (Hi) and (Hii), the boundary value
problem (5.14)-(5.15) has a solution (x(t, e), y(t, e)) for e sufficiently small
which is such that, for each integer N > 0, x(t, e) = ,I 1 Xj(t) + j(t/e))ei + e x+' R(t, e) ,qo(t/e)
At' e) =
e
+ I (Y (t) + qj+l (t/e))ei + ex+1 S(t, e),
where both R(t, e) and S(t, e) are uniformly bounded throughout 0 < t < 1.
Remarks
1. The asymptotic solution (x, y) is not uniquely determined because several values X0(l) may be possible in hypothesis (Hi) and,
for each X0(l), several values A may satisfy hypothesis (Hii). For example, the problem dx dt = y
e d! = -1(1 + 3x2)y x(0, e) + ey(0, e) = 0,
x(1, e) = 0
has (X0(t), Y (t)) = (0, 0) as the unique solution of its reduced problem. (The outer expansion has all terms zero.) However, the corre-
sponding Q(A) = fo (1 - J(l + 3r')) dr has the three zeros A = 0, A = 1, and A _ -1, each of which satisfies hypothesis (Hii). The
134
5. NONLINEAR BOUNDARY VALUE PROBLEMS
three corresponding asymptotic solutions are
x(t,e) _ Ae1/2e[(1 + A2) - 2et/e1-1/2 y(t,e) _ --(1 + A2)e-1/2e[(1 + A2) - x2e-t/e]-3/2.
They all converge for t > 0 to the trivial solution of the reduced problem. Related boundary value problems with multiple solutions are discussed in Chapter 7, and by Chen (1972) and O'Malley (1972c). We note that problems where X0(1) is not a simple root of P(y) and/ or where X0(0) is not a simple root of Q(A) are also tractable, but the resulting expansions will necessarily be more complicated than (5.20).
2. A proof of asymptotic correctness is given by O'Malley (1970c).
In cases where g2 (x, t, 0) > 0, one can obtain similar results with nonuniform convergence at t = 1. 3. Several related problems can be solved analogously. For example, if a2(--)
e
as
j=0
a -> 0
with a20 0 0, the problem (5.14)-(5.15) can be asymptotically solved
using only hypothesis (Hi) [see O'Malley (1970c)]. Likewise, if f2(X0(0),0,0) = f2x(X0(0),0,0) = 0 and a2(0) 0 0, the problem can also be solved asymptotically under hypothesis (Hi) with 0(T) = 0.
4. Earlier work on boundary value problems for quasi-linear systems includes that of Harris (1962) and Macki (1967). Some general results for vector systems are given in Hoppensteadt (1971).
4. AN EXTENDED DISCUSSION OF A NONLINEAR EXAMPLE Consider the two-point problem
ex" = (x')2,
x(0) = 0, x(1) = 1
(5.35)
4. AN EXTENDED DISCUSSION OF A NONLINEAR EXAMPLE
135
whose solution is readily found to be
x(t, e) = -e In (1 - t + to
(5.36)
Away from t = 1, note that x(t, e) converges to the limiting solution
X(t,e) = -e In (I - t)
(5.37)
as the small positive parameter e tends to zero [and that 1(t, 0) = 0 satisfies the reduced equation]. This limit, however, cannot approximate the solution near t = 1, where it blows up logarithmically. Thus,
nonuniform convergence of the solution must occur at t = 1 as e
0.
If we did not know the exact solution, how might we attempt to find an approximate one? Since the reduced equation (x' )2 = 0 has only constant solutions, no solution of it can be a uniformly valid limiting
solution throughout 0 < t < 1. Moreover, the results of Vishik and Lyusternik (cf. Section 5.1) indicate that endpoint nonuniformities are possible with derivatives being unbounded there like O(ec/e) for some C > 0. [Indeed, x'(1, e) = eeVVe - e.] It would be reasonable to also
study the possibility of nonuniform convergence within 0 < t < 1, but we shall not do so. How to predict that the nonuniform convergence occurs at t = 1 is not obvious. We note, however, that the differential equation implies that x is nondecreasing. By the boundary conditions, then, z(1) must be positive. Noting that the linear equation ex = az for a = i(l) > 0 allows boundary layer behavior at the
right endpoint of any interval, it is reasonable to anticipate the limiting behavior obtained. Thus, we shall ask that the solution x(t, e) be asymptotically represented in the form x(t, e) = X (t, e) + S (K,.-),
(5.38)
where the outer solution X(t, e) converges to the limiting solution -e ln(1 - t) as e -> 0 and the boundary layer correction at t = 1, tends to zero as the stretched variable Ic = (1 for some -O(e) = o(1), tends to infinity. Vishik and Lyusternik's results indicate that a selection
0, would be appropriate
136
5. NONLINEAR BOUNDARY VALUE PROBLEMS
since x'(1, e) - (0, e)Thus, we will proceed with K
1-t
(5.39) = e-h/e ' noting that the usual technique involving use of principal limits does not seem to provide the necessary stretched variable in this example.
A new feature here is that the outer solution blows up near the point t = 1 where the boundary layer occurs. Since x(1) = 1, howevmust also blow up at c = 0 er, the boundary layer correction in such a way that the singularity of the outer solution is canceled. Similar behavior is known to occur in certain physical problems [cf.,
e.g., Dickey (1973) for a problem involving the inflated toroidal membrane]. Clearly, the outer solution X (t, e) must satisfy the system for t < 1
and the initial condition. Thus, we have eXrr - Xr2 = 0,
X(0, e) = 0.
When e = 0, Xr (t, 0) = 0 = X(0,0),
so X (t, 0) =
(5.40)
0. Thus, it
is
natural to set X(1'8) = eZ (t, e),
where
Zrr = Z'2'
Z (0, e) = 0.
(Solving the differential equation for Z is no simpler than solving the
original equation for x, but our point here is to illustrate that our asymptotic representation-outer solution plus correction-still holds.) Integrating, then, we have the outer solution X (t, e) = eZ (t, e) _ -e In
c(
\ (e t ) '
(5.41)
where c(e) is an undetermined smooth function of e. For Z(t, e) to be
defined throughout 0 < t < 1, we require that c(e) > 1. [Since the outer solution is asymptotically given by (5.41) while x(l) = 1 is prescribed, we have another indication that an exponential stretching near t = 1 is necessary.]
4. AN EXTENDED DISCUSSION OF A NONLINEAR EXAMPLE
137
Having obtained the outer solution [up to c(e)], we shall now seek the boundary layer correction. Note that the differential equation [cf. (5.35)] implies that eXrr
+ eel/,L
= X2
-
2
e = 0, then, K(K, 0) = 0 and, since = 0. Thus, we set
(K, e) =
-> 0 as K -> 00, (K, 0)
e),
where
'IKIntegrat
2c(e)
ry,2
x - 'IKK +
K + (C(e)
-
ry,
l)e'/F
then, v = 1/i satisfies vK -
2c(e)
K + (c(e)
-
Iv - -1. 1)eh/`
Thus, if (c(e) - 1)eh/E -> oo as e -> 0, the limiting equation is V,(K,0)
= -1 and i(K,0) = 1 - In (K - 12) for constants 1 and 12. This is unsatisfactory, however, since i must remain bounded as K -> oo. Thus, we take
c(e) = 1 + o(e I/e)
(5.43)
for which the limiting system vK (K, 0) - (2/K) v(K, 0) _ -1 implies
n(K, 0) = 1 + In (
K
K + l2
,
(5.44)
where 1 and 12 are undetermined constants. Since i -> 0 as K -> oo,
5. NONLINEAR BOUNDARY VALUE PROBLEMS
138
however, 11 = 0. So far, then, we have
x(t,e) = -eln(1 and, expressing t in terms of K, the singularities at t = 1 and Ic = 0 cancel and we have x(t,e) = 1 - e In (K + 12) + o(e).
To satisfy the boundary condition at t = l(ic = 0), however, we must pick 12 = 1. Thus, we have
x(t, e) _ -e In(1 - t) + e In
\
1
KJ
+ o(e)
(5.45)
+ uniformly in 0 < t < 1, in agreement with the known solution (5.36). Moreover, the outer solution, which is asymptotically valid for t < 1,
is given by X(t,e) = -e In (I - t).
CHAPTER
6
THE SINGULARLY PERTURBED LINEAR S TATE REG ULA TOR PROBLEM
We shall consider the linear state regulator problem consisting of the system
Tt =
Al (t, E)x + A2(1, 8)Z + BI(t, E)u
e T, = A3 (t, E)x + A4 (t, 8) Z + B2 (1, E)u
on the interval 0 < t < 1 (or, equivalently, on any closed bounded interval), the initial conditions x(0, E) = x° (E)
- z'(--), 139
.2) (6.2)
140
6. THE LINEAR STATE REGULATOR
and the quadratic cost functional J(e)
=
Ly'(l,e)V(e)y(l,e) + f' [ y'(t, e) Q(t, e) y(t, e) + u'(t, e) R(t, e) u(t, e)] dt
(6.3)
which is to be minimized. Here, the prime denotes transposition,
y=
[x], Z
Q=
[Qi; Q2'
Q2], Q3
and
IT = I
171
M2],
M2' 3
where x is an n-vector, z is an m-vector, u is an r-vector, and the matrix R is symmetric and positive definite while the matrices Q and 7 are symmetric and positive semidefinite throughout 0 < t < 1 for e > 0 small. Crudely, the object is to find a control u which is not too
large which will drive the state y toward zero, especially at the terminal time t = 1. When e > 0 is fixed, this elementary control problem has a unique optimal solution which minimizes J [cf., e.g., Athans and Falb (1966) or Anderson and Moore (1971) for appropriate background material]. We are interested, however, in obtaining the asymptotic solution of the problem as the small positive parameter a tends to zero. Such singular perturbation problems are of considerable importance in practical situations where a represents certain often neglected "parasitic" parameters whose presence causes the order of the mathematical model to increase [cf. Sannuti and Kokotovic (1969) and Kokotovic (1972)]. In particular, we note that Hadlock et al. (1970) give an example for the optimal tension regulation of a strip winding process where asymptotic results are far superior to the physically unacceptable results obtained by setting e = 0. We will assume that the matrices A,, Bi, x°, z°, 7 Ti, Qi, and R all have asymptotic power series expansions as e --> 0 and that the coefficients
in the expansions for the A,, Bi, Qi, and R are infinitely differentiable
functions for 0 < t < 1. To obtain necessary and sufficient conditions for an optimal
THE LINEAR STATE REGULATOR
141
control, we introduce the Hamiltonian H(x, z, u, p1, P2 , t, e)
_ J(x'Q1 x + 2x'Q2z + z'Q3z + u'Ru) + p (A1 x + A2 Z + B, u) + p2 (A3 x + A4 Z + B2 u). (6.4)
Elementary calculus of variations [cf. Kalman et al. (1969)] implies that along an optimal trajectory
aH
(6.5) = Ru + B'Ipl + B2p2 = 0, where the costates p, and ep2 are defined to be solutions of the u
differential equations
dd' _
-aH--Q1x-Q2z-Aipi-Asp2
edd2=-aH=-Qix-Q3z-A2Pi-A4P2 on 0 < t < 1 and the terminal conditions
(x(l,e)l
( P,(l,e) l _ \eP2(l,e)/
-e)\z(l'e)/
Pl(l,e) = 77j(E)X(1,e) + e 2(E)Z(1,e) P2(l,e) _ 'T2 --)X(','-) + lr3(e)Z(l,e)
We note that the state equations (6.1) can be rewritten as dx
aH
dz
dt W edt
aH
T analogous to the usual Hamilton-Jacobi theory [cf. Courant and Hilbert (1962)]. Further (6.5) implies the control relation
u(t,e) =
B2(t+e)P2(t,e))
(6.8)
and since a2H/au2 = R is positive definite, this optimal control will minimize J(e).
