Solution of differential equation models by polynomial approximation

1 d dp - -(n ) = - - = constant r dr rz dz One example is the power law fluid described by the constants m and n [Bird (1960), p. 11]: (9)

where Vmax and Vav have the same meaning as in equation (8) and M = (n + 1)/n. In laminar flow of a pseudo-plastic material (n < 1 or M > 2) the velocity profile is steeper than the parabolic velocity profile (8) for a fluid with Newtonian properties. A fluid with dilatant behavior is characterized by n > 1. Fluid flow in the turbulent regime is described empirically by a relation similar to (11) but with M » 1. Vz is independent of r except close to the wall r = R. In packed beds we shall assume the velocity profile to be flat since the effect of an r-dependent axial velocity is probably small in comparison with other effects-at least for a large length-to-diameter ratio. · With the velocity profile given by (8), (11), or Vz = constant, we now treat the steady state solution of (1) and (2) for a homogenem~s fluidphase reaction in a tubular reactor. k and D are assumed to be independent of cA and T, but not necessarily the same in the r- and z-directions. D and k will usually be empirical quantities describing axial mixing in a packed bed or a mixed solid-fluid heat conductivity. The conditions for v are those used above to solve the trivial versions of Navier-Stokes equation for unidirectional flow. With these remarks the steady state mass and heat balances for sections 1.2 to 1.4 can be set up.

In tube flow, dvz/ dr is always negative and the absolute value sign in (9) can be eliminated: T,z

=

(11)

ml- ~;1"-l(- ~;)

=

m(- ~zr

Now the third equation of (3) becomes m]_ !!._[j _ dvz)n] = _ dp r dr '\ dr dz

(10)

(12) ,1

aT

a ( aT) + k z a2T 2 + Q

pc vz- = k - - r v az r ar aR

az

(13)

where superscripts r and z are used for radial and axial transport properties, respectively. V is based on the empty tube velocity whether the bed is packed with solid particles or not and RA is conversion per unit volume packed bed. 2

10

Mathematical Models from Chemical Engineering

Chap. 1

Sec. 1.2

Steady State Homogeneous Flow Model

11

. In~egrating (14) ~nd (15) over the cross section, neglecting the axial dtffus10n terms and Imposing a condition of zero flux across the cylinder axis, yields

1.2 Steady State Homogeneous Flow Model for a Reacting System

I

1.2.1. Derivation of a one-dimensional model

i

0

(15)

Vz acAd

-- X 0 Vav a(

Dimensionless independent variables { = z/ L and x = r/ R are introduced in (12) and (13): (14)

1

1

2

(acA) = 2LDr -2R

Vav

2

x=1

.!!:__ aT dx 2 = 2 2Lk r (aT) Vav a( R VavPCp ax x=1

0

+2

Vav

X

A

i~ 1

0

dx

Qx dx

VavPCp

(16)

(17)

For an impermeable wall the concentration gradient at x = 1 is zero while by the film theory the temperature gradient is proportional to th~ difference between the reactor wall temperature T w and the fluid temperature Tx= 1 just inside the wall.

kr(aT\

=

R a-;J x= 1

These equations are extremely simple compared to (1) to (3): The momentum balance has been solved independently of the mass and energy balances, the time dependence of cA and T has been dropped, and the transport properties are assumed constant. Even so, the resulting model consists of two coupled nonlinear partial differential equations (RA and Q depend nonlinearly on cA and T); its complete solution is a major numerical enterprise that is beyond the scope of this text. Simplifications of the two equations are indeed desirable and often possible in practice. Here we shall discuss how these simplifications can be brought about by discarding small terms and by suitable averaging methods that transform (14) and (15) into two coupled nonlinear firstorder ordinary differential equations (25) and (26). In subsection 1.2.3, the axial dispersion terms on the right-hand sides are reintroduced-but the model still consists of ordinary differential equations. Finally, in subsection 1.2.4, the total set of side conditions of (14) and (15) are discussed and a particularly simple variant of the equations is treated a little further. In equations (14) and (15) the coefficients of the second-order derivatives with respect to { are frequently several orders of magnitude smaller than the coefficients of the first-order derivatives. Consequently these terms can frequently be neglected. On the other hand, the coefficients of the radial diffusion terms are quite large si\ce L/ R is usually » 1. T?us, e~cept for unusually r.apid reactions, one may assume that the radial vanatwn of cA and T IS of minor importance; this indicates that the mean values of cA and T over the cross section of the reactor may be used rather than the x- and (-dependent variables in (14) and (15) if the purpose of the calculations is to compute the conversion of a given feed in a reactor of given length L.

ax

f -LR 1

-

(18)

U(Tw- Tx=I)

Inserting in (16) and (17) yields

2

I

1

0

II

Vz acA L --;xdx = -2 -RAxdx Vav a!:> o Vav

(19)

aT LU I --Qxdx L =2 . (Tw- Tx=I) + 2 I -Vz -;:xdx a!:, RvavPCp VavPCp 1

2

0

1

Vav

0

(20)

The average concentration and temperature in the cross section of the reactor are defined by CAVav

=

I I

1

2vzCA ((, x)x dx

(21)

2vzT({, x)x dx

(22)

0

1

Tvav =

0

and fo~ sn:all radial concentration and temperature gradients the righthand Side mtegrals of (19) and (20) are well approximated by

I

2

0

1

2

f

1

L -RAx dx Vav

L

L

= -RA (cA,

_ T)

(23)

Vav

L

_

--Qxdx = --Q(cA, T) o VavPCp VavPCp

(24)

Finally we wou~d.like to s~bstitute T w - Tx=I in (20) by a corresponding temperature '!nvmg force m terms of the average temperature f of (22). Now ITw- Tl is necessarily greater than ITw + Tx=II and in order to compensate for this a modified heat transfer coefficient [J (which is smaller than U) is introduced at the same time.

Mathematical Models from Chemical Engineenng

12

Chap. 1

Sec. 1.2

Our model (14) and (15) n_?W reduces to a one-dimensional form in terms of the variables cA and T defined by (21) and (22):

Steady State Homogeneous Flow Model

13

~; = P Day~ exp [ ~ 1 - ~) J- Hw(IJ -

Ow)

(32)

Lko n-1 D a= - c 0

(25)

Vav

{3 = c 0 (-dH) pcPT0 (26)

Hw =

20 __!:__ R

A suitable choice of

0

is obtained by (27)

where the chapter 6. Let us where the

best choice of a depends on the velocity profile as discussed in For a flat velocity profile, a = i is best. . . . consider (25) and (26) for an nth-order Irreversible reaction rate constant k is given by an Arrhenius form: RA

= kcA = a exp (- ~T)c':..

(28)

The Damkohler number, Da, is a measure of rate of reaction at inlet conditions, y is a dimensionless activation energy, {3 is the adiabatic temperature rise relative to inlet temperature T 0 if all the reactant is consumed, and Hw is a dimensionless heat transfer coefficient (the number of heat transfer units in the reactor). Thus the resulting reactor model is a set of two nonlinear first-order ordinary differential equations that are much easier to solve than (14) and (15). The major model simplification lies in the averaging process (16) and (17), which can also be used if the axial diffusion (or dispersion) terms of (14) and (15) are retained. In certain cases (31) and (32) can be solved in closed form. Thus for an adiabatic reactor ( 0 = 0), multiplication of (31) by {3 and addition to (32) yields

and the heat production term Q arises due to the heat of reaction

d d((O

(29)

In (28), a is a temperature-independent constant, E is the activation energy, and R 0 the universal gas constant. . Let the reactor inlet temperature and concentration f = To and cA = co, respectively. RA can be rewritten into a dimensiOnless form:

?e

CpPVav

+ {3y) = 0

or () + {3y is equal to a constant, which is determined from the inlet conditions 0 = y = 1 to be 1 + {3. Now() can be eliminated from (31): dy

n

d( = -Day exp

[

1

y{3(1 - y) + {3(1 _ y)

J=

-F(y)

or (30)

where()= T/To, y = cA/co, k(To) = ko, and 1' = E/RaTo. Equations (25) and (26) now become

~=

-

Da y" exp [

~ 1 - ~) J

(31)

C-

Jl

du

Y

F(u)

(33)

For each value of y the integral (33) must be found by a numerical method (except for yf3 = O) but tabulation of ( as a function of y is nevertheless a trivial problem. If y is desired at a fixed grid of C-values, (33) is interpreted as an algebraic equation in y. Inverse interpolation in a table of ( versus y may give sufficiently accurate values for y at the grid points.

Mathematical Models from Chemical Engineering

14

Chap. 1

The final step in the solution of (31) and (32) for an adiabatic reactor is to determine the temperature profile from the algebraic equation () = 1 + {3[1 - y(()], which, of course, does not present any problem once y (() is known. It is of some interest to note that an analytical solution of (31) and (32) is possible also for a nonadiabatic reactor if E 0, i.e., if the rate constant is temperature independent as is the case in the high temperature S0 2 -converter beds. After solving the mass balance and inserting y(() in the energy balance, this becomes a first-order inhomogeneous equation with constant coefficients. A "hot spot," i.e., a maximum in 0((), may well occur also when E = 0. After the hot spot, all reactant has been burnt off and the reactor acts as a simple heat exchanger. Exercise 1.1 treats a reactor where this simple solution is applicable.