6. THE LINEAR STATE REGULATOR
142
Using (6.8), the m + n state equations (6.1) and the m + n costate equations (6.6) can be rewritten as the linear system dx =A1x+A2z-S1p1 -Sp2 T, ddl e
j
edd2
(6.9a)
= -Q1x- Q2z-Alp1 -A3P2
(6.9b)
= A3x + A4z - S'P1 - S2P2
(6.9c)
= -Q2 'X - Q3z - Aip1 - AaP2,
(6.9d)
where S(t, e) = B1(t, e) R-1 (t, e)B2 (t, e) and
i = 1, 2. We must solve the system (6.9) subject to the 2m + 2n boundary conditions provided by the initial conditions (6.2) and the terminal Si (t, e) =
conditions (6.7).
Since the order of the system drops from 2m + 2n for e > 0 to 2n for e = 0, we have a singularly perturbed problem. In particular, the system obtained when e = 0 cannot be expected, in general, to satisfy
all the limiting boundary conditions. Thus, even if the limiting solution as e -> 0 satisfies the limiting system, nonuniform conver-
gence near the endpoints must be expected as e -> 0. From our previous work, we are led to seek an asymptotic solution of the form x(t, e) = X(1, e) + em1(ic, e) + en1(a, e)
(6.10a)
z(t, e) = Z (t, e) + m2 (K, e) + n2(0,8)
(6.10b)
P1 (t, e) = P1 (t, e) + -P1 (K, e) + 871 (0, e)
(6.10c)
P2 (t, e) = P2 (t, e) + P2
e) + 72 (a, e),
(6.10d )
where K and a are the stretched variables K
= 1/81
a
(0, e), Z (t, e), P1 (t, e), P2 (t, e))
(6 . 11 )
(6.12)
THE LINEAR STATE REGULATOR
143
will be the outer solution; (Eml (K, E), M2 (K, E), -PI (K, E), P2 (K, E))
(6.13)
will be the boundary layer correction at t = 0; and e), n2 (a+ E)+ EYI (a+ E)+ Y2 (a, E))
(6.14)
will be the boundary layer correction at t = 1. The three matrices (6.12)-(6.14) will all have asymptotic power series expansions. Further, the terms in the expansions for the boundary layer corrections will tend to zero as the appropriate stretched variable tends to infinity. Away from t = 0 and t = 1, then, the solution will be asymptotically given by the outer solution, while at these endpoints convergence will be nonuniform as e -> 0. Clearly, then, we must ask that the outer solution satisfy the linear system (6.9). Setting e = 0, the leading term of the outer solution must satisfy the reduced system dd ° = A10X0 + A20 Zo - S10 P10 - SO P20
ddlo
= _Q10X0 - Q20Zo - A'10 P10 - A30P20
(6.15a)
(6.15b)
0 = A30X + A40 Zo - SoP10 - S20P20
(6.15c)
0 = -QmX0 - Q30Zo - A2'0P10 - A40P20.
(6.15d)
Applying two of the boundary conditions when e = 0 further requires that X0(0) = xo
(6.16)
P10(1) _ ITi0X0(1)
To solve (6.15}{6.16), it is necessary to solve Eq. (6.15c,d) for Z0 and
P20 as linear functions of X0 and P10 and to solve the remaining boundary value problem for X0 and P10. This will require two matrices to be nonsingular. As our first assumption, then, we will ask:
144
6. THE LINEAR STATE REGULATOR
(Hi)
The reduced problem dx = A1(t,0)x +A2(t,0)z - B1(t,0)R-'(t,0)B1(t,0)p1 dr - B, (1, 0) R'(1, 0) BZ (t, O )P2
(6.17a)
dd' = Q1(t,0)x - Q2(t,0)z - A,(t,0)p1 - A3(t,0)p2
(6.17b)
0 = A3(t,0)x +A4(t,0)z - B2 (t, 0) R- 1 (t, 0) B'1 (t, 0)p1 (6.17c)
- B2 (t, 0) R-' (t, 0) BZ (t, 0)p2
0 = -Q2(t,0)x - Q3(t,0)z - A2' (t,0)p1 - A4(t,0)p2 X(0) = x$,
P1(1) - Ir1(0)x(1) = 0
(6.17d) (6.17e)
has a unique solution (X0(1), Z0(t), P10(1),1'20(1))
throughout 0 < t < 1. In particular, then, Zo
A.
P20
-Q30
(-A30 \ Qzo
-5201 -A40/
SO,
A2'0) (P,X0 0
and there will remain the linear system
X} / Poo
A10
-Q1o A40
-510
-Aio)
+
A20
-So
(-Q2o -A30
-S20)-' -A30
X (-Q3o -AaoC
Q2'0
SO
Az0)
X0 P10
subject to the boundary conditions on X0 and P10. Note that, since ZO(0) 0 A and P20(1) 0 orZOXo(1) + 1T30Z0(l), in general, nonuniform convergence generally occurs at both t = 0 and t = 1. Higher-order terms (X (t), Zi(t), P1j(t), P2J(t)) of the outer expansion
will satisfy nonhomogeneous forms of (6.15)-(6.16) which can be uniquely solved successively up to specification of X(O) and P1j(1)
145
THE LINEAR STATE REGULATOR
- Ir10X(l). Since X(0,8) = x°(e) - em1(0, e)
and P1(l,e) - 77j(e)X(1,e)
=
e(-Yl (0, e) + IT1(e)n1(0, C) + IT2 (e) (Z(1, e) + n2 (0, e))),
the terms of the outer expansion can be determined successively once lower-order terms in the initial values of the boundary layer corrections m1 (0, e), Y1 (0, e), n1 (0, e), and n2 (0, e) are known.
Since the boundary layer correction at t = 1 (or 0) is asymptotical-
ly negligible at t = 0 (or 1), the boundary layer correction at t = 0 must satisfy the linear system
da11 = eA1 (.-K, e)m1 + A2(eec, e)m2 - eS1 (elc, -)P1 dp,
d_
-Q1(elc, .-)m1
2
(6.18a)
- Q2 (.-K, .-)M2 - eA'1 (eK, e)P1 - A3' (.-K, e)P2 (6.18b)
da22 = eA3 (elc, e)m1 + A4 d
- S(elc, e)P2
e)m2 - -S'(.K, e)P1 - S2 (eK, e)P2
(6.18c)
-eQ2 (.-K, e)m1 - Q3 (.-K, e)m2 - -A2' (.-K, 8)P1 - A4 (.-K, e)P2
(6.18d)
while the boundary layer correction at t = 1 must satisfy dn,
do = -eA1(1 - ea, e)n1 - A2(1 - ea, e)n2 + eS1(1 - ea, 071 + S(1 - ea,e)72
(6.19a)
da = eQ1(1 - ea, e)n1 + Q2 (1 - ea, e)n2 + CA, (1 - ca, e)Yl
+A3(1 - ea,e)72 do
(6.19b)
-eA3(1 - ea,e)n1 - A4(1 - ea,e)n2 + -_S'0 - CC, 071 + S2 (1 - ea, e)72
(6.19c)
ddo = eQ2(1 - ea,e)n1 + Q3(1 - ea,e)n2 + eA2(1 - ea,e)Yl
+ A4(1 - ea,e)Y2.
(6.19d)
146
6. THE LINEAR STATE REGULATOR
When e = 0, then, we ask that dm10 =
A2o(0)m2o - So(0)P20
(6.20a)
= -Q2o(0)m2o - Aso(0)P2o
(6.20b)
= A4o(0)m2o - S20(0)P20
(6.20c)
= -n- (01 o
(6.20d)
dK dd oo
ddK20
dP20
dK
-20
- A ' o(0) 1-20
In particular, note that the last two equations of (6.20) form a system of 2m linear equations and that the initial condition (6.2) yields the mvector m40(0) = zoo - Z0(0).
(6.21)
In order that m20 and P20 -> 0 as K -> 00, it is necessary that the matrix
G(t) =
A4o(t)
-Q30(t)
-S20(t) -Aao(t)
have m eigenvalues with negative real parts at t = 0. Analogously, in order to define appropriate determinations of n20(a) and ?20(a), it is necessary that G(l) have m eigenvalues with positive real parts. In order to obtain the expansions (6.10), we will use the three additional assumptions: (H2)
The 2m X 2m matrix
G(t) _ r A4(t,0)
-Q3(t,0) has m eigenvalues A, (t),
. . . ,
0 < t < I and m eigenvalues there.