Sec. 1.2

15

Steady State Homogeneous Flow Model

is found by solution of (31) with () = 1 and n

1:

y(( = 1) = exp (-Da)

2. In the absence of radial diffusion (D' = 0), a fluid element retains its identity throughout its passage of the reactor and

=

i

1

y(( = 1) = 4

o

x(l - x 2 ) exp (

2(1

Da

2 )

dx

-X )

The ave:age ~o~centration at the rea_ctor outlet obtained from (34) for a fimte D hes between the solution of these limiting cases that are compared in table 1.3 for various values of the Damkohler number. TABLE

1.3

AVERAGE OUTLET CONCENTRATION FOR INFINITE AND ZERO

1.2.2. Limiting solutions to the full fluidphase model

RADIAL DIFFUSIVITY

Da

The maximum effect of the simplifications leading from (14) and (15) to (31) and (32) can be estimated by a study of two limiting cases-either infinite or zero radial diffusivity. This will be done for an isothermal first-order reaction where (14) and (15) as well as (31) and (32) are particularly simple to solve. The full model for laminar Newtonian flow with first-order isothermal reaction in an empty tube is 2 ay D'L 1 a ( ay) 2(1 - x ) - = -2- - - x- - Day

a(

R

Vav X

ax

(34)

ax

when the axial diffusion is neglected. The side conditions of (34) are

y

CA =-

= 1 at ( = 0

0.1 0.2 0.5 1.0 2.0

D'

~

oo

D'

0.905 0.819 0.607 0.368 0.135

=

0

0.910 0.832 0.649 0.443 0.219

It is seen that the model simplification is without practical consequences when the reaction is slow since both limiting models give the same result for small values of Da. It is shown at the end of the next subsection that a finite radial dispersion can be treated by means of an equivalent axial dispersion term; even though this introduces a secondor_der derivat~ve i? (31),. the resulting model is still considerably simpler than the partial differential equation (34).

Co

ay

ax

= 0 at x = 0 and x

1 for (

2::

0

The averaging process of subsection 1.2.1 can be performed even though the axial diffusion terms are retained. One obtains the following set of second-order nonlinear equations:

Two limiting cases are considered. D'L 1. R 2 ~ ~ oo

or

y((, x) = y(()

for all (

With no radial gradients the solution of (34) and (31) are identical and the average outlet concentration

Y(i:

= 1) =

1.2.3 Axial dispersion of mass and energy

r

2

4(1 - x )y(1, x)x dx

p~M ~;; - ~~ -

1d e

Day" exp

2

de

[ (

PeH d(z- d( + {3 Dayn exp 'Y

H 1

1-01)] -

%) ] = 0

(35)

Hw(O- Ow)= 0 (36)

16

Mathematical Models from Chemical Engineering

Chap. 1

The two dimensionless groups PeM and PeH are axial Peclet numbers defined by _ VavL Pe M - - D

Sec. 1.2

17

Steady State Homogeneous Flow Model

Mass balances for the two outer sections are

1 d 2y Pe_ d( 2

-

dy d~

= 0 for ~ < 0 with y = 1 at ~ ~ -oo

-

dy d~

= 0 for ~ > 1 with y finite at ~ ~ oo

and 2

1 d y

Pe+ d~ 2 The transport coefficients D and k which enter into the Peclet numbers are empirical quantities which bear no relation to the molecular properties. Experiments for a packed bed [Gunn {1971)] show that PeM is between 0.8 and 2 times the bed length per particle diameter ratio L/ dP for various packings. PeH is smaller, but of the same order of magnitude. Since L/ dP is very large in most industrial reactors, the numerical coefficient of the second-order terms are small. Since the second derivatives are of the same order of magnitude as the first derivatives, it is certainly reasonable to neglect the axial dispersion terms in a steady state calculation as we did in subsection 1.2.1. Models (35) and (36) are interesting, however, from a mathematical point of view; as we shall see at the end of the subsection, inclusion of an axial diffusion term may lead to a useful interpretation of the often far more important radial diffusion terms. Solution of the set of two second-order differential equations (35) and (36) requires two side conditions for each dependent variable. The correct specification of these side conditions is by no means an easy task and it might be helpful to consider the slightly simplified model where Hw = 0. Here y(~) is a monotonically decreasing function of ~ and () (~) is an increasing function when the reaction is exothermic. Mass or energy is carried toward ( = 1 by convective transport. The diffusion mechanism acts in the direction of decreasing y and () and consequently by this mechanism energy is carried toward ~ = 0 while mass is carried toward ( = 1. Both effects tend to decrease y and increase () near the reactor entrance and it would certainly be incorrect to take the side condition y = () = 1 at ~ = 0 from section 1.2 since obviously y < 1 and () > 1 at ~ = 0. Bischoff {1961) gives a very clear presentation of the side condition problem, and his procedure will be briefly reviewed for the mass balance since the resulting side conditions apply to the present example as well as to a more cpmplicated example in the next subsection. Assume *p;;tt the reaction section 0 ::::; ~ ::::; 1 is preceded by an entrance section -oo<~ < 0 and followed by an exit section 1 < ~ < oo. The Peclet numbers are, respectively, Pe_ and Pe+ in these sections, and no chemical reaction occurs except in 0 ::::; ~ :S 1-a situation that may be visualized by a radiation-catalyzed chemical reaction where shielding prevents reaction for ~ < 0 and ~ > 1.

The solutions of these equations are y

= 1 + A 1 exp (Pe_ ~) for ' < 0

y

= A 2 + A 3 exp (Pe+ ~) for

~

(37)

> 1

where A 3 = 0 to satisfy the condition at~~ oo. The solutions for the outer sections tell us that y decreases from 1 to 1 + A 1 when~ increases from -oo to 0 and that y is constant (=A 2 ) for ~ > 1. Mass balances for differential volumes around ~ = 0 and ~ = 1 are

1 dy_ Y- - Pe_ d~

=

1 dy+ Y+ - PeM d~ {38)

1

dY_ 1 dY+ y_ - PeM · d~ = y+ - Pe+ d~

where Y+ and Y_ are values of y inside the reactor, Y- = 1 + A 11 Y+ = A 2 , and the derivatives in the outer sections are found by differentiation of {37): dy _ _ dyj dr - drl ~

~

(-0-

_ - A1 Pe_

and

dY+ _ dyj d - d I (

~

_

- -

(-1+

0

Inserting these results into {38) one obtains

1- Y+

- _1_ dy+ PeM d(

1 dY_ y_ - PeM d~ = Az

{39) (40)

We finally apply that the concentration profile is continuous across the boundaries at ~ = 0 and ~ = 1. Thus Y+ = Y- = 1 + A 1 and Y_ = Y+ = A 2 whereby {40) is reduced to 1 dY_ PeM d( = 0

or

dY_ d~ = 0

for any finite PeM

(41)

Equations (39) and {41) are the required set of boundary values for {35).

18

Mathematical Models from Chemical Engineering

Chap. 1

The boundary conditions are independent of the rate expression in (35) and also independent of the specific values of Pe+ and Pe_, i.e., on the conditions in the outer sections. An identical derivation for the heat balance shows that the side conditions for (36) are completely analogous to (39) to (41) with (0, PeH) instead of (y, PeM)· If Pe_ ~ oo, y very rapidly attains the value 1 for ( < 0 - which is the reason why (39) is often described as implying a discontinuity of y (or 0) at this point. The dependent variables are both continuous of course but the first derivatives are discontinuous except when Pe_ = PeM, i.e., when the diffusivity is the same in the entrance sector and in the reactor. It is also noted that the gradient at ( = 1 is zero only when PeM is finite and in fact nothing may be concluded in advance about the gradient at ( = 1 when PeM ~ oo. This is in full agreement with our expectations: (39) degenerates for PeM~oo to y = 1 at ( = 0 and (35) degenerates to the first-order equation (31) for which only one side condition is necessary. It is finally noted that direct integration of (35) and (36) for Hw = 0 and insertion of the side conditions derived above yields

1 - y(( IJ((

~ 1) ~ Da

= 1)- 1 =

r

exp [

k ~) k

~ 1 - ~)

fJ Dar exp [

~ 1-

d( d(

or 0

= 1 + {3(1 -

y)

at ( = 1

Furthermore if PeM and PeH both approach zero, the model degenerates to the adiabatic stirred tank model for which 0 and y are constant and equal to their exit values 0(( = 1) and y(( = 1). Consequently the model with side conditions (39) to (41) and Hw = 0 correctly reduces to the adiabatic stirred tank model in the limiting case (PeM, PeH) = 0:

Sec. 1.2

Steady State Homogeneous Flow Model

any reactor with axial dispersion andy(() must cross the plug flow profile exactly one time in 0 < ( < 1. The nonisothermal reactor might behave quite differently since the transport of energy toward ( = 0 greatly enhances the chemical reaction in the inlet section of the reactor and the conversion may in fact become higher than in a plug flow reactor. It is well known that the limiting case (42) of an adiabatic stirred tank reactor may exhibit three steady states even for first-order reactions. The other limiting case (31) and (32) is an initial value problem with no singularities in the derivatives and it has one and only one solution whatever the values of the parameters. Consequently one suspects that for a small value of the Peclet number and large enough values of the heat sensitivity parameters {3 and y, several solutions of the adiabatic reactor model with axial mixing may exist in a certain range of Da-values. Over the last 10 years numerous articles from chemical engineering literature have treated the two equations (35) and (36) and many interesting phenomena of multiple solutions have been found also for Hw = 0. A short list of references to some interesting papers on this subject is given at the end of the chapter. In table 1.3 we considered two limiting solutions of the partial differential equation (34), wJ:lich describes first-order isothermal reaction for laminar flow of the reactant in an empty tube. The case D' ~ oo corresponds to a plug flow reactor while the caseD'= 0 corresponds to a strong apparent axial dispersion of reactant: The fluid that reaches ( = 1 along an x -coordinate close to 1 has had an extremely long residence time in the reactor while the fluid in the center of the tube reaches the outlet after L/2vav time units. The result of the uneven distribution of residence times is clearly a much smaller conversion for large values of Da. The interpretation of a less than perfect radial mixing in terms of an axial dispersion coefficient was originally given by Taylor (1953). For flow in empty tubes a one-dimensional model approximation for a system with laminar flow requires the use of an "effective" axial diffusivity given by Dz ~ Dz eff-

0 = 1

+ {3(1 -

19

+ _...!:_ R 48

2

Va/

D'

1 1 1 VavR 2 or - - = - - + - - PeM,eff PeM 48 D'L

(43)

(42) y)

Axial mass diffusion in an isothermal reactor implies that y(( = O) < 1 and that the concentration gradient is less steep than in an ideal plug flow reactor. The plug flow reactor gives a higher exit conversion than

If we consider the effect of the last term alone, we may compare the average concentration at the reactor outlet as calculated by (34) and by (35) for various values of the dimensionless group DrL/vavR 2 • Table 1.4 shows the result of a calculation by the complete model (34) and by the "improved" averaged model (35) for Da = 1.

20

Mathematical Models from Chemical Engineering

TABLE

Chap. 1

0.1 0.2 0.5 1

2

Yc=I (full model)

Yc=I [by (35) and (43)]

0.4038 0.3931 0.3809 0.3750

0.4179 0.3982 0.3818 0.3752

Steady State Homogeneous Flow Model

21

Boundary conditions at ( = 0 and at ( = 1 are taken from the discussion in subsection 1.2.3.

1.4

IMPROVED ONE-DIMENSIONAL REACTOR MODEL

D'L/VavR

Sec. 1.2

c = 0: l

1 ay y---=1 PeM ac

and

1 ao ()---=1 PeH ac

ay = ao = 0 ac ac

= 1:

(45)

Symmetry across the cylinder axis gives It is seen that the Taylor dispersion concept permits a much better model approximation than the crude averaging process of subsection 1.2.1 that leads to either Yc= 1 = 0.3679 or to Yc= 1 = 0.443 for all entries of table 1.4 depending on whether complete mixing or no mixing at all is assumed. Wicke (197 5) reviews the application of Taylor dispersion for packed beds where numerical coefficients are selected that are different from those of (43).

1.2.4 Boundary conditions for twodimensional model-the extended Graetz problem

-Day n exp

=0 (44)

ao = vz(x)ac

2 2 L 21a ( xao)] + {3 Day n exp [~ 1 - -1)] - 1 [a- 02+ PeH a( R X ax ax ()

= 0

The assumption of Dz = D' and kz = k' is not correct for flow in a packed bed and the resulting model will become considerably more complex than (44).

ao ax

ay = ax

=

o

at

x= o

(46)

0

at x = 1

(47a)

The side conditions at x = 1 may be much more complicated for the energy balance. Assume that the tube wall is insulated from ( = 0 to C = ( 1 and that heat transfer to an exterior medium of temperature T w(C) is admitted for C1 ~ C ~ 1:

ae =

[~ 1 - -1)] ()

=

The tube wall is assumed to be impenetrable for the reactant

-

After having discussed some important and (from a practical point of view) very reasonable simplifications of the complete model for the fluid phase, we shall briefly return to (14) and (15) and review the boundary conditions of these equations. For laminar flow in an empty tube, D and k can be taken to be equal in the radial and axial directions and they are molecular transport properties. The dimensionless model is

2 2 ay 1 [a y L 1 a ( ay)] Vz(x)-=----+---xac PeM a( 2 R 2X ax ax

ay ax

ax

0

.

at

X

= 1 for 0 :::; ( < (1 (47b)

Ow may be a constant or it may vary with (. For a given mass flow of coolant and a given inlet temperature T w(C = 0) or T w(C = 1), a total heat balance over part of the reactor plus heat exchanger gives an ordinary differential equation to determine T w(C) simultaneously with the solution of the partial differential equation model for the reactor. This type of side condition is discussed in chapter 4 for a considerably simpler model. It should be noted that (44) with its side conditions (45) to (4 7) is a boundary value problem in ( as well as in x. A considerable simplification is obtained if the axial diffusion terms, which are often quite insignificant, are neglected; the equations then become of first order in ( and can be solved by a marching technique from ( = 0. The somewhat complicated boundary condition (4 7b) at x = 1 presents no difficulty, the adiabatic and the nonadiabatic reactor differ only in the side condition for the energy balance and not in the structure of the differential equations.

22

Mathematical Models from Chemical Engineering

Chap. 1

It may finally be observed that the two equations (44) as well as the

side conditions can be decoupled if PeM = PeH. This leads to another significant simplification; however, this is not based upon a true physical description of the system since PeM is usually considerably greater than PeH. We shall repeatedly have occasion to discuss the solution of the linear partial differential equation that appears when the heat generation term is dropped from the heat balance (15). 2vav0 - x2)aT = az

+[_!_ ~(x R pep x ax

aJ\ + a-;J

R2a2~]

(48)

az

Sec. 1.3

solve the model with negligible axial conduction, Pe --? oo. The equation is a representative model for a large number of systems such as tracer dispersion and gas absorption; this explains its frequent appearance in the chemical engineering literature. The quantity of primary interest is the Nusselt number for ( 2::: 0; it is defined in the usual way by the first relation in (53): Nu (() = 2Rh = k

2(aTjax)lx=l T-Tw

= -1(dii/ d() + (2/Pe )(d I d( ) J~x() dx 2

(49)

for all (

~

oo

(50)

(51)

!~=0 at x= 1

,<0

(52a)

() = 0 at x = 1

(2:::0

(52b)

ax

2

1 J V TdA T=-AA Vav

(53)

2

-

if

=

~ ~ ~: = 4

The linear model (49) with boundary conditions (50) to (52) may be called the extended Graetz problem since (in 1885) Graetz was the first to

t

(54)

1

2

x(l - x )1J dx

The last relation in (53) is found by integration of the energy balance (49) over the cross section. The boundary condition at x = 0 and the definition of if are inserted and the numerator of the last expression appears. For Pe --? oo a very convenient expression for Nu (() is obtained:

PeR

= o at x = 0

2

h is a rigorously defined average heat transfer coefficient based on a driving force f - T w:

Nu (() =

z

a8 ax

2(a8jax)lx=l 8

()

A detailed analysis of this problem may give some insight into the difficulties of solving the complete model (14) and (15) without neglecting the axial diffusion terms. As a reference for the numerical solution of (48), which will be started ./ in chapter 4 and continued in chapter 9, we shall here give an outline of the dimensionless quantities that can be derived from the equation. The boundary conditions are imposed at z ~ -oo and at z ~oo and we shall assume that the tube is insulated from z = -oo to z = 0 and that the wall temperature is constant, T w for z 2::: 0. The Newtonian fluid enters the tube at z ~ -oo with T= T 0 , flows in fully developed laminar flow through the tube, and attains the temperature T = Tw when z --? oo. The absence of a natural scale for the z -direction permits a small simplification of the model in comparison with (44) if the dimensionless variables are chosen as follows:

() = 1 for (--? -oo and () is finite (~o) for (

23

Steady State and Transient Models for Solids

(dii/d() 20

(55)

1.3 Steady State and Transient Models for Solids 1.3.1 Catalyst pellet model

Whenever we have referred to a packed bed in the previous section, we might have thought of an ion exchange column, a dehumidifier, a heat recuperator, a fixed bed reactor, or a number of other important chemical units. The solid component of the bed is, of course, an active partner in the process that we wish to simulate but mass and energy balances have been written in section 1.2 for the fluid phase alone, as if there was no coupling between the phases. It has implicitly been assumed that the temperature and concentration at any point in the solid at tubular position (r, z) is equal to the temperature and concentration in the fluid at

24

Mathematical Models from Chemical Engineering

Chap. 1

the same position. This is certainly not true: The concentration of adsorbent is lower in the fluid phase inside the pore system of the adsorbent than in surrounding bulk fluid phase-otherwise the fluid phase would not be purified during its passage of the adsorption column. In steady state operation the temperature on the surface of a catalyst particle is higher than in the fluid at the same tubular position when the reaction is exothermic, and the external surface is again cooler than the interior of the particle since the major part of the conversion takes place on the large interior surface of the catalyst pellet. In this section we shall review the most common mass and heat transfer models for steady state and transient behavior of porous particles. This is a necessary preliminary step if the reactor model of the previous section is to be refined-and the arguments for this refinement seem pretty strong. It is, however, important to strike a balance between degree of refinement and computational work. Model (44) for the fluid phase is quite complicated even without axial dispersion, and the occurrence of a complicated coupling mechanism between fluid phase and solid phase may push the model beyond the capability of even a large computer-especially for transient studies or in steady state optimization of a reactor operation. We shall visualize the catalyst particle as a porous solid with twisting pores of different diameter and length, with crevices, dead-end pores, and perhaps a micropore system imbedded in a system of larger pores. These large pores serve as main passages for the fluid reactant that penetrates the solid and reacts on the pore walls. The heat of reaction Q = ( -dH)RA -which may be positive or negative-is conducted through the pore system and the solid matrix to the pellet surface and from there into the surrounding bulk fluid phase. Formally the mass and energy balances for pure diffusional flow of one reactant in a large surplus of inert are derived from (1) and (2) when the convective terms are dropped.

Sec. 1.3

Steady State and Transient Models for Solids

V is the Laplacian operator for the solid phase. It may contain one, two, or at most three spatial coordinates in the pellet. For onedimensional diffusion, (57)

where ~ = 0, 1, an? 2 for planar, cylindrical, and spherical geometry, respectively, and x Is the ratio of the distance from the center measured relative to the pell~t radius or (for slabs) the half thickness of the pellet. The accumulatiOn term on the left-hand side of the mass balance contains the pellet porosity e since it is assumed that the accumulation occurs in the ga~ phase onl~. Correspondingly, p and cP are pellet density and heat capacity, respectively, and not properties of the solid matrix alone. A number of other physical processes of considerable interest to

ch~mical engineering, e.g., leaching of solids, adsorption of vapors on sohds: an~ thermal treatment of solids, are described by one or both equatiOns m (56). The detailed treatment that we give these equations in

sev~ral ~hapter~ of this text is well justified by their importance to engmeenng design and also by the importance of linear variants of the equations in theoretical numerical analysis. The pr_?duction. terms - RA and Q are, of course, not always caused a chemical reaction. In a physical adsorption process, RA is the rate of mcrea~e of adsorbed material on the solid and Q is the heat of adsorption per_ umt adsorbed material. The physical adsorption may be described as an mstantaneous reaction-the immobilized material on the solid surface is ~l~ays in equilibrium with material in the fluid phase at the same positiOn.