-B2(t,0)R-'(t,0)Bz(1,0)1 -A4(1,0)
(6.22)
A,,, (t) with negative real parts throughout (t), ... , A2,i(t) with positive real parts
THE LINEAR STATE REGULATOR
(H3)
147
The linear system
edt = G(t)c
(6.23)
has m linearly independent solutions of the form cj(t,e) = Dj(t,e)e)J(0)u/`,
j = 1, 2, ..., m,
(6.24)
j = m + 1, ... , 2m.
(6.25)
and m linearly independent solutions
cj(t,e) = Dj(1 - t,e)e-\j('X-1)/`, (H4) Setting
D (0,0)
_
J)
DD.
for each j where the DJ 's are m-dimensional vectors, suppose that the m X m matrices D' = (DII DZ
(6.26)
and D2 = ((Dm +1 - ir30Dm+1) ... (D2m
- i30Dim))
(6.27)
are both nonsingular.
Remarks
1. Because of the special form of the matrix G(t), the existence of m eigenvalues A with negative real parts actually implies the existence of
m other eigenvalues -A with positive real parts. Further, when Q3 = 0, (H2) follows provided the real parts of the eigenvalues of A4(t,0) are all nonzero throughout 0 < t < 1.
2. Hypothesis (H3) follows readily if Eq. (6.23) has no turning points in [0, 1] [cf. Turrittin (1952) and Wasow (1965)]. The construc-
6. THE LINEAR STATE REGULATOR
148
tion of the asymptotic solutions is analogous to that in Section 3.2. Weaker hypotheses are also possible [cf. O'Malley (1972d, 1974) and Kung (1973)].
Since (6.23) can be rewritten do/dic = G(eK)c, the linear system for m20 and P20 [cf. (6.20)] has the solution mzo(K))
Pzo(/ K)
=
I
a>o D>(0, 0)eaJ(o)x
i=I
for constants M j, but, since Am+1,
(6.28)
... , X2m have positive real parts, we
must pick Nm+1 = Nm+2 = "' = N2m = 0
in order to eliminate exponentially growing terms. Further, with m20(0) specified by (6.21), 801,,802
......8m0 are uniquely determined
since, by (6.26), the matrix of coefficients D' is nonsingular. Thus, P20(K) and, by (6.20), dm10/dK and dplo/dK are exponentially decaying as K -> 00. Defining
mio(K) = -
f f
(6.29)
dK(S)ds
the lowest-order terms in the boundary layer correction at t = 0 become completely specified as exponentially decaying terms. Analogously, we have (n20(a)) Y2o(a)
aD(0,0)e J°,
= jGm+1
(6.30)
where the a°'s are uniquely determined by the initial condition Y2o(0) - 73020(0) = 1Ti0X0(1) + -304(1)
(6.31)
since the matrix D2 is nonsingular. It follows that n10(a) and Y1o(a) are also uniquely determined as exponentially decaying terms.
149
THE LINEAR STATE REGULATOR
Higher-order coefficients in the boundary layer corrections are determined as solutions of analogous nonhomogeneous problems. For example, from (6.18), we find that m2j and P2j satisfy a system dm2j
d dK
dK'
_ A4o(O)m2J - S20(0)P2j + -Yi-I (K) 2
= -Q30(0)m2j - A' 40(O)P2j + lj-1(K),
where s1 -, and 1n-j-1 are exponentially decaying and successively known and
m2j(0) = z° - Zj(0).
(6.33)
Note that Zj(0) is known successively since the jth term of the outer expansion is determined by preceding terms of the boundary layer corrections. Since m2j and p2j tend to zero as K -> oo, they must be of the form
m2J (K) P2j (K)
_I ,
(M2'(K)> P2j(K)
1
>
(6.34)
where
P2j(K)/
;_' D,(0,0)e-'(O)KJOK Q,(s) 2m
K A,(s)
+ 2 D,(0, O) eai(O)K im+1
L s(S)
s(s) is the determinant s(s) = det(DI(0,0)e-`I(O)K ...D2m(0,0)e-\-(O)K),
and A1(s) is the determinant obtained from s(s) by replacing its ith column by the exponentially decaying vector (s)
Pj-1(S)
6. THE LINEAR STATE REGULATOR
150
Since the matrix D' is nonsingular the initial condition (6.33) determines (6.34) uniquely. Furthermore, hypothesis (H2) and the fact that eM(')d;(s)/4(s) is exponentially decaying imply that m2; and p2) will also decay exponentially as K --> oo. Integrating the differential equations for dm1;/dK and dp1j/dK from K to infinity, then, we determine the
exponentially decaying terms mij and pi;. Proceeding, we can also determine the jth terms in the boundary layer correction at t = 1. Knowing the expansions (6.10) for the costate vectors, the control relation (6.8) implies that the optimal control has an expansion of the form U(1,8) = U(1,8) + v(K, E) + W(0,8),
(6.35)
where U, v, and w have asymptotic power series expansions as e -> 0 and
U0(t) = -Ro`(t)[Bio(t)P10(t) + B2o(t)P2o(t)]
is the control corresponding to the solution of the reduced problem (6.17). Knowing the asymptotic expansions for the optimal trajectories implies that the corresponding power y'(t, E) Q(t, E)y(t, E) + u'(t, E) R(t, E) u(t, E)
is of the form L1 (t, e) + L2 (K, E) + L3 (a, e),
where the L;'s have asymptotic power series expansions such that L2(or L3) tends to zero as K (or a) tends to infinity and with L10(t) = Xo(t) Q10(t)Xo(t) + 2X0'(t) Q20(1) Z0(t) + Z0' (t) Q30 (1) Zo(t) + Uo(t) Ro(t) U0(t).
Thus, the optimal cost can be written as J * (E) = We) + I f' L, (t, E) dt +2 fo'o L2(K,E)dK +
2f L3(a,e)do
(6.36)
THE LINEAR STATE REGULATOR
151
and it will have an asymptotic expansion 00
J* (e) ^,
Ji* e,
i=0
with leading term Jo = X01(l)ITIOX0(1) +
I f1
L1O(t)dt
being the cost corresponding to the solution of the reduced problem. Boundary layer corrections, then, contribute leading order terms to the state vector z and to the optimal control u, but their influence on the state vector x and the optimal cost J' (e) is in higher-order terms.
Thus, we have formally obtained the following theorem whose proof is given by O'Malley (1972a). Under hypotheses (Hl)-(H4), the optimal control problem (6.1)-(6.2) has a unique solution for e sufficiently small, such that, for every integer N > 0, the optimal control u(t, e) and the corresponding trajectories x(t, e) and z(t, e) satisfy THEOREM 6.
u(t,e) = I (u,.(t) + vj(ic) + wj(a))ei +
O(eN+1)
I=0
x(t, e) = X0(t) + 2 (X (t) + m1,J_1 (K) + n1,_1
O(eN+1)
I=1
Z(t,e) = 2 (Zj(t) + m2j(ic) + n2J(a))ei + 0(eN+1) j=0
as e -> 0 uniformly in 0 < t < 1. Here the terms which are functions of K = t/e [or a = (1 - t)/e] decay to zero exponentially as K (or a) tends to infinity [i.e., away from t = 0 (or 1)]. Moreover, the optimal cost has an asymptotic expansion such that N
J' (e) _ I J,* .-l + 0(rN+1) i-0
ase-p0.
152
6. THE LINEAR STATE REGULATOR
As an example, consider the scalar problem dx
x
dt
edt = -x-z+u on the interval 0 < t < 1 with x(O) = x0 and z(0) = zo being dz
prescribed constants and with the quadratic cost (x2(t) + u2(t)) dl
J(e) _ (x2(1) + ez2(1)) + to be minimized. The appropriate Hamiltonian is
H(x,z,u,p1,p2) _ 21 (x2 + u2) +PIx + p2(-x - z + u) and the Euler-Lagrange equations for minimum cost are dx
T,
=x
dp1
dz
edt =
-x-z+u
edd2 = P2
with the relation aH/au = 0, or u = p2,
and the terminal conditions P1(1) = x(1),
P2(1) = z(1).
The general solution of this constant coefficient system is
x(t) _ P1 (t) =
Z(t) = P2 (1) =
-2c2(.-)(I + e)e' C1(e)e-'
+ c2(e)(1 + e)e' 2c2(e)et
+ 2eC3(e)C-(1-t)/e
- c3(e)(1 +
e)e-(1-t)/e + c4(e)e't/e
2c3(e)(1 + e)e-(1-t)/e,
THE LINEAR STATE REGULATOR
153
where the c;(.-)'s are arbitrary. Applying the boundary conditions, we have the optimal solution x(t) = x0 et Z(t)
=
u(t) =
X0
1+e 2
[-et + e-1- + 3 e-0-019 l + z0e-de + O(e-11e)
x0e
3(l+e) e-0-01e + O(e-11e)
t
The solution of the control problem with dx/dt - x, x(0) - 1; edz/dt - -x - z + u, z(O) - 2; and J(e) -;(x2(1) + ez2(1)) + } fo (x=(t) + u2(t))dt, e - 0.1. FIGURE 17
6. THE LINEAR STATE REGULATOR
154
with the corresponding cost
+E)zoo
+4ez(1
Jo
4 (3e2 - I +E 9(1
e -2v
do]
+z
E)z
This solution is pictured in Fig. 17. It can also be simply obtained by the expansion procedure outlined above. Here, the reduced problem has the solution Xo(t) = xoe`
Zo(t) _ -xoe'
U0(t)=0 with the corresponding cost
Jo =
z
(3ez - 1).
4 Remarks
1. In models for practical control problems, one must expect several small interrelated parameters to enter the problem in both regular and singular ways. Techniques appropriate for such problems could be developed as for the initial value problems in Chapter 4.
2. Linear state regulator problems for time invariant systems on infinite time intervals are important in applications. Controllability assumptions are needed [cf. Kalman et al. (1969)] and the singular perturbation problem is more subtle [cf. Hoppensteadt (1966)]. Considerable progress on this problem has been made using the Riccati
THE LINEAR STATE REGULATOR
155
matrix formulation [cf. Kokotovic and Yackel (1972) and O'Malley (1972b)]. Likewise, other regulator problems could be considered by proceeding analogously.