?Y

For one spatial variable x the side conditions at x = 0 and x = 1 are acA aT =- = 0 ax ax

-

aT_ - -kev~zT Pc pat - R 2

atx = 0

(56)

+0

The two transport coefficients De and ke, which are considered to be independent of spatial position in the pellet (either directly or implicitly through cA and T), are composite properties that reflect the different transport mechanisms inside the complex pore structure. Experimental values for De and ke are found in Satterfield (1970) pp. 65 and 171.

25

2

acA = RhM i); De [cb - cA(X = 1)] aT Rh ax = -,z-[Tb - T(x = 1)]

l

(58)

at x = 1

(59)

Tb a?d cb are p:op~rties of the bulk fluid phase. They are generally functiOns of positiOn m the reactor and of time but in the present section we shall regard them as known quantities. hM and h are mass )l.Dd,.""",#.

l

//

~

~. . ··-' .•·~·:f<~~t.Jl!;. '

26

Mathematical Models from Chemical Engineering

Chap. 1

transfer coefficients for the film that surrounds the pellets. The two dimensionless groups

Sec. 1.3

Steady State and Transient Models for Solids

27

activation energy, respectively. 2 2 R [ aexp ( - -E-)] Con-1 =De RaTo

(60)

Le = DePCp

and

ke£

(65) (61)

are called the Biot numbers for mass and heat transfer, respectively. The pellet model is seen to be a set of coupled nonlinear partial differential equations, which are again coupled to the fluid phase mass and energy balances through the boundary conditions (59). The fluid phase balances are themselves coupled partial differential equations with time and one or two spatial coordinates as independent variables. The solution of the complete set of balances for the coupled phases is a considerable task even in the steady state case, and development of a simplified pellet model is certainly warranted. Our objective is (1) to obtain a solution of the pellet model for the steady state problem, and (2) to discuss model simplifications for the transient calculations. One dimensionless form of (56) is

ay

2

2

aT = V y - exp

[

~ / 1 - 1)] y n 1'\ 0

(62)

(63)

E

y=--

RaTo

Finally, the dimensionless spatial boundary conditions are

ay __ ao __

ax

(axay)

ax x=l

0

at

.

= BIM (yb

x = 0

(66)

-

(67)

Yx=l)

(68)

1.3.2 The steady state solution of the pellet model

We shall first consider the steady state solution of (62), (63), (66), (67), and (68). For convenience (c 0 , T 0) are chosen as (cb, Tb) and the steady state model is

for an nth-order irreversible reaction. CA

y=Co

and

(69)

T

T=-

To

where c0 is a reference concentration and T 0 a reference temperature. is a dimensionless time given by

(70)

T

ffi

(64)

X=

0:

X=

1:

dy = d(} = 0

dx

dx (71)

The dimensionless groups are the Thiele modulus <1>, the Lewis number Le, {3, and y; {3 and y are the dimensionless heat generation and

:: = BiM (1 -

y)

and :: = Bi (1 - 0)

28

Mathematical Models from Chemical Engineering

Chap. 1

In particular we are interested in the so-called effectiveness factor ,, the volume averaged reaction rate relative to the rate at bulk phase temperature and concentration. 'Y/

e

J~ <1> 2 rate (y, 8) dxs+l s+l = J~ <1>2 rate (1, 1) dxs+I = J0 rate (y, 8) dx

(72)

In section 1.4 we use 'Y/ wherever possible to represent the solution of the pellet problem when the combined fluid-phase pellet model is solved. For industrially important problems, Bi is quite often small and the intraparticle temperature gradient may be negligible. This leads to a significant simplification of (69) and (70): Integrate (70) over the pellet volume to obtain (s

+ 1) Bi(1

-

ii)- -{3
r

y" exp [

~ 1 - ~)] dx>+l (73)

In (73) a driving force (1 - if) has been used rather than (1 - 8x=I) which contains an additional unknown temperature 8x=I· We compensate for the use of a larger temperature driving force by using a smaller value Bi for the transport coefficient, quite in analogy with (27).

1

h= h+

R aP ke

or

Bi

(74) in terms of y and the constant average temperature if. For a first-order reaction, (74) can be solved analytically to give, e.g., for spherical geometry,

i = <1>2 exp [

(75)

1

y( 1 - ~) J.

Equation (7 5) is

inserted into (~3) to give an algebraic equation in if: Bi ( 1 _ B) ·

=

·\P(8

+ f3y) =

o

or

d8 dx

dy

+ {3- = c 1 = 0 [from (71)] dx

A further integration yields

+ {3y =

C2

-{3 Bi M

(76)

where the integration constant is related to the surface concentration and temperature by

or (77)

= 1 + aP Bi

3 BiM <1> 1 coth <1> 1 - 1 ,=-i <1>1 coth <1>1 + BiM -

29

An important feature of this greatly simplified pellet model is that it may have more than one steady state solution, just as the true model (69) and (70). For n "::fi 1, (74) is easily solved numerically, perhaps in parallel with (73), and quite accurate values of 'Y/ are obtained even for rather large values of {3. Numerical solution of the complete model (69) and (70) is, however, not in principle more complicated. Multiplying (69) by {3 and adding the result to (70) yields

Bi

where aP is chosen as 1/(s + 3) as shown in chapter 6. The mass balance becomes

'Y/ is a function of if since

Steady State and Transient Models for Solids

8

= -{3i7J(8 = if)

1

Sec. 1.3

1 coth <1> 1 - 1 <1> 1 coth <1> 1 + BiM - 1

The solution if is inserted into (7 5) to give the value of 'Yl·

Equation (77) is used to eliminate 8 from (69), which can now be solved with its two boundary conditions. Finally 8(x) is found from (77) and the solution of (69). Equation (77) is independent of the surface concentration in two cases: 1. Bi = BiM 2. BiM » 1 and Bi » 1 in which case 8x= 1 = Yx= 1 = 1 In both situations (77) simplifies to 8

=

1

+ {3(1

- y)

(78)

The solution of the steady state model is consequently seen to present only minor numerical problems. Whether the full model or a fairly accurate approximate model (73) and (74) is used, one obtains the effectiveness factor 'Y/ as a function of the bulk fluid phase properties (cb, Tb), the two dependent variables of the fluid-phase steady state model that are again functions of one or two spatial variables in the reactor. The transient equations (62) and (63) for the pellet are much more difficult to solve; when these are coupled to the transient, mass and

Mathematical Models from Chemical Engineering

30

Chap. 1

energy balances for the fluid phase, an almost unsurmountable numerical task is at hand. Thus simplifications of the dynamic pellet model are certainly desired. 1.3.3 Linearized dynamic model for the

For small deviations from a steady state solution, the rate expression in (62) and (63) may be linearized around this steady state. Let (Yss' Bss) be the solution to (69) to (71) and define deviation variables y(x, T) = y(x, T)- Yss(X)

2 y" exp [

~1 -

=

(79)

B(x,T)- Bss(x)

these circumstances the steady state (Yss' Bss) is said to be asymptotically stable. This linear stability analysis is treated in detail in chapters 5 and 9.

If the effective heat conductivity ke of the solid is sufficiently large in comparison with the film transfer coefficient h (i.e., if Bi is small), the whole temperature increase occurs over the film, and the energy balance in (56) can be integrated over the system volume to give an ordinary differential equation in the average pellet temperature f:

-

dT pcP-d = (s

t

!J)] = y;,exp [ ~ 1- ~)] + YRy +OR, 2

where (80)

+

-

h 1)-(Tb - T)

R

(81)

The linearized transient equations are t'7 2

v

Y" - R yY" - R oe"

ao 2 Le- = V B + {3Ryy + {3R 6 8 A

A

(82)

A

aT

with boundary conditions

ay = ao = 0

ax

ax

at x

=0

atx

=

-ay + s·lMY"' = o

ax

ae -+BiB= 0

(83) 1

A

ax

Substitution of an exponential time dependence for y and 0 yields an eigenvalue problem. If all eigenvalues have negative real parts, the solution (y, 0) of (82) will decrease to zero for all x when T ~ oo. Under

+

Il (-Il.H)RA dx

s+l

o

(84)

where the replacement of Tb - Tx = 1 by Tb - f may as in the steady state model be compensated for by using a smaller value h of the heat transfer coefficient 1 1 R h ==-+a- or h = - - - h h P ke 1 + ap Bi The mass balance cannot be

ay -aT

31

Steady State and Transient Models for Solids

1.3.4 Simplifications of the nonlinear transient model

pellet

B(x,T)

Sec. 1.3

si~ilarly

simplified since BiM » 1:

BiM = hMR = .!(2hMR)D0 = _! Sh D 0 De 2 Da De 2 De The Sherwood number (based on the reactant diffusion coefficient D 0 through the film) is never less than 2. D 0 may be several orders of magnitude larger than De, the "effective" diffusion in the porous pellet, and consequently BiM » 1. Simplification of the transient mass balance is, however, possible but for quite different reasons. The parameter Le is generally very large, of the order of 10 to 1000, for most industrial reactions. This means that the concentration changes much more rapidly than the temperature of the pellet. Under these circumstances (pcP/ke)R 2 is a more natural time scale than (e/ De)R 2 that was used in (64), and (62) and (63) will appear as

1

1)]

(62a)

~ 1 - !J) J

(63a)

ay = V 2 y - 2 y n exp [~ 1 - - -Le aT1 B

:~ =

2

V 1J

+ {3 Tt

2

y" exp [ I ke R peP

= -z-

32

1

Mathematical Models from Chemical Engineering

Chap. 1

Our assumption that the concentration profile is instantaneously established-the so-called quasi-stationary assumption for the mass balancemeans that the left-hand side of (62a) can be equated to zero. The energy balance (63a) is handled by the previous simplification (84) and we obtain the following set of equations:

Sec. 1.3

Steady State and Transient Models for Solids

With these data one obtains Re =

2 · v 0 Rpb = 77 J.Lb ' .