3. The method given here has been extended to certain nonlinear
problems. Much progress has been made by Kokotovic and his students [cf. Sannuti and Kokotovic (1969), Kokotovic and Sannuti (1968), Hadlock (1970), and Sannuti (1971)]. They also discuss models
leading to such problems. O'Malley (1972d) further discusses some quasi-linear problems. 4. Note that the hypotheses made in solving the preceding problem involve systems of dimension less than (2m + 2n) X (2m + 2n). This order reduction is of great practical importance when working with large-order models. Alternative Riccati matrix approaches to the same problem involve additional computational advantages and feedback control [cf. Yackel and Wilde (1972) and Kung (1973)].
5. The limiting solution (X°, Z0, U0, Jo) obtained is determined from the reduced problem (6.17) corresponding to the two-point problem for the system (6.9). It has not been shown that this limiting solution could be obtained from the reduced problem consisting of the system dx
= A,(t,0)x + A2(t,0)z + B,(t,0)u dt 0 = A3(t,0)x + A4(t,0)z + B2(t,0)u, the initial condition x°(0), and the cost functional J(0). This alternative reduced problem seems to be the more natural one since it is immediately obtained. For certain problems, the equivalence of the two limiting problems has been shown [cf. Haddad and Kokotovic (1971), O'Malley (1972b), Kokotovic and Yackel (1972), and Kung (1973)]. The most general results, due to Sannuti, are unpublished. Problems without the e multipliers of ore and 7r3 in the terminal weighting matrix or(e) can also be solved [cf. Kung (1973)]. The natural reduced problem is not then appropriate, however.
156
6. THE LINEAR STATE REGULATOR
6. The boundary layer behavior obtained in these control problems might be predicted by introduction of delta functions. We note that the function a e-ot/E, E
0< t< 1, a> 0
behaves like the delta function S(t) as E -* 0; i.e., for any differentiable f (t), J.1 o
f(t)ae-atledt = f (0) - f (1)e a/E + E
f
fa(t)e-of/Edt
o
= f (0) + O(E). Likewise, ae-0-t>/E,
0 0
behaves like the delta function 8(1 - t) as E -* 0. Recalling the preceding example, we note that the boundary layer behavior of the optimal control near t = 1 forced the nonuniform behavior of the resulting trajectory z(t) there. This delta function formulation could be further utilized here and in other singular perturbation calculations.
7. Additional use of singular perturbations in optimal control problems includes the work of Lions (1972, 1973) on partial differen-
tial equations, Collins (1972) on bang-bang control, Jameson and O'Malley (1974) on singular arcs, and Bagirova et al. (1967).
CHAPTER
7
BOUNDARY VALUE
PROBLEMS WITH MULTIPLE SOLUTIONS ARISING IN CHEMICAL REACTOR THEORY
Boundary value problems of the form
eu"+u'=g(t,u), u'(0) - au(0) = A
0
u'(1) + bu(1) = B arise in the study of adiabatic tubular chemical flow reactors with axial diffusion [cf., e.g., Raymond and Amundson (1964), Burghardt
and Zaleski (1968), and Cohen (1972)]. We shall be interested in 157
158
7. BOUNDARY VALUE PROBLEMS WITH MULTIPLE SOLUTIONS
obtaining asymptotic solutions of the problem as the positive parameter e tends to zero. Physically, this means that the Peclet number is
becoming large (as it would, for example, when the diffusivity becomes small). Mathematical aspects of the problem have been studied by Cohen and others. We shall take a, b, A, and B as constants
and shall assume that g(t, u)
is
infinitely differentiable in both
arguments. As E -+ 0, our experience with singular perturbation problems leads
us to expect nonuniform convergence at t = 0 with convergence elsewhere to a solution of the reduced problem Uo(t) = g(t, Uo) U0'(1) + b Uo(l) = B,
if it exists. Thus U0(l) would satisfy the nonlinear equation
g(l,ao) + bao = B.
(7.2)
Corresponding to each root ao of (7.2), then, we shall assume that there is a unique solution of the corresponding nonlinear terminal value problem Uo(t) = g(t, Uo)
(7.3)
U0 (1) = ao
which exists throughout 0 < t < 1. We shall then seek asymptotic solutions u(t, e) of (7.1) converging to
the solution U0(t) of (7.3) for t > 0. We shall give a detailed discussion of the three cases: (a) ao is a simple root of (7.2), (b) ao is a double root of (7.2), and (c) ao is a triple root of (7.2). The general case where ao is a root of arbitrary finite multiplicity follows analogously [cf. Chen (1972)].
CASE a:
In Case a, ao is a simple root of (7.2) so
P(ao) = g(l,ao) + bao - B = 0
(7.4a)
PROBLEMS ARISING IN CHEMICAL REACTOR THEORY
159
while
P'(ao) =
b
0
(7.4b)
and we shall seek a solution u(t, e) of (7.1) of the form u(t, E) = U(t, E) + Ev ('r, E),
(7-5)
where the outer solution is expanded as 00
U(t, E)
I U(t)E'
j=
and the boundary layer correction as 00
v(T, E) - 2 vl (T)EJ j=0
with the coefficients vj('r) tending to zero as the boundary layer coordinate T = t/E
(7.6)
tends to infinity. If this scheme is possible, u(t, e) will converge to U0(t)
as E - 0 throughout 0 < t < 1 while u'(t, E) will converge nonuniformly at t = 0. Thus, the outer solution U(t, E) must satisfy the differential equation and the terminal boundary condition of (7.1), i.e.,
cu" + U' = g(t, U) U'(1,E) + bU(1,E) = B.
These equations are automatically satisfied when E = 0 since U0(t) satisfies the reduced problem. They imply that higher-order coefficients U(t) with j > 0 must satisfy linear terminal value problems U(t) = gw(t, U0)U + gj_1(t)
U'(1)+bU(l) = 0
(73)
160
7. BOUNDARY VALUE PROBLEMS WITH MULTIPLE SOLUTIONS
on 0 < t < 1, where gj_1 is known successively. Thus, the terminal value U;(1) is uniquely determined by the linear equation
P'(ao) U(l) = (gw(l,ao) + b) U(1) _ -g;-1(1)
(7.8)
when ao is a simple root of (7.2). When P'(ao) = 0, we generally encounter a difficulty, and the outer expansion cannot be obtained as a power series in E. In Case a, however, the outer expansion U(t, E) is uniquely and straightforwardly determined. The boundary layer correction v(T,E) must account for the nonuniform convergence near t = 0. Since the outer solution is known, the boundary layer correction must satisfy the initial value problem
V, + v, =
g(ET, U(ET, E) + Ev(T, E))
- g(ET, U(ET, E))
(7.9)
v,(0, E) = A - U'(0, E) + aU(0, E) + eav(0, e)
and v must tend to zero as T tends to infinity. When e = 0, then, vo,,r + vo, = 0
vo,(0) = A - U0'(0) + aUo(0). Thus, we have
Vo(T) = -vo,(0)e ".
(7.10)
Similarly, higher-order coefficients v;(T) must satisfy vi" + vj,, = Gj_ I (T)
viT(0) = Bi-,, where Gi_I and h,-I are known successively and O for any 8 > 0 as T -+ oo. Thus,
vi(T) = -Bj_le T -JT = O(e(1-a)T)
Jo
(T) = O(e ('-ah)
e ('-') Gj_I (s)dsdr
as T -+ oo;
(7.11)
PROBLEMS ARISING IN CHEMICAL REACTOR THEORY
161
i.e., the boundary layer correction v(T, E) can be uniquely determined termwise with coefficients tending to zero as r tends to infinity. CASE b:
Now ao is a double root of (7.2) so
P(ao) = g(l, ao) + bao - B = 0
b=0
P'(ao) =
(7.12a) (7.12b)
and
P"(ao) = g.(l,ao)
0
(7.12c)
and we know that an outer expansion U(t, e) cannot generally be determined as a power series in E. We avoid the possible problem by seeking an outer expansion of the form 00
U(t, 1E) "'j0 Y. U (t) (1)',
(7.13)
where f > 0 and U0(t) again satisfies (7.3) throughout 0 < t < 1. Since the outer solution must satisfy the differential equation and the terminal boundary condition of (7.1), successive coefficients of (,/ )' imply that Ui (t) = g. (t, U0)U1
U;(1) + bU1(1) = 0,
Uz(t) =
U0)Uz + igww(t, U0)U2 - U11
U2'(1) + bU2(l) = 0, U'3 (t) = gu(t, U0)U3 + g,ru(t, U0)U1 U2 1
+
31
gg. (t, UO)
UU3
-
U'i
U3(1) + bU3(l) = 0,
etc. By (7.12), note that
b)U1(l) = 0, so the terminal
condition for U1 is satisfied whatever U1(1). Likewise the boundary
162
7. BOUNDARY VALUE PROBLEMS WITH MULTIPLE SOLUTIONS
condition for U2 becomes g,,,,(1,ao) U11 (1) = 2U'o(1).