jH

jD

=

Pr = (J.L;p) b = 0.94,

[Satterfield (1970), p. 82] =

Jv

(85)

33

eb

Sc =

(..1!:_) pD

b

1.26

0.357 Re 0 .359 = 0.19

1.37 for most gaseous reactants [Satterfield (1970), p. 83]*

h = jH(pcPhvo Pr- 213 = 2.97

X

10-3 caljcm 2 s °K

. Rh BI = k = 1.27 e

Solution of (85) analytically (for n = 1) or numerically gives the concentration profile as a function of if just as in (74) and (75). The integral in (86) is evaluated and the dynamics of the pellet is described by a single ordinary differential equation in ii. This extremely simplified-and yet realistic-model retains several important characteristics of the complete model (62) and (63). For example, it allows a "quenched" state with a small reaction rate to slide into an "ignited" state with a large reaction rate and a significant temperature drop across a boundary film. 1.3.5 Pellet parameters for a sulfuric acid catalyst

A numerical example will indicate typical values of the pellet dimensionless groups for a gaseous reactant. Villadsen (1970) and Livbjerg and Villadsen (1972) present the following data for a silica-based so2 oxidation catalyst: Pellet: £ = 0.4, De = 5.2 X 10-2 cm2 /s, ke = 7 X 10-4 cal/s em °K, 3 (pep) = 0.4 cal/ cm °K, R = 0.3 em Reaction conditions: Inlet of an industrial converter. Pressure 1 atm, Cbo = Csoz = 1.6 X 10-6mol/cm3for 10 vol 0/o in air, T 0 = 77\°K, superficial gas velocity v 0 = 100 cm/s, and bed porosity Eb = 0.4, heat of reaction, 23,100 cal/mol Gas-phase properties at 500 °C: Ph = 4.57 x 10-4 g/cm 3 , (cph = 0.24 cal/g °K, Db = 0.655 cm2 /s, J.Lb = 3.58 x 10-4 g/cms. kb = 9.19 x 10- 5 caljcm s oK

so2

BiM _ hM ~ _ (Pr\ 213 jD ke Bi - h D - &J -:- (p )· n }H

e

{3

=

CP /Y-Je

ke

= 5450De = 73

c (-dH)D bO k T. e = 3.6 X 10-3 e

0

DePCp

L e = - - = 74 kee

The value of {3 is so small that. in the absence of heat and mass film resistance the pellet will be at substantially the fluid-phase temperature To (d T = 2.~ oK if all reactant is co~verted in the pellet). For spherical e_ellets and B1 = 1.27 we predict an h value of 0.80h, while the value of h/ h that is obtained from the full model is 0. 78 (see section 6.2). Thus our model sim~li.fication (84) is fully justified. BiM is large and the simpler boun~ary cond1~10n. y = 1 at x = 1 could be used rather than (67)-the resultmg reduction m complexity of the model is, however, small. If the catalyst is extremely active, the reaction occurs on the pellet surface and ~ --- 0 at any point in the pellet, including x = 1. jj is given by (77) and It can be as large as 1 + {3(BiM/Bi) = 1 + 73{3 = 1.26-or d T = 200 °K across the film. This maximum temperature increase is seen to be {3 BiM

B 1.

=

23 (cb0(-dH)) ~ jv(Pr) . S (p ) 0.6f3bo 1

]H

C

CP bOTbo

where f3bo is the dimensionless adiabatic temperature rise for a pure gas-phase conversion of the reactant. In practice this large temperature gradient across the film is of course never obtained since the reaction proceeds at a finite rate. ' ' *Bird (1960, p. 647) uses iH

=

fv·

Mathematical Models from Chemical Engineering

34

Chap. 1

The parameter Le for the pellet dynamics is very large and our assumption of quasi-stationarity for the mass balance (62a) is certainly justified. Many theoretical studies of the dynamic behavior of catalyst pellets have been made with Le < 1 where-as introduced in chapter 9-most of the mathematically interesting phenomena occur. If for a moment we assume that Le « 1, the energy balance (63) instantaneously follows even rapid variations in the concentration profile. Now, consider a pellet whose pore volume is filled with reactant at concentration cbo· The pellet is suddenly ignited and an instantaneous temperature rise

is registered if the rate of reaction is very fast. The transient temperature rise may be much larger than {3 when Le is close to zero; but as we have seen from the numerical example, the Lewis number is in reality much larger than unity and the energy balance reacts sluggishly to changes in concentration.

1.4 Heterogeneous Model for a Reacting System In section 1.2 we derived homogeneous models for a reactor, that is, models where pellet reactant concentration and temperature are the same as in the fluid outside the catalyst pellet. In section 1.3, steady state and dynamic models for the pellet were discussed. A combination ~f res~lts from the two sections leads to a heterogeneous reactor model m wh1ch pellet phase and fluid phase may have different reactant concentration and temperature at the same position in the reactor. The two phases are coupled through a set of boundary conditions. 1.4.1 Parameters and variables in heterogeneous model

The dyq.amics of the reactor bed are characterized by two time constants tu~and tH. The first time constant tM is the ratio of the free reactor volume to the volumetric inlet flow rate, i.e., a measure of fluid residence time. The second time constant tH is the ratio of the reactor heat capacity to the heat capacity of the inlet stream. tH is consequently a measure of response time to thermal disturbances and is called the

Sec. 1.4

35

Heterogeneous Model for a Reacting System

thermal residence time. Leb

(M=-

Vo

Subscript b refers to the fluid phase and subscript 0 to inlet fluid conditions. When the same quantities are used for both fluid and solid phases, they are shown without subscript for the solid phase. The bed porosity eb is between 0.3 and 0. 7 and with a reactor pressure of the order of 1, the last term in tH is several orders of magnitude larger than 1. Hence the thermal residence time is much larger than the fluid residence time tM and in practice all capacitance terms except the thermal capacitance of the solid phase may be neglected. We shall see that this leads to a tremendous model simplification for transient calculations. It will be assumed that axial dispersion can be neglected and that radial gradients can be accounted for by using an effective wall heat transfer coefficient Ub defined by (27) with a = !. The fluid phase is taken to be a nearly ideal gas and the pressure drop in the reactor is neglected. For '!n equimolar chemical reaction the linear fluid velocity at reactor temperature Tb is v

= v 0 -Pbo = Pb

Tb Tbo

Vo-

1.4.2 Models for the two coupled phases

For spherical pellets the fluid-phase mass and energy balances are 2 a 2acb -7TRb-(vcb) = eb7TRbaz at

+ (1

2a[ (p ] 2a(pcpTh -7TRb- v cPTh = eb7TRb az at

2

3

- eb)7rRb-hM(cb R

+ (1

2

C8 )

3

- eb)7TRb-h(Tb - Y's) R

+27TRbUb(Tb - Tw)

(I's, cs) are temperature and concentration, respectively, of the reactant on the surface of the pellets. (Tb, cb), the fluid-phase variables, are assumed to be constant in the cross section of the bed. Dimensionless variables Yb = cb/cb 0 , ()b = Tb/Tbo, and ( = z/L are introduced. v = v 0 (Tb/Tb 0 ); (cph is independent of temperature and

Mathematical Models from Chemical Engineering

36

Chap. 1

Sec. 1.4

37

Heterogeneous Model for a Reacting System

and the energy balan_ce becomes

reactant concentration.

or 1-eb -peP aii = 3h-1-- -eb- ( ( 1 )b -eb

The solid-phase balances are (56): ae = ---zV D 2 e - RA(e, T ) eR at aT k 2 peP---;;( = R 2V T

+ (-liH)RA (e, T)

The subscript e that was used to denote effective transport coefficients ke and De has been dropped here to avoid confusion of nomenclature. Boundary conditions for the solid-phase balances are

(pepho at

R

eb

-

_

())

1 - eb (-liH)ebO R AO + __

(pepho

eb

Tb 0 (pepho

rr;;;:;;

The f~ctor (-liH)ebo/ !'bo(peP ho is the dimensionless adiabatic temperature ~1se f3bo ~efined m subsection 1.3.5. The factor of aiij at was defined m subsectiOn 1.4.1 as (tHftM) - 1. Note that ii is a function of t as well a~ of ~he bed a~ial variable C, which explains why the pellet energy balance IS still a partial differential equation even after the averaging process over the pellet variable x. W_e shall finally introduce Hb and Hw, two dimensionless groups that contam the number of heat transfer units, fluid to pellets and fluid to reactor wall, for the reactor of length L: Hb = tM 1 - eb _1_ 3h = (1 - eb)L 3h eb (pepho R v 0 (pepho R Hw = tM 1 2 Ub = 2LUb Vo(pcphoRb eb(pepho Rb The combined fluid-phase and pellet model is now

where x is the dimensionless pellet coordinate. The model simplification (85) and (86) is introduced into the solid balances:

D 2 R 2V e = RA (e, T)

at = -R 3h(Tb pepat

- + (-liH) - n

I

1

o

-

RA (e, T) dx

3

The mass balance can be solved first, and the concentration profile e (x) is obtained as a function of eb and f. An effectiveness factor 17 is defined by the following relation:

I

1

11RA (ebo, Tbo) = 11RAo =

RA (e, t) dx

3

-() ayb aob ayb baY Yb aY - tM-at ~

~

aob - ac = Hb(Ob - 0)

(tH -

ao

1 - eb

RAo ebo

ayb at

+ --tMrr-- = tM- + Da 11 eb

+ Hw(Ob - Ow)

-

tM ) at = Hb(()b - 0)

(87)

+ f3bo Da 11

where Da = [(1 - eb)/eb]tM(RA 0 /eb 0 ) is the Damkohler number as defined in section 1.2 but based on the catalyst volume of the bed and the inlet superficial gas velocity v0 • The steady state model that is obtained by dropping the time derivatives in (87) is dyb aob -()b dC - Yb--;;f = Da 11

0

Now the mass balance can be formally integrated- over the pellet volume

D hMR 3 R2 v(eb - es) = 11RAo

40b

-

- dC = Hb(Ob - 0) 0 = Hb(()b -

+ Hw(Ob - Ow)

ii) + /3bo Da 11

(88)

38

Mathematical Models from Chemical Engineering

Chap. 1

The model consists of two coupled first-order ordinary differential equations coupled to an algebraic equation. The pellet steady state model is hidden in YJ = YJ(O, yb), which is obtained by (73) and (74) or for large values of Bi from the solution of (69) to (71). If the reactor is adiabatic, further simplification of (88) is possibl~ similar to (33) in section 1.2. Since Hw = 0 for an adiabatic reactor, (} can be eliminated between the second and third equation to give d(}b

- - = -f3bo Da YJ d{

The first equation is multiplied by f3bo and the two equations are added: d (f3bo0bYb d{

+ (}b)

=

0

and (}b can be eliminated to give a final model that consists of one ordinary differential equation for Yb and two algebraic equations for (}b and if. The coupling of a set of ordinary differential equations with algebraic equations or the coupling of partial differential equations with ordinary differential equations in the boundary conditions as seen in subsection 1.2.4 is a very common feature of chemical engineering models. The transient model (87) is frequently-and with full justification according to the arguments of subsection 1.4.1-simplified by dropping the accumulation terms with factor tM. The resulting model is aYb aob -(}ba( - Yba( = Da YJ - aob a{

= Hb(Ob -

ao

if)

+ Hw(Ob - Ow)

(89)

-

- - = Hb((}b - 0) + f3bo Da YJ at/tH

The model is still a set of three coupled partial differential equations with yb, ob; ,&and if as dependent variab.les and with the pellet dynamics hidden in YJ, which is given by the solutiOn of (85) and (86). The time constant tp for transport of- heat from the pellet to the fluid phase is obtained if (86) is divided by 3 Bi for spherical pellets: 2

R P_P C d0 - + -= {3 cf.> exp [ - = ( ob - (}) 3h dt 3 Bi

'~Y{\ 1 - -=1 ) JJ

1

0

o

y n dx 3

Sec. 1.4

Heterogeneous Model for a Reacting System

39

or t P

= Rp~p =

tH

3h

Hb

as also seen from the last equation of (89). tP may well be much smaller than tH if Hb is large but it is certainly much larger than the internal pellet time constant, which has been neglected in our previous averaging process (84). If tP is much smaller than tH, that is, if we assume if to follow (}b instantaneously-the so-called "chromatographic assumption" of Aris and Amundson (1973, p. 35), because it corresponds to an instantaneous local equilibrium between the solution and adsorbed species in a chromatographic column-then additional simplification of the model (89) is possible. Subtraction of the middle equation of (89) from the last equation yields (90) The consequence of this last simplification is best seen by consideration of a nonreactive system (Da = O) where (90) is a first-order linear partial differential equation in (}b· This is solved by the method of characteristics as described in Aris and Amundson (1973) to give a temperature front that marches with constant velocity litH through the reactor without changing shape. Thus by neglecting the heat transfer resistance between pellets and fluid phase, our dynamic model becomes unable to predict the broadening of sharp temperature disturbances that is observed in experimental reactor dynamic studies. Slowly changing temperature disturbances may still be accurately simulated but the model is unable to cope with highfrequency disturbances. A dispersive element may be included in the transient reactor model even 'when the pellet dynamics is neglected. Most industrial reactors are built of high heat capacity material and the dynamics of the reactor wall may well be much more important (large time constant) than the pellet dynamics [see, e.g., Hoiberg and Foss (1971)]. This coupling element has not been considered in our models but it may easily confound a laboratory study of reactor dynamics. It should finally be mentioned that axial dispersion can be included in the model without increasing its complexity. Some investigators, e.g., Wrede Hansen (1974) and S0rensen (1976), collect all dispersive effects (pellet fluid, fluid wall, etc.) into an effective axial heat dispersion term in the same way as real gradients are handled in subsection 1.2.3. If a suitable Pe-number can be derived from experiments to describe this

40

Mathematical Models from Chemical Engineering

Chap. 1

lumping process, the method leads to the same dynamic results as in the present treatment; this does, however, have the benefit of operating with real physical properties.

Sec. 1.5

Hollow-Fiber Reverse-Osmosis Systems

41

2. Assumptions in the mathematical model: The arrangement of fibers is shown in figure 1-1(a). Variations in concentration and

1.5 A Model for Hollow-Fiber ReverseOsmosis Systems The flow systems of the previous sections were all characterized by a unidirectional convective flow. This greatly simplified the solution of Navier-Stokes equations and the complications arose due to diffusional effects in the fluid or in a solid phase that was coupled to the fluid phase. In this and the next section we deviate somewhat from the main stream of our text, which by and large deals with solution of models taken from the previous sections. The excursion into reverse osmosis here or into extrusion of polymers in the next section is not motivated by a desire to treat these subjects on their own merits. Rather we wish to stress that model building is certainly more complicated than we have shown so far. 1. Description of the process in a hollow-fiber membrane: Heavywalled hollow cylindrical membranes with an internal diameter of 50-20 microns are spun from a semipermeable material and imbedded in an epoxy tube sheet that is housed in an exterior shell. Pressurized feed solution flows over the fibers in the shell. The product water permeates the fibers and flows inside them toward the open ends, which are at atmospheric pressure. A very large membrane area per unit volume is provided and concentration polarization-a rna jar source of efficiency loss in reverse osmosis (or ultrafiltration)-is negligible due to the permissible small product flux. These features of the hollow-fiber membrane make it popular in a variety of situations: desalination and waste water treatment in general-as well as in biological separations where the treatment of uremia by hemodialysis is only one example. The setting up and analysis of a model for the highly complicated flow pattern in a hollow-fiber agglomerate is important to establish rational design criteria for wall thickness, outside fiber radius, and fiber length and to determine the capacity at given feed flow rate, operating pressure, and inlet feed ~ncentration. Optimal values of outside fiber radius r0 and fiber length L can be determined for given operating conditions and this has been the object of the study by Gill (1973). His model looks considerably more complicated than the one that is derived below, but when the various assumptions that he makes through his derivation are introduced immediately, the results become identical.

(a)

----------------------------Phase 1

-------------------(b) Figure 1-1. (a) Hollow fiber reverse osmosis. (b) Arrangement of fibers and fluid flow directions through phases 1-3.

pressure are small around the periphery of the imaginary hexagonal boundary th~t surrou~ds each fiber, and this boundary is replaced by an eqmvalent cucle having the same annular area between it and the outside fiber wall. ~e radius re of the equivalent annulus is determined by the porosity of the bed s and the outside fiber radius r 0 : - ( -1-) 1- c

r = e

1/2

r

0

(91)

42

Mathematical Models from Chemical Engineering

Chap. 1

The velocity vector v for the fluid flow through the system shown in figure 1-1 (b) has two components Vr and Vz· Since the system is composed of three phases, (1) outside the fibers (i = _1), (2) in the fiber wall (i = 2), (3) in the passage on the product si~e of the fiber (i = 3), and the two velocity components are used m all three phases, it is convenient to use u; for Vzi and vi for vri to avoid double indexing. It is assumed that the fluid is Newtonian and that steady state laminar flow exists in the system. Both pressure and concentration decrease significantly in the axial z -direction due to friction loss and product removal. The simultaneous solution of nine coupled nonlinear partial differential equations (two equations for v and one mass balance f~r e~c~ pha~e) is an almost impossible task. Luckily a number of Simphfymg circumstances exist that drastically reduce the computational work.

Sec. 1.5

Hollow-Fiber Reverse-Osmos1s Systems

zero means that the solute is strongly retained (absorbed) by the fiber material. K is clearly an empirical constant for the fiber membrane. It is found to vary only a few percent with the main design objective, which is the productivity <1>, defined by the area-averaged linear velocity in phase 1,

Assumption 2: Axial diffusion terms are negle_cted in compariso_n with axial convective terms, and the radial velocity component V1 Is thus independent of axial distance z. The resulting velocity field (92) and (93) for phase 1 contains considerably fewer terms than the parent equations (7): u 1 au 1 + v au1 = _ .!_ ap1 + v_! ~(r au1) 1 az ar p az r ar ar v 1 av1 ar

=

_.!_ ap1 + v p ar

.!(.! ~(rv1)] ar r ar

(92) (93)

(94)

K is called the rejection coefficient of the membrane. If K is close to unity, almost no solute enters the fiber wall; a value of K close to

U1av(Z) -

U1av(Z

= 0) (95)

= 0)

and the representation of c1 w by (94) with a constant K is reasonable for less than approximately 0.5-0.6. ~e area-averaged axial velocity u 1av is defined similarly to (94) by a ~md_ b_alance from z = 0 to z using the wall radial velocity v 1 w, which IS mdependent of z by assumption 2.

(z )=_!_J (r, A u1

1

z)

_

dA1 -

U1av(Z

=

2r0 0)- -2--2v1wZ

At

~quation (96) shows that mserted into (94) to give

du 1 av/ dz

Ye - Yo

(96)

= -[2r0 /(r; - r~)]v 1 w, which is (97)

Integration of (97) from z = 0 to z with the initial condition c = c 0 and U1av = U1aiz = 0) at z = 0 immediately gives the concentration of solute at the wall as a function of z: C1w(z)

Co

Assumption 3: The small flow v 1 w out of phase 1 probably rules o_ut the possibility of concentration polarization that would set up radml concentration gradients. Thus c 1 is assumed to be independent of r and a mass balance over the annular volume V1 = 1r(r; - r~) dz will give a relation between the loss of solute from the bulk phase 1 and the am~nt that has penetrated the membrane at radius ro: 2 d(u1avC1w) 1r(r e2 - Yo) = - ( 1 - K)2 1TYoV1wC1w dz

=

U1avCz

U1av

Assumption 1: The equation of motion can be solved s_eparat~ly. The present system admits to the simplification discussed _m sect10n 1.1 since v is independent of concentration and the process Is assumed to be carried out isothermally.