(7.14)
Thus, to obtain a real value U, (1) we ask that (7.15)
U0(l)g.,,(l,«0) > 0
[anticipating a later difficulty if U'o(1) = 0]. Under this assumption, two distinct terminal values U, (1) are possible: namely U, (1) = ±a,, where a, = [2U'o(1)/g,,,,(l,ao)]'/z > 0. Thus, there are two determinations of U, (t) as unique solutions of the linear problems UI(t) = g. (t, U0)Ui
(7.16)
U,(1) = ±a,;
i.e., U, (t) = ±a, exp [ f ` g (s, Uo (s)) ds]. Continuing,
the boundary
condition for U3 becomes
gww(1,«0)X3(1) - v;(1) = 0.
g,,,,(1,«0) U1(1) U2(1) +
Since (7.15) implies that ao) U, (1) 0 0, this equation yields a unique determination of U2(1) = a2. Thus, U2 satisfies the linear problem
Uz(t = 9.01 Uo)Uz +
Uo) Uz - U0 (7.17)
U20)
=
a2
1
g"2
= 901%) +g.(1
,«o)[(lao)
-
Continuing, we can successively determine all initial values U(1) and higher-order coefficients U(t) uniquely once U, (1) is selected. Thus, two outer solutions U(t, f) can be formally obtained corresponding to the double root ao of (7.2) which satisfies the restriction (7.15). Corresponding to each of these outer solutions U(t, V E-), we shall seek an asymptotic solution of the form
u(t,E) = U(t, f) + Ev(T, Vc), (7.18) where the boundary layer correction v(T, f) has the asymptotic expansion 00
v(T, Vt) - I vj(T)(V`)' j-0
PROBLEMS ARISING IN CHEMICAL REACTOR THEORY
163
and the coefficients v; - 0 as T = t/E -+ oo. It follows that v(T, J E) must satisfy the initial value problem uTT + VT = g (ET,
U(ET, J) + Ev (T, V ° )) - g (ET, U(ET, \/)) (7.19)
vT(0,f) = A - U'(0,f)+aU(0,VE-)+Eav(0, f). The coefficients in the expansion for v can be uniquely obtained as in Case a. CASE c:
Here ao is a triple root of (7.2), so
P(ao) = g(l,ao) + bao - B = 0 b=0
P'(ao) =
(7.20a) (7.20b)
P"(ao) = g,,,,(l,ao) = 0
(7.20c)
P (ao)
(7.20d)
= g.(1, ao) # 0,
and we will attempt to obtain an outer expansion of the form 00
U(t,E'/3) "' 2 U(t)(Eil3), i=0
(7 . 21)
where U0 satisfies (7.3) and E'/3 > 0. The differential equation and the terminal condition of (7.1) then imply that Ui(t) = g. (t, U0)U1
U;(1) + bU1(1) = 0, Uz(t) = gw(t, U0)U2 + jgww(t, U0)U2
U2'(1) + bU2(l) = 0, U3(t) = gu(t, U0)U3 + gw(t, U0)U1 U2
+ 3 i g. (t, Uo) U13 - Uo
U(1) + b U3(1) = 0, U4(t) = g (t, UO) U4 + g,,,, (t, UO) (U1 U3 + Uz )
+ gwr (t, UO) Ul U2 + 41 g.. (t, UO) U14 - U'
U4(1) + bU4(1) = 0,
164
7. BOUNDARY VALUE PROBLEMS WITH MULTIPLE SOLUTIONS
etc. Note that (7.20) implies that the terminal conditions for U1 and U2 are satisfied whatever U1(1) and U2(1). Also the boundary condition for U3 becomes 3!gn,,,(1,«0) U1 (1) - U011 (1)
= 0.
(7.22)
Thus, we ask that U0 (1)
*0
(7.23)
and we obtain only one real nonzero determination U1 (1) = a1. Then U1 is the unique solution of Ui(t) = g. (t, UO)U1 U1(1)
-
(7.24)
(6U'0(1) 11/3
a1
g. (1,ao)
.
Continuing, the terminal condition for U4 becomes
9 0, a0)U4(1) - U1(1) = 0. (7.25)
g. (1, «o) Ui (1) U2(1) +
4I
Since (7.25) implies that U1(1)
0, we can uniquely determine
U2(1) = a2 and U2 must satisfy U'2(t) = 91,01 UO) U2 + igww(t, UO)U12
(7.26)
U2(1) = a2.
Continuing, we can uniquely obtain higher-order coefficients U(t) in a straightforward fashion. Here we obtain the behavior near t = 0 by seeking an asymptotic solution u(t, E) such that u(t,E) = U(t,E1/3) + Ev(T,E1/3),
(7.27)
where the boundary layer correction v(T,E1/3) has an asymptotic expansion V (T,
E1/3) .,. ' Uj (T)Ell3 j =0
PROBLEMS ARISING IN CHEMICAL REACTOR THEORY
165
such that the vj(T) -+ 0 as T = t/E -+ 00. Though ao is a root of (7.2) of multiplicity three, satisfying (7.23), note that we have only obtained a single corresponding asymptotic solution of (7.1). Generalizing slightly, we have THEOREM 7:
Consider the boundary value problem
0
Eu" + U' = g(t, U),
u'(0) - au(0) = A u'(1) + bu(l) = B. Let ao satisfy the nonlinear equation
P(ao) = g(l,ao) + bao - B = 0 and suppose that the reduced problem
Uo(t) = g(t, Uo) Uo(1) = ao
has a unique solution throughout 0 < t < 1. Then, for e sufficiently small, (a) corresponding to every (simple) root ao of multiplicity m = 1, the boundary value problem has a solution u(t, e). (b) corresponding to every ao of even multiplicity in such that U1
> 0,
(7.28)
there are two distinct solutions u(t, e). (c)
corresponding to every root ao of odd multiplicity m > 1 such that
U" (1) # 0,
(7.29)
166
7. BOUNDARY VALUE PROBLEMS WITH MULTIPLE SOLUTIONS
there is one solution u(t, E). In all cases, the solution u(t, E) is of the form u(t, E) = U(t, el/m) + Ev(t/E, El/m);
i.e., for every integer N > m, N
u(t, e) _ 2 U ' (t)eJ/m + j=0
j= m
(U(t) + vj_m(t/E))Ejlm
+ (N+l)/mR(t, e).
(7.30)
where R(t, e) is bounded throughout 0 < t < 1 and the vj_m t/e -9 00.
s -0 0 as
Remarks
1. The extra hypotheses (7.28) and (7.29) on U'o(1) when m > 1 allow us to solve the boundary condition bUm(l) = i Um (1) am
a(uma0)
- U'o(1) = 0
for U(1) and later boundary conditions for successive Uj(l)'s since amg(l,a0) # 0. au'"
The results of Parter (1972) indicate that such conditions may be necessary for existence.
2. The proof of asymptotic correctness follows simply from the previous results for nonlinear initial value problems (cf. Chapter 4). It has not been shown, however, that the asymptotic solutions constructed are the only solutions existing. Finally note that all these solutions
converge to a solution U0(t) of the reduced problem throughout
0 < t < 1 while derivatives generally converge nonuniformly as
e-90at t = 0.
PROBLEMS ARISING IN CHEMICAL REACTOR THEORY
167
3. Analogous results have been obtained for the more general problem eu" + f (u, t, E)u' = g(u, t, E),
0
m(u(0), u'(0), E I = 0
n(u(l),u'(l),E) = 0 [see O'Malley (1972c)]. For these problems, multiple solutions for the boundary layer correction (corresponding to any outer solution) are also possible. Generalizations to two-parameter problems and vector equations are discussed by Chen (1972). Keller (1973) discusses the
stability problem for the diffusion equation for which the ordinary differential equation (7.1) provides the steady state limit. A number of related problems in chemical reactor theory are discussed by Cohen (1972a). We note that the techniques developed here for problem (7.1) and the resulting multiple solutions may actually be more relevant to batch reactors and to tubular reactors with recycle than to (7.1) itself.
CHAPTER V
SOME TURNING POINT PROBLEMS
1.
A SIMPLE PROBLEM Consider the special linear boundary value problem
ey" + 2a xy' - a/3y = 0,
-1 < x < 1
with y(± 1) being prescribed
(8.1)
for a and $ constants with a 0 0 and for e a small positive parameter. Note that the results of Section 3.1 are not applicable here since
a(x) = 2ax
has a zero within [-1,11. Further, note that the reduced problem
obtained by setting e = 0 in (8.1) is singular at x = 0. Near x 168
1. A SIMPLE PROBLEM
169
= -1(+1), however, a(x) is negative (positive) if a > 0, so those earlier results lead us to expect uniform convergence to occur at both
endpoints when a > 0. On the other hand, if a < 0, a(x) is positive (negative) near the left (right) endpoint, so we then expect nonuniform
convergence at both x = ±1. Thus, one might expect the limiting solution for a > 0 to satisfy 2xz'l -,8z, = 0,
z,(-1) = y(-1)
2xz2 - az2 = 0,
z2(1) = y(1)
[-1,0)
on
and
on (0, 1],
with possibly unbounded behavior at the "turning point" x = 0, depending on the sign of a. Likewise, one might argue that the limiting solution within (-1, 1) for a < 0 should be the trivial solution of the reduced equation. These conjectures are correct except for a
countable number of $ values when a "resonance" phenomenon occurs.
The problem (8.1) can, fortunately, be solved in terms of the appropriate Weber (or parabolic cylinder) functions [cf. Whittaker and Watson (1952)]. [In certain very special cases (e.g., when Q = 0) the equation may be simply integrated and special functions need not be used.] Note that, if y satisfies (8.1), u = y exp(ax2/2E) satisfies 2
E2d
2
= (a2x2 + Ea(l + a)
u
and, since the parabolic cylinder functions
(it) are
and
linearly independent solutions of d2W
+ (n +
at2)w = 0,
dt2
the general solution of the differential equation of (8.1) is
y(x) = exp (
a 2e2
/
[Cl
((2a)h/2 E
x/ + C2
Dip
(\
a )1/2)] 2E
(8.2)
170
8. SOME TURNING POINT PROBLEMS
for arbitrary C, and C2. For (8.1), then,
Y(+1) = exp(-2E)
[Cl
+CZD4±i(
_io(±(2a)1/2
D-
2a '/z E
))
(8.3)
11
provide two linear equations to determine C, and C2. To solve these equations for small e, we need the following asymptotic approximations for D,, (z): 4z
)z"(1 + 0(1)),
exp (-
as
arg zI < 4
Izi -* oo,
4z
exp(- )z"(1 + 0(1)) D. (z) =
- (27r)'/2 e"° exp(zz/4) (1 r (-n)
z"+'
as
+0(l))
I z I - oo,
4
< arg z <
S4 .
(8.4)
Note that D (z) is exponentially large when I zi -* co and arg z = or unless n is a nonnegative integer. Then D. (z)
= exp (- Z2 4) He (z),
n = 0, 1, 2, ... ,
where He,,, the nth Hermite polynomial, satisfies
He. (z) = z"(1 + 0(1))
as
Izi -* oo.
is exponentially small as Thus, when n is a nonnegative integer, Izi -* oo with arg z = or. This difference has a substantial influence on the behavior of the asymptotic solutions to (8.1). For this reason, we shall separately consider four cases.
1. A SIMPLE PROBLEM
CASE (i):
171
a > 0, $ 0 -2n, n = 1, 2, .... Using the appropriate
asymptotic expansions from (8.4), the linear equations (8.3) have a unique solution with the asymptotic determinations e a/
C1(E) =
2«
[y(-1) - (-1)R/zy(l) + 0(1)]
,/Z
and
e-ae
C2(E) =
Dip(-,(2E
[(-1)R/zy(1) + 0(1)]
,/z
) ) Thus, (8.2)-(8.4) imply that for x > 0, Y(x)
- $/2)(y(-1) - (-1)R/2y(l) + 0(l))exp(-ax2/E) - r(-1 2( 7ra )1/2
X1+fl/2
+ ((-l)a/zy(l) + 0(1))(-x)$12 or, for any 8 > 0,
y(x) = O(exp[-axz(l - 8)/E]) + (y(1)x$/2 + 0(1)), so
y(x) -* zz(x) = y(1)x$12
as
e--->O
for
x > 0.
(8.5)
Likewise, for x < 0,
y(x) _ (-x)R/2[(y(-1) - (-1)a/zy(l) + 0(1)) + ((-1)a/zy(l) + 0(1))] or
y(x) - Z, (x) - y(-1)(-x)0/2
as
E-0
for
x < 0.
(8.6)
8. SOME TURNING POINT PROBLEMS
172
Finally s
$/2r(y(-1) - (-1)/2y(l)) r(1 + $/2)
Y(0) _ (-() ) +
1/2'
L
I'(1 + 8/4)
21+,8/4
J y(l)(-i2)R/4 +0(1) ] = O(Efl/°)
r(j - Q/4) since
2n/2
Dn(0) _
r((1 - n)/2)*
Thus, if $ > 0, y converges uniformly on [-1, 1] since y(O) ---> 0 and z, (0) = z2 (0) = 0. (Derivatives of y may still converge nonuniformly
at x = 0, however.) If $ = 0 and y(l)
y(-1), convergence is
nonuniform at x = 0 since
y(o) - (y(1) +y(-1)), while z, (x) = y(-1) and z2 (x) = y(1). Lastly, if Q < 0, but not an even integer, z, (0) and z2(0) are both undefined while y(O) becomes unbounded like 0/4. In summary, then, the anticipated convergence
to the solution of the appropriate reduced problem occurs on the
intervals -1 < x < 0 and 0 < x < 1. Note that complete asymptotic expansions for the solution y of (8.1)
would be obtained if we used complete expansions for the Weber functions and the coefficients C; in (8.2). CASE (ii): imply that
a < 0, $
C, _
and
C2 =
2m, m = 0, 1, 2, .... Here (8.3) and (8.4) e°/2e
1),112y(- 1)
a
-
1/2
+ 0(1))
2a /2 (y(1) + (_I)P/2y(-1) + 0(1)).
1. A SIMPLE PROBLEM
173
Thus, (8.2)-(8.4) imply that
y(x) =
xi
1112 (y(1) + o(l))exp[a(l - x2)/E] (-x)11
y(x) =
+s/2
for
x > 0,
(y(-1) + o(l))exp[a(l - x2)/E] + 0(e-0-01-) for
x < 0,
and
y(O) = O(e`x('-1)/e)
for any 8 > 0. Since a < 0, it follows that
y -+ 0
within
(-1, 1) as e -a 0
(8.8)
with nonuniform convergence generally occurring at both endpoints. CASE (iii):
a < 0, Q = 2m, m = 0, 1, 2, .... Here (8.2) becomes
y(x) = C,exp(-
2E2)D_l-m(i(
2a)1/2x
Hem(-(_2a)1/2x)
+ C2 and e-«/2e
C'
2a
D-1-m
)) 1/2
r[,'(y(1)
-
(-1)my(-1))
l
+ o(l)]
while C2 =
1
Hey,(-(
u2
e) )
[I(y(1) + (-1)my(-1)) + 0(1)]
8. SOME TURNING POINT PROBLEMS
174
Thus, for x # 0, Y(x) _ [(Y(1)
-
(-1)-Y(-1))eXp[a(1
-
x2)/E]
+ (y(1) + (-1)my(-1))xm] + 0(1) and
y(0) = O(ea{1-8>le)
for any 8 > 0.
Since a < 0,
Y(x) - I(Y(1) +
(-1)mY(-1))xm
as
E -> 0
away from x = ±1 where nonuniform convergence generally occurs. Note that, again, the limiting solution satisfies the reduced equation. Here, however, the limiting solution is nontrivial unless (-1)m-Iy(-1)
Y(1) =
CASE (iv):
a > 0, /3 = -2n, n = 1, 2, .... Here (8.2) has the form Y(x) =
)1/Zx) + CZexpl - 2EZ>p "(1(26 >1/Zx),
where a/e
C1
(
ll
(-1)nY(-1))
+ 0(1)J
and ea/2e
C2 =
(Y(1) + (-1)"y(-1)) + 0(1)
.
175
1. A SIMPLE PROBLEM
Thus, for x # 0, Y(x) =
exp[a(z'_n x2)/E] [I("
'1) - (-1)ny(-1)) + o(l)]
+ x-n[I(Y(l) + (-l)ny(-1)) + 0(1)] or
y(x) = O(exp [a(1 - X2)161)
(8.10)
so y becomes exponentially large as a -> 0 away from x = ± 1. This unpleasant behavior can be illustrated by the special equation
sy" + 2xy' + 2y = 0, which can be directly integrated without use of special functions. We summarize our results in Table 1. TABLE 1
Limiting solution as e -* 0
Case
y(-1)(-X)"/2,
(i) a > 0, /3 0 -2, -4, -6,
-1 < X < 0
O(efl/'),
Y(I)X"/',
(ii) a < 0,
0,
/300,2,4,...
X=0
0<X
-1<X<1
i(y(1)Xen + y(-1)(-X)e 2),
(iii) a < 0,
/3=0,2,4,... (iv) a > 0,
-1 <x< I
Solution becomes exponentially large within (-1, 1).
/3 = -2, -4, -6,...
The limiting solution of
ey"+2axy'-a$y=0, y(± 1)
is given in the table.
prescribed
-1 <x< 1
176
8. SOME TURNING POINT PROBLEMS
Examples illustrating Cases (i)-(iv) are given in Figs. 18-22.
YA 2
V
1
FIGURE 18
The solution of ey" + 2xy' = 0, y(-1) _ -1, y(l) = 2, e
small.
YA
2 t
1 t
T -1
FIGURE 19 = 2, a small.
The solution of ey" + 2xy' - 2y = 0, y(-1) = -1, y(l)
1. A SIMPLE PROBLEM
177
1
-1
0
x
1
v
FIGURE 20 = 2, a small.
The solution of ey" + 2xy' + y = 0, y(-1) = -1, y(1) y
2 1
-1
0
1
x
-1
FIGURE 21
- 2, e small.
The solution of ey" - 2xy' + y - 0, y(-1) _ -1, y(l)
178
8. SOME TURNING POINT PROBLEMS
Y
a
4 0
1
T -1
FIGURE 22 The limiting solution of ey" - 2xy' = 0, y(-1) _ -1, y(l) = 2, a small.
2. A UNIFORM REDUCTION THEOREM We wish to study the asymptotic behavior of solutions to boundary value problems for linear equations of the form
ey" + 2x A(x, e)y' - A(x, e) B(x, e)y = 0
(8.11)
on the interval -1 < x < 1, where A and B are holomorphic (i.e., single valued and analytic) in x and a and have asymptotic power series expansions 00
A(x,e) - 2 Aj(x)ei
j0 00
B(x, e)
2 Bj(x)eJ
j0
as a --> 0 for x in some complex neighborhood of the interval -1 < x < 1 and for a in some complex sector S = (0 < jej < e0,
179
2. A UNIFORM REDUCTION THEOREM
jarg eI < 90, Bo > 0). We shall also suppose that A and B are real and
that
Ao(x) # 0
(8.12)
for -1 < x < 1. The point x = 0, then, is a turning point of Eq. (8.11) and a singular point of the corresponding reduced equation. We shall not give a general definition of turning points. [The interested
reader should refer to Wasow (1965), Olver (1973), and McHugh (1971) for general discussions of turning point theory.] We expect that our results in Section 1 where A and B were constants will be useful in obtaining the limiting behavior in the more general case. Let us first introduce the new variable 71 = 71(x) = { a Jox sA0(s)
ds)
1/2
I
(8.13)
where a = Ao(0) and x71(x) > 0. Note that 71'(x) = xAo(x)/r1(x)a
> 0 for -1 < x < 1 and that 71(x) is a monotonically increasing function on [-1, 1] with 71(0) = 0. Thus x can be uniquely expressed as an increasing function of 71. With respect to 71, y satisfies
ey n
+ 120M Ax, e) _ eri (x) ] L A0(x) (i1,(x))2Jy
-
(a2712A(x, s)B(x, s) 1
`
x2Ao(x)
Jy
-0 (8.14)
for r_ = r1(-1) < 71 < 71(l) = 71+. [We note that this equation is slightly simpler than {8.11) since the coefficient of the first derivative is now 2ar1 when e = 0.] We shall now formally obtain THEOREM 8:
There exist holomorphic functions M(71, e), N(71, e), and
a(e) such that any solution y of (8.4) can be expressed in the form y(r1, e) = M(ri, e)w + eN(ri, e)w,
(8.15)
180
8. SOME TURNING POINT PROBLEMS
where w(71) satisfies the comparison equation ew,, + 2a7gw, - (a/3 + ea(e))w
=0
(8.16)
for /3 = Bo(0). Here M, N, and a have asymptotic power series expansions 00
M (71, e) "' 2 M (71)ei
jO 00
N(71, e) ,.. I N(71)ei J o 00
a(e) -- 2 aj ei jo
ase-->0inSwith M(0,s)= 1. Remarks
1. Note that the comparison equation (8.16) is closely related to the
original equation (8.11). In particular, if A(x, e) and B(x, e) were replaced by a = A(0,0) and $ = /3 + ea(e)/a in (8.11), the equations would coincide upon substituting 71 for x. As shown in Section 1, the general solution of (8.16) has the form w(71) = exP 1 -
)1/271)
a 2t
) + C2 D
/ [Cl
2e /'/271
i \ 2e
\
(8.17)
where C, and C2 are arbitrary constants. Differentiating, then,
(,),2 w° (71)
_
E
,q/
[_CIQ_iA eXp
\
2t
/
2)1/2,q)
\\ lV2
( (
(8.18)
[cf. Whittaker and Watson (1952) for the appropriate differentiation formulas]. Thus the theorem implies that the general solution of (8.11)
2. A UNIFORM REDUCTION THEOREM
181
(written as a function of 71) will be y(71) = C1exp
-s
arlZ 2e
lr
/L(M(,q,s)D
((2a i-}R\\ E
2E 1I2N(f1,
+CZexp(- «
s)D (( )''Zf) J
Z
)[M(s)D(i(-_-)2«
2e yz ,q)
+
for q- < n < q+. (8.19)
2. Sibuya (1962) showed that expansions analogous to those for M, N, and a are asymptotically correct in 71 sectors with central angle less that IT. The uniform validity of these expansions in a full neighborhood of 71 = 0 was established by Lee (1969). Such results are very difficult and have been achieved only for equations reducible to Airy's equation [cf. Wasow (1965, 1968)] or Weber's equation. It is likely, however, that certain new work [cf. Sibuya (1973)] will be useful in obtaining more general results.