43

=

[U1av(Z

=

O)]K

(98)

U1av(z)

The wall velocity V 1 w can also be represented in terms of the difference in fluid pressure across the fiber wall, p 1 w - p 3 w, modified by the corresponding difference is osmotic pressure, 7T1 w - 1r3 w:

V1w = -A[{p1w - P3w) -

(7T1w -

1T~w)]

(99)

where A is a permeability coefficient for the membrane.

Assumption 4: The flow through the membrane is in the radial d~r.ection only-a reasonable assumption considering its low permeabdtty. Consequently, by a simple geometric argument, (100) (101)

Mathematical Models from Chemical Engineering

44

Chap. 1

. s.· The variable osmotic pressure difference in (99) is • . . · 1 eliminated by the assumption that osmotic pressure IS proportwna to concentration of solute or c c 1T = -1T(Z = O) = -1To c0 Co (102) A ssumpt wn

Vtw = -A(Ptw - P3w - ::

Kc,w)

Phase 1: To obtain the velocity field [u(T, z!, v(T)], it is necess.a~y t~ solve equations (92) and (93) using the followmg boundary conditions.

aT

ui UI

= 0 at

T

=

Te

[no shear at "free surface"]

= 0 at T = To [no slip at fiber wall] = u (z = O). h(T) at z = 0 [where h(T) is the inlet

• • . h 1 ] velocity distnbutiOn m t e annu us

Iav

vi

= 0 at r = re [no material flow across

VI

=

VIw

at T

=

T

=

Flow of Polymer Melts in Extruders

45

In conclusion, the design procedure of Gill (1973) reduces to the solution of (92), (93), and (104), with boundary conditions given by (103). AU remaining variables are found by simple manipulations using the various expressions (98) to (102) that were derived when the assumptions of the model were cataloged.

1.6 Flow of Polymer Melts in Extruders

We can now take an overview of the resulting computational work.

aui

Sec. 1.6

(103)

Te]

To

The pressure p is eliminated by introducing the equation of continuity (4) in the system of equations. . For constant density V · v = 0 or from Bud (1960, P· 83),

(104)

The mass balance equation (2) will not be set up since the important quantity ciw(z) has already been found in (98).

Phase 2·

Assumptions 4 and 5 make a closer study of this phase v w, c 3 W' and P3w are all known by (100) to (102) as 3 functiQnS of phase 1 variables.

superflu~us since

Phase 3: The flow in the hollow fibers may be count.er~ur~ent [u (L T) = 0] or concurrent [u 3 (0, T) = 0], but the flow field msi~e t~e m~mbrane need not be calculated since the product concentratiOn m the fluid that leaves the system at the fiber end can be calculated from c3w and VIw and these are already known.

The purpose of an extruder is to force molten polymer at a controlled rate through a die. Solid polymer is fed through a hopper at the rear of the screw and conveyed forward in a channel of gradually decreasing depth. Heat is transferred to the material from the barrel wall, and in the forward section of the screw the pressure increases due to the channel geometry. The polymer starts to melt and internal friction generates further heat. The last section of the extruder, the metering section, where all polymer is molten, is of particular interest in the control of the extruder operation. Zamodits (1969) has set up a model for this metering section and by solution of the equations of motion and energy he is able to predict the effect of geometrical parameters, operating conditions, and melt rheology on the extruder performance. Trowbridge (1973), in a study of criteria for multiplicity of the solution to nonlinear boundary value problems, makes an almost successful attempt to restate this model and he CQmputes approximate regions of uniqueness for the solution. We shall generally follow the development of Zamodits (and Pearson), which brings out some very interesting aspects of model formulation, especially with respect to the final problem statement in a form which is different from the explicit differential equations that we have encountered previously. A screw and barrel system with constant helix angle a is shown in figure 1-2. The screw diameter is D, the largest distance between screw and barrel is h, and the flight walls of height (h - 8) almost touch the barrel. The distance 8 is small enough to exclude any flow over the top of the flight wall. Consequently the polymer flows in a shallow channel of breadth Band depth ~h, which can be unrolled to a long straight channel when h « D (or B), in which case curvature effects in the (x, z)-plane are neglected. The components of the velocity vector are called u, v, and win the x-, y-, and z-directions. v is zero (except very near the flight edges) and the transversal (u) and down-channel (w) velocity components are independent of x and z in the unrolled channel of constant depth h and constant pressure gradients apjax = Px and apjaz = Pz in the x-, and z-directions. Thus the velocity field (u, w) is a function of y alone. It is described by the relative motion of the top and bottom surfaces of the channel and

46

Mathematical Models from Chemical Engineering

Chap. 1

Sec. 1..6

Flow of Polymer Melts in Extruders

We shall consider an insulated screw and a barrel with constant temperature T 0 - a situation likely to occur in practice. The flow of heat in the shallow unrolled channel is in they -direction only. The assumption of no heat transfer over or through the flight walls is consistent with the approach to the flow of melt in the channel.

1I

-D--

l

dT dY (a)

= 0 at Y = 0,

Figure 1-2. Screw and barrel system for polymer extrusion.

by the local pressure gradients, the constants Px and Pz of which only Pz is known from pressure measurements along the channel. . Inertia forces (centrifugal forces in this case) are small and m t~e unrolled channel model they are neglected. Now the sc_rew may. e regarded as stationary and the barrel as rotating (N revolutiOns per tlm~ unit) without altering the problem. ~ith no slip at the screw and barre surfaces, the velocity boundary condttlons are

= 0,

u

=

= 0 at y = 0 1rDN sin a, w = 1TDN cos a

u=

at Y = 0,

0

h

udy =

=-

dTyx dy

du

-Q

W = 1 at Y

=

fl UdY

=

0

dw

Apart from this heat generation term the equation of energy only contains the conductive term in the y-direction. With a temperatureindependent heat conductivity k one obtains

at y = h

u/ (1rDN sin a), 1

(105)

(106)

0

which expresses that no material is transferred perpendicular to the flight walls.

(108)

= Tyx dy + Tyz dy = TyxUy + TyzWy

d 2T k-2 + Q = 0 dy

Furthermor~,

f

(107)

The equation of energy (2) is also free of convective terms and Q is the (often appreciable) heat generated by internal friction. Bird (1960), p. 731, gives the following expression for the heat generation term Q = - ( T: Vv) for our velocity field:

w

or in terms of the dimensionless variables Y = YI h, U = and W = wj(1rDN cos a) that we shall use later, W = 0

= T 0 at Y = 1

__ dTyz Pzdy

(b)

u

T

With our present assumptions on the flow field (u and w are only functions of y and v = O) it may be checked from the expression for v • Vv in (3) that the equation of motion only contains viscous terms and that we only need to consider the change in the y-direction of x- and z-momentum flux in they-direction (Tyx and Tyz): Px

r; =

47

(109)

with boundary conditions (107). Zamodits used a two-dimensional form of the power law expression (9) for the nonzero components of the shear stress:

= - C[(U2y + W2)1/2]n-1 y Uy Tyz -_ - C(( Uy2 + Wy2)l/2]n-l Wy Tyx

(110)

We are now able to write the balances (108) and (109) in terms of the velocity gradients uy and wy: (111)

48

Mathematical Models from Chemical Engineering

= _ dryz = ..!!_[C( 2 +

Chap. 1

2)(n-1)/2 ] Wy Wy

(112)

2 k d T + C(u2 + w2)(n-l)/2(u2 + w2) = 0 dy2 y y y y

(113)

Pz

dy

dy

Uy

ih

wdy

Flow of Polymer Melts in Extruders

(114)

0

Direct integration of (111) and (112) from y 1 to y and from y 2 to y gives px(y - Y1) = C(u~ + w~)(n-l5/ 2 Uy _ C(u2y + w2)(n-I)/2w y y Y )Pz (y - 2

49

Each of the equations is squared. The squared equations are added and the quantity V is found: w2y cos 2 a V = U 2·2 y sm a +

=

where C may be a function of T -and thus indirectly a function of y since Tis a function of y. uy and wy are first calculated from (111) and (112). The result is (117) and (118). Next, with uy and wy formally expressed as functions of y, (113) can be solved to give (120). Unfortunately the solution (117) and (118) of the equation is very untidy. It contains three unknown parameters: P 1 = PxiPz (pz can be measured but Px is unknown), y 11 and Y2· The last two parameters are the values of y where, respectively, uy and wy are zero. The three unknown parameters in the end are expressed through the boundary conditions (105) and the total mass balance (106) but the result (121) to (123) is not a neat one. The stress neutral surfaces uy = 0 and wy = 0 deserve a physical interpretation. uy = 0 is uninteresting-it occurs for some value y 1 E (0, h). wy = 0, which occurs at y 2 , is more interesting. For a large N, a small value of p2 , or a small value of h, the extremum of w is outside the channel (y 2 < 0). But for small N (large Pz or h), Wy may change sign for y E (0, h). In the first case, w decreases from 7TDN cos a at y = h to zero at y = 0. In the second case, w passes through a minimum and since w = 0 at y = 0, it is negative close to the channel bottom. This flow reversal is of course unwanted since the volumetric output q from the extruder is directly related to w: q = B

Sec. 1.6

(~z)2/n(7T~~-2[Pi(Y-

Y1)2

+ (Y- Y2)2r/n

Now from (115) Uysina =

h~1 pz(Y- y1 )(7T~D)-ny-(n-1)/2 (117)

Insertion of V in (116) gives Wycosa = (Y- Y2) 7T~N(~z)l/n[Pi(Y- Y1)2

+ (Y- Y2)JI-n)/2n (118)

Finally V is inserted in the energy balance (109) and a temperature dependence for C is introduced: C = Co exp [ -A(T- T 0 )] = C 0 exp (-0)

(119)

d2(} h2 dY2 + kACo exp (-O)(u; + w;yn+I)/ 2

= d2(} dY2 +

Ah (3n+I)/np~n+I)/n -I/n ( (}) k C0 exp ;;