The idea that solutions of complicated equations with turning points might be analyzed in terms of solutions of simpler equations with analogous turning points was much developed by Langer and others [see, e.g., Langer (1931, 1949), McKelvey (1955), Cherry (1950), and Wasow (1965)]. This work tends to justify the JWKB approxima-
tions widely used by physicists [cf., e.g., Olver (1973)]. For a wide variety of second-order equations, Lynn and Keller (1970) have given formal procedures for representing solutions asymptotically in terms of solutions of simpler comparison equations. Hanson and Russell (1967) have obtained related results for second-order systems. We now give a formal proof of Theorem 8. First, rewrite (8.14) as eY,R +
(2a-q + e (rt, e)) y,
- (a# + tai 8(rt) + eS(ii, e)) Y = 0,
(8.20)
182
8. SOME TURNING POINT PROBLEMS
where x(71, e)
=
(71
))) + EA0(x (A(x, e) - Ao(x)),
B(71) = 271(x) 1
(afl2(x)Bo(x) x2Ao(x)
and 8(ij, e) are holomorphic functions of 71 such that power series expansions as a -> 0 in S. Setting
and 8 have
y = Mw + eNw,, differentiating and using Eq. (8.16), we have
y, _ (M, + (a/3 + ea)N)w + (M + eN, - 2a7jN)w,. Differentiating again and substituting into (8.20), we obtain C(1, e)w + e D(71, e)w, = 0.
Setting C and D separately equal to zero finally implies a pair of differential equations for M and N. Thus, we have C(71,e) = 2a7l(M, - OM) + e[(a - S)M
+ (a/3 + ea) (2 N, + N) + M, + Mm] = 0
(8.21 a)
and
D(71, e) _ -2a(71N,
+Nm]=0
(8.21b)
for 71_ < 71 < 71+. When e = 0, C = 0 implies that
Moll -8(71)Mo=0 and, since Mo(0) = 1, we have Mo(71) = exp [
f'
e(t) dt].
(8.22)
2. A UNIFORM REDUCTION THEOREM
183
Further, when e = 0, D = 0 implies that 2a((rgNo),, + ijNo(9(71) + X0(71)))
=
(71)M0 + 2M0,
=
29(71))
x exp [ Jo'7 9(t) dt].
Integrating, then,
2arIN0 (71) = exp [o Jy 9(t) dt] - k exp [- J"o (B(t) + do (t)) dt]
for some constant k. In order to avoid a pole at 71 = 0, however, we must select k = 1, so No(71) =
2a71[Mo(71)-M(71)
(8.23)
In general, for each j > 1, we equate coefficients of ej successively to zero in (8.21). Thus, M must satisfy 2a71[M, - 9(71)M] = -aj_l Mo + Kj-j(71), where
is known successively; for example,
KO(71) = 80(71)M0
- 0(71)[Mo, + a/3N0] - M - 2a$N0ij.
In order to obtain a holomorphic solution M(71), the right side of the differential equation must have a zero when 71 = 0. Thus, we must select
aj-1 = Kj-1(0) and, defining
Ij-x(71) =
a-[Kj-1(71) - aj-IMo(71)],
we have
M,,-9M =
4-1.
(8.24)
8. SOME TURNING POINT PROBLEMS
184
Since M(0) = 0, then, M (?I) = fo'" Lj-jt) exp [ f B(s) ds]dt.
(8.25)
Further, N must satisfy an equation of the form 2a(7j N, + N + 71N (8(71) + o (q))) = o (71) Mj + 2 Mj, + Jj-, (71), where Jj-, (q ) is known successively, e.g.,
J.(71) = (ao
- 8.(71))N. + i (71) (2arjNo + Mo) + o No, + Nom .
Thus, for N to be holomorphic at 71 = 0, we must select
1f
NJ(71) = 2a7l
n
[(fi(t) + 29(t))Mj(t) + 2Lj-,(t) + Jj-,(t)]
o
X exp [- f ' (B(s) + j0 (s)) ds] dt.
(8.26)
Hence, the expansions for M, N, and a can be determined recursively.
The uniform validity of these results follows from the work of Lee (1969).
3. THE BOUNDARY VALUE PROBLEM In this section, we wish to apply our results to the boundary value problem ey' + 2 x A (x, e)y' - A(x, e)B(x, e) y = 0,
-1 < x < 1
y(-1), y(l) being prescribed constants,
(8.27)
where A and B satisfy the same assumptions as in Section 2. As a consequence of Theorem 8, we have: COROLLARY 1:
Suppose Ao(x) > 0 and Bo(0) _ /3 # -2n, n = 1, 2,
.... Then the boundary value problem (8.27) has a unique solution y(x)
3. THE BOUNDARY VALUE PROBLEM
for e sufficiently small such that
y(x) =
-1 < x < 0
for for for
Z, (x) + o(1) O(efi14)
Z2(x) + o(1)
x=0 0 < x < 1,
(8.28)
where Z, (x) and Z2 (x) satisfy the reduced problems
-1 < x <0
2xZ' - Bo(x)Z1 = 0,
Z, (-1) = y(-1)
(8.29)
and
0<x<1
2xZz - B0(x)Z2 = 0,
(8.30)
Z2(1) = y(1);
Z1,2(x) = y(+1)exp
L
f
B20 (s) x
PROOF: According to (8.19), the solution y as a function of 71 is of the form y(71) = M001)w01) + 0(1)
as a -> 0 in S whenever w, = o(1/e). Here w(71) = exp1 -
)1/2+ f
[CiD_I_JR((2e
)1 Z,q/ C2DjA(i\2e
J
satisfies (8.16). Further, since %3 = /3 + (e/a)a(e) # -2, -4, ... for e
sufficiently small, Section 1 implies W, (71)
w(ri) -
O(e$14) W2 (q)
for for for
ri_ < 71 < 0 'n = 0
0 < ri < ri+,
186
8. SOME TURNING POINT PROBLEMS
where w, and w2 satisfy
y(-1)
(71-) =
271w,n - /3w, = 0,
W,
27jw2 - /3W2 = 0,
w2(71+) =
Mo (r1- )
and
Al) Mo(71+)
and wln(71_) and w2 (71+) are bounded. Thus,
A-1)
/-Mo(71)71,1l2
for ri_ < 71 < 0
Mo(71_)(71_)
y(71) -*
for 71 = 0
0('61"),
MA
Mo(q)71fl12
for 0 < 71 < 71+.
Mo(71+)(71+)0/2
Note, however, that (71))\71±/1/Z=exp
Mo
= exp
f
dq](71±)1lZ
-a> IX71(x)Bo(Z)71,(x) dx]
J
+1 2Ao(x)x L J_x
r
= expEJxB2ss)ds], +,
x
0
so the results follow. An example illustrating Corollary I is given in Fig. 23. Likewise, we have
Suppose that Ao(x) < 0 and Bo(0) 2m, m = 0, 1, 2, .... Then the boundary value problem (8.27) has a unique solution
COROLLARY 2:
y(x) for e sufficiently small such that
y(x) -* 0
as
a -> 0 for
-l < x < I .
(8.31)
3. THE BOUNDARY VALUE PROBLEM
187
y A
2T
x
FIGURE 23 The limiting solution of ey" + 2xy' + 3(1 + x2)y = 0, y(-1) = -1, y(1) = 2, a small.
PROOF: Since all solutions of
ev, +2arlvn-a(/3+ea)v=0 v(71±)
bounded
and their derivatives tend to zero as a -> 0 when a -t 0 and 0 + ea # 0, 2, 4, ..., the results follow from the theorem. For further discussion of these results, see O'Malley (I 970d).
The most interesting and obstinate case occurs when Ao(x) < 0 and BO(0) = 2m, m = 0, 1, 2, .... As we found when A0 and B0 were constants, it is then possible to obtain a nonzero limiting solution within (-1, 1). Such "resonance" phenomena are potentially useful in applications. For example, boundary value problems such as (8.27) arise in the study of flow at high Reynolds number between counter-
rotating disks [cf. Watts (1971)]. Ackerberg and O'Malley (1970) showed that resonance will occur if the coefficient a(e) in the comparison equation (8.15) is appropriately exponentially small as
8. SOME TURNING POINT PROBLEMS
188
e - 0. If, however, any term in the asymptotic expansion of a(e) is nonzero, the limiting solution within (-1, 1) will be trivial. Thus, the example
ey"-xy'+(m+x)y=0 used by Watts has the trivial limiting solution since ao and note that Ko(0) # 0] while the "nearby" equation
0 [cf. (8.24)
ey"-xy'+(m+x-e)y=0 allows nontrivial limiting solutions. Since we are merely able to determine the asymptotic expansion for a(e) termwise, we have not given a satisfactory sufficient condition for resonance, though aj = 0, j = 0, 1, 2, ... , provides a countable infinity of necessary conditions. It would obviously be worthwhile to have an easy-to-check sufficient condition.
When resonance does occur, the limiting solution will satisfy the reduced equation. Thus, Ackerberg and O'Malley showed that, when f!I xAo(x)dx > 0, the limiting solution will be Zi(x) = y(- 1) (- x)m exp
LJ
xi
(Bo(s)2s 2m ) ds]
throughout -1 < x < 1 while, when fl, xAo(x) dx < 0, the limiting solution will be
Z2(x) - y(l)xmexpIJx
(Bo(s)2s
2m) ds]
on -I< x < 1. When fl, xAo (x) dx = 0,71- = -'h+, and y(71)-M0(ij)v('7)
on 'I-<71 <71 ,
where V(71)
)') =21(y(1)+y(rlm 71* 71-
m+o(1)
189
3. THE BOUNDARY VALUE PROBLEM
and vn(ij±) = O(1/e). Thus, nonuniform convergence occurs at both endpoints and
y
- Cx°' exp [fox
(Bo(s)2s
2m) ds]
within (-1, 1) for some constant C # 0 (in general). An expression for C can be easily obtained. It shows that the limiting solution will not be the average of Z, and Z2 (as was incorrectly claimed by Ackerberg and O'Malley). Attempts to solve these problems with resonance using matched asymptotic expansions have not succeeded. By combining matched expansion techniques and the WKBJ method it has been possible to obtain solutions exhibiting resonance (cf. Ackerberg and O'Malley).
The results of Kreiss and Parter (1973) further indicate the subtle nature of the resonance phenomena. A more unpleasant unbounded resonance can occur when Ao(x)
> 0 and Bo(0) = -2, -4, -6, ... as
(8.10) suggests.
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INDEX Casten, R. G., 16, 17, 195 Cauchy problems, 85, 102
A
Abrahamsson, L. R., 60, 190 Ackerberg, R. C., 187, 188, 189,
Chen, J., 73, 102, 134, 158, 167,
190
191
Chemical reactors, 157 Cherry, T. M., 181, 191 Cochran, J. A., 22, 39, 41, 191
Airy's equation, 181 Amundson, N. R., 157, 198 Anderson, B. D. 0., 140, 190 Angular solution, 119 Arbib, M. A., 141, 154, 194 Aris, R., 86, 194 Asymptotic approximation, 13 Asymptotic equality, 13 Asymptotic expansion, 13, 15 Asymptotic sequence, 12 Asymptotically negligible, 13 Athans, M., 140,190
Coddington, E. A., 36, 50, 116, 120, 191
Cohen, D. S., 116, 157, 167, 191 Cole, J. D., 10, 22, 191 Collins, W. D., 156, 191 Comparison equation, 180, 181 Composite expansion, 17, 18 Controllability, 154 Cooke, K. L., 106, 191 Courant, R., 141, 192
B
Bagirova, N. Kh., 156, 190 Bang-bang control, 156 Bellman, R., 33, 58, 190 Bjurel, G., 60, 190 Bobisud, L. E., 60, 190 Borel-Ritt theorem, 14 Boundary layer, 8 Boundary layer correction, 8, 80 Boundary layer jump, 78
Boundary layer thickness,
D Dahlquist, G., 60, 190 Delay equations, 102 Delta function, 156 Desoer, C. A., 60, 192 Dickey, R. W., 136, 192 Difference equation, 111 Differential-difference equations, 102-114
75,
DiPrima, R. C., 73, 192 Dorr, F. W., 9, 12, 192
101
Boundary layer type, functions of, 85 Boyce, W. E., 60, 191 Burghardt, A., 157, 191
E Eckhaus, W., 16, 192
Eigenvalue problems, 606 Enzyme kinetics, 86 Erd6lyi, A., 2, 12, 15, 22, 23, 31, 39, 84, 94, 120, 192 Exceptional case, 49, 71
C Calvert, J., 60, 190 Cancellation law, 49, 71 Carrier, G. F., 12, 191
203
INDEX
204
Inner and outer expansions, see
Falb, P. L., 140, 141, 154, 190,
Matched asymptotic expansions
194
Fast-time, 22 Feshbach, H., 32, 196 Fife, P. C., 119, 123, 125, 192 Fraenkel, L. E., 16, 17, 192 Friedman, A., 60, 192 Friedman, B., 56, 193 Friedrichs, K. 0., 15, 193
Fundamental set of asymptotic solutions, 46, 61, 68, 69 G
Genetics, 86 Geometrical optics, 48 Gilbarg, D., 97, 193 Gol'denveizer, A. L., 17, 47, 193 Gordon, N., 102, 193
H Haber, S., 119, 193 Haddad, A. H., 155, 193 Hadlock, C. R., 140, 155, 193 Hale, J. K., 78, 107, 193 Hamiltonian, 141 Hamilton-Jacobi theory, 141
Handelman, G. H., 60, 65, 191,
Inner expansion, 17 Inner region, 17 Interior nonuniformities, 117
J Jameson, A., 156, 194 Jamshidi, M., 140, 193 JWKB approximation, 181 K
Kalman, R. E., 141, 154, 194 Kaplun, S., 15, 194 Kasymov, K. A., 116, 194 Kato, T., 32, 194 Keller, H. B., 60, 167, 190, 195 Keller, J. B., 48, 50, 59, 60, 65, 181, 193, 195, 196, 198 Kevorkian, J., 22, 195 Kokotovi6, P. V., 140, 155, 193, 195
Kollett, F. W., 22, 195 Krein, S. G., 85, 195 Kreiss, H. 0., 60, 189, 190, 195 Kung, C-F, 155, 195
193
Hanson, R. J., 181, 193
Harris, W. A., Jr., 60, 73, 134, 193, 194
Heinekin, F. G., 86, 194 Hilbert, D., 141, 192 Hilbert space, 60 Hoppensteadt, F. C., 85, 86, 134, 154, 194
Hukuhara, M., 59, 194 Hutchinson, G. E., 111, 194 I Imanaliev, M. I., 156, 190 Implicit function theorem, 78 Infinite interval problems, 4, 85, 154
L
Lagerstrom, P. A., 15, 17, 195 Langer, R. E., 181, 195 Latta, G. E., 195 Lee, R. Y., 181, 184, 195 Levin, J. J., 195
Levinson, N., 36, 50, 116, 119, 120, 191, 193, 195, 196 Lewis, R. M., 48, 195 Limiting solution, 7 Lions, J. L., 156, 196 Lindberg, B., 60, 190 Linde, S., 60, 190 Lynn, R. Y. S., 181, 1%
Lyusternik, L. A., 85, 116, 200, 201
205
INDEX
M McHugh, J. A. M., 179, 1% McKelvey, R. W., 181, 1% Macki, J. W., 134, 1% Mahony, J. J., 22, 1% Matched asymptotic expan
Order symbols, I Outer expansion, 16, 81 Outer region, 16 Outer solution, 16, 81
sions, 15-21 Matching, 17, 81
Parabolic cylinder functions, 169 Parasitic parameters, 140
Maximum principle, 9 Mazaika, P. K., 60, 198 Meerov, M. V., 47, 1% Method of perturbation, I Meyer, K. R., 103, 191 Meyer, R. E., 8, 16, 196 Miranker, W. L., 85, 1% Modulo arithmetic, 50 Moore, J. B., 140, 190 Morrison, J. A., 22, 196 Morse, P. M., 32, 1% Multiple solutions, 165, 167 Murphy, W. D., 86, 1% Murray, J. D., 10, 196
Parter, S. V., 9, 12, 166, 189, 193,
P
198
Patching, 16 Peclet number, 158 Perko, L. M., 22, 198 Pittnauer, F., 14, 198 Population models, 111 Power series, 14 Prandtl, L., 8, 198 Protter, M. H., 9, 198 Puiseaux polygon, 59
Q Quasilinear equations, 119-124 Quasilinear systems, 125-134
N
59, 60, 66, 71, 73, 75, 101, 102,
R Rayleigh, Lord, 60, 198 Raymond, L. R., 157, 198 Reddy, P. B., 111, 199 Reduced problem, 7, 49, 50, 72 Regular perturbation method, 1, 27-34 Regulator problems, 139-156 Reiss, E. L., 17, 23, 198, 199 Rellich, F., 32, 199 Retarded argument, 109 Resonance, 187 Riccati matrix, 154 Riekstins, E. Y., 15, 199 Russell, D. L., 181, 193
103, 109, 119, 120, 123, 134, 148, 151, 155, 156, 167, 187, 188, 189, 190, 191, 193, 194,
Sannuti, P., 111, 140, 155, 195,
Nau, R. W., 60, 197 Nayfeh, A. H., 22, 197 Newton polygon, 59 Nonexceptional case, 49, 71 Nonuniform convergence, 4 Numerical techniques, 86
0,1
0,1 Oden, L., 60, 190
Olver, F. W. J., 2, 12, 179, 181, 197
O'Malley, R. E., Jr., 22, 41, 50,
197, 198
Optimal control, 111, 139-156 Order reduction, 155
S 199
Schmidt, H., 15, 199 Searl, J. W., 22. 23, 199
206
Secular terms, 23 Shampine, L. F., 9, 12, 192 Shensa, M. J., 60, 192 Shock layer, 97 Shock phenomena, 10 Sibuya, Y., 181, 199 Simmonds, J. G., 60, 197 Singular arcs, 156 Singular perturbation, I Slow-time, 22 Stengle, G., 59, 199 Stiff differential equations, 60, 86 Stoker, J. J., 15, 193 Stretching transformation, 16
Stretched variable, 81, 95, 99, 101, 116
T Transition layer, 119 Trenogin, V. A., 59, 85, 199, 200 Tsuchiya, H. M., 86, 194 Tupciev, V. A., 125, 200 Turning point, 44, 168-189 Turrittin, H. L., 48, 59, 70, 147,
INDEX
V
Vainberg, M., 59, 200 van der Corput, J. G., 15, 200 Van Dyke, M., 8, 15, 16, 18, 200 Vasil'eva, A. B., 16, 73, 103, 125, 190, 200
Vibrating string, 60-65 Vishik, M. I., 85, 116, 200, 201 W Walker, R. J., 59, 201 Wasow, W., 12, 14, 16, 44, 45, 73, 120, 147, 179, 181, 201 Watson, G. N., 169, 180, 201 Watts, A. M., 187, 201 Weber functions, 169, 170 Weinberger, H. F., 9, 198 Whitham, G. B., 22, 201 Whittaker, E. T., 169, 180, 201 Wilde, R. R., 155, 201 Willett, D., 31, 94, 120, 201 Wyman, M., 15, 192
199
Two-parameter problems, 66-75, 94-102 U Undetermined coefficients, 53, 62
Uniform reduction, 178-184
Y
Yackel, R. A., 155, 195, 201 Yarmish, J., 116, 125, 201 Z Zaleski, T., 157, 191