(120)

x [Pi(Y- Y1)2 + (Y- Y2)2](n+I)/2n = 0 Inte~ation of (117) and (118) from Y = 0 to Y gives U and W as functions of Y. When Y = 1, U = W = 1 and two equations in the unknown quantities Y 11 Y 2 , and P 1 result.

or in dimensionless form (121) (115)

(122)

50

Mathematical Models from Chemical Engineering

rLy

Chap. 1

Finally from (106) 0

=

exp (~){Y- Yt)[Pi(Y- Yt) 2

+ (Y-

Yz) 2](t~n)/Zn dY dY (123)

If () = A(T- T 0 ) is zero for 0 ::s Y ::s 1, the problem is completely defined by the three definite integrals (121) to (123) to determine Yb ¥ 2 , and Pb and the integrals of (117) and (118) to give U and Was explicit functions of Y. () may be zero because T = T 0 , in which case the energy balance is unnecessary. If A = 0 (a temperature-independent viscosity), (120) is reformulated in terms of T - T 0 , and the temperature increase from Y = 1 to Y = 0 can be found from the resulting linear differential equation. The real difficulty appears when the effect of a temperaturedependent viscosity (A -::/= 0) is desired. In this case the system of definite integrals (121) to (123) cannot be calculated unless the nonlinear boundary value problem (120), (107) has been solved for 8(Y); since (120) contains all three unknowns Pb Yb and Y2 , the final model formulation looks like a minor version of the Gordian knot. Zamodits has guessed Pb Yb ¥ 2 and has solved (120) by finite differences. The problem is excellently suited for the numerical methods presented in this text. A reasonably accurate solution is obtained by the one-point collocation method of chapter 6 in which 8( Y), the solution of (120), is expressed by means of a parabola in Y. The integrals are evaluated by quadrature formulas and finally the whole problem is resolved in terms of four nonlinear algebraic equations.

1.7 Solution of Linear Differential Equations The mathematical models that occur in the current chemical engineering literature are by and large nonlinear. It is the nonlinear character of the models that is interesting, and the research potential of linear models has largely been exhausted, especially by an intensive effort in the field of linear control theory. Very f$ of the models of the previous sections have been tractable by the analytical methods that constitute the main body of typical courses in mathematics or "applied" mathematics. The idea of extracting useful information from a simplified, linear version of the model should, however, always be kept in mind. In section 1.2 the case of zero activation energy produced a linear model with all the interesting features of the

Sec. 1.7

Solution of Linear Differential Equations

51

nonlinear par~nt model, and in section 1.3 the linearized heat and mass bala~c~s. provided useful information on the behavior of the systems in

the VICinity of a steady state. The impo.rtance of linear methods reaches far beyond a preliminary study of nonl~near models. In the final analysis all the numerical methods of the fol.lowmg_ chapter~ reduce to a few standard linear problems: the c?mputatwn _of the solutwn to a set of linear algebraic equations and the eigenvalue/ eigenvector analysis of a matrix. Intermediate stages in this process are the. study of coupled first-order differential equations with c.onst~nt coefficients and. the solu~ion of linear partial differential equations m the form of an eigenfunctiOn expansion. . In the present section these two main subjects will be briefly reviewed With no atte~pt to prese.nt the subject in depth. A comprehensive treatment of linear. al~ebrmc p.roblems is given by Wilkinson (1965), and ~he fundam~ntal pnnciples of lmear algebra applied to chemical engineermg models Is covered by Amundson (1966). 1. 7.1 N-coupled first-order differential equations with constant coefficients

A short notation for N -coupled equations with constant coefficients is dy dt = Ay

+ b,

y(t = 0) = Yo

(124)

~is a~ (N X N} matrix with constant elements Aih the coefficient of yj(t) m the .zth _equatiOn. b is a constant vector, and a particular solution Yt of (124) IS giVen by (125) if all eigenvalues of A are different from 0 If dyI dt ~ 0 for t ~ oo, Yt is the final (or steady state) value of y. . (125) The general_ solution of a linear differential equation L(y) = 0 is the sum of t~e. soluti~n of the homogeneous equation-i.e., the equation with all explicit functiOns of t deleted-and a particular solution. The solution of the homo~eneous equati~n ~sa linear combination of N linearly independent solutiOns, ~ac_h multiplied by an arbitrary constant ci. The arbitrary cons~a~ts ar~ elimmated from the general solution by means of the side conditions-m the case of (124) from an initial condition y(t = O) = whe · o ther Situations · · . reas m the N side conditions may be given Yo, at different values of t. A trial solution of the homogeneous equation dy j dt = Ay ·s t k u exp (At): I a en as

uA exp (At) = Au exp (At)

52

Mathematical Models from Chemical Engineering

Chap. 1

or (126)

Au= Au

(126) is the algebraic eigenvalue problem for matrix A and it is seen that each of N different eigenvalues of A produces one independent solution of the homogeneous differential equation. In all the problems considered in this text, A is diagonable and the solution of (124) is

Sec. 1.7

Solution of Linear Differential Equations

1 1 Th . fA · A - 1 . e mv_er~e ~ IS -1 = UA- u- . Yt is first constructed in three steps. multiplicatiOn of U by -b to form Vt, division of the ith element of V1 by At ~o for~ v~, and m~lJiplication of U by v2 to form Yt· . Yo - Yt IS ~ultiphed by U to form v3 , which is multiplied onto the dtagonal matnx exp (At) to g~ve v4 with ith element v 3 i exp (.Ait). Finally the component Yi of the solutiOn vector y is obtained as the scalar product of v5 = eTU and v4 :

Yi =

Uilv31

exp (.A1t)

+ ll; 2 v 32 exp (A 2t) + ... (131)

N

y(t) =

I

ciui exp (Ait)

+ Yt

53

(127)

1

The arbitrary constants c can be found as the solution of N algebraic equations:

Yo- Yt = Uc

(128)

where U is a matrix composed of the eigenvectors ui of A. Yt is the solution -A-lb of (125). Very efficient computer codes exist for diagonalization of arbitrary real matrices A: A= UAU- 1

(129)

In a QR algorithm (described in chapter 4) the eigenvalues Ai and the eigenvectors ui of A are found simultaneously with the eigenvectors of AT (or the eigenrows of A). If the eigenvalues are all real, A is a diagonal 1 matrix with .Ai in element Aii. U is the eigenvector matrix and u- the eigenrow matrix. If A is symmetric, all eigenvalues Ai are real. A nonsymmetric matrix may have complex eigenvalues and if an eigenvalue, say A1 is complex, A is not diagonal but contains the real and complex part of A, and its complex conjugate in a 2 x 2 block centered on the diagonal (see exercises 1.11 and 4.14 ). When (129) is inserted into (124), one obtains dy = UAU- 1y

dt

+b

d(U-ty) = A(U-ly)

An .Mth-order differential equation with constant coefficients is conver~ed mto a system of M first-order equations by introduction of a new vanable for each derivative y(l) , y(2 ) , ... , y(M- 1) . I n th"Is manner N c.oupled second-order equations are converted into 2N first-order equations: (132) or d~

-=BO+Q~

dt

(133)

d6

-=I~

dt

d\fl =

dt

M =

Mtfs

(134)

r: ~J

which is a matrix of order (2N x 2N). Equation (134) is solved by the method of (129) and (130).

+ u-tb 1.7.2 A linear partial differential equation

dt 1

y :J~
E~uation .(49) i~ a fairly general example of a linear partial differential equa~10n. It Is typically solved by separation of variables assuming that a solution On of the equation can be written as the product of a function of x alone and a function of {alone:

+ Y1

= Y 0 - Y1 and Y = exp (At)Y0 + [I - exp (At)]Y1 y = U exp (At)U- 1y0 + U[I- exp (At)]U- 1y1 = U exp (At)U- 1 (y0 + A - 1b) - A - 1b c

(130)

(135)

54

Mathematical Models from Chemical Engineering

Equation (135) is inserted into (49): GC 1) = G nV 2F n (1 - X2)Fnn G(l) n

= dG and d(

+ Pe- 2Fnn GC 2)

GC2) n

Chap. 1

dFn

= d2G

= O) = 1 =I b~Fn(X) 1

d(2

d [ p(x) dx dy] - [q(x) +A W(x)]y = 0 dx

(143)

(138)

+ a2Fn = 0 atx = 1

[(1 - x 2 )An - Pe- 2 A~]

-

1 d (x dFn) - - (1 - X2 )AnFn x dx dx

=

atx = 0

'

0

Fn = 0

atx

dy f31dx + a1y = 0

atx =a

dy f32dx + a2y = 0

atx = b

p(x ), q(x ), W(x) are given continuous functions of x. Assume Yi (x) and yj(x) to be solutions of (143) for A = A1 and A" respectively:

(145)

(146) Multiply (145) by yj, (146) by Yh and subtract

l?is equation is integrated over the finite interval a to b-using integration by parts on the left-hand side:

=1

yields the set of eigenfunctions and eigenvalues for (49) when Pe --? oo. The complete solution of (49) is a linear combination of the linearly independent solutions ()n = FnGn or

1

= [YiPYi >- y,py?>JJ!-

r

py:l)yj'> dx +

a

00

I 1

b~Fn (x) exp (An()

(144)

(139)

(140)

() =

(142)

must be used. The method of computing b~ from (142) with Fn given by (140) appears as one of the results of the theory of Sturm-Liouville equations. A. very brief review of the major features of this theory may serve to classify many of the differential equations to be treated in the following chapters. For a satisfactory general treatment the reader is referred to either Courant and Hilbert (1953) or to Titchmarsh (1946). The general Sturm-Liouville problem is

+ a 1Fn = 0 at X = 0

The nontrivial solution Fn that is obtained for each of these values of An is an eigenfunction. Equation (139) has infinitely many eigenvalues and corresponding eigenfunctions. In fact each region C < 0 and C > 0 produces a separate set of eigenfunctions for the problem of (49) and (51) and (52). When Pe --? oo, only the region ( > 0 is of interest and () = 1 for C = 0, a8jax = 0 at x = 0, e = 0 at x = 1 specifies the required side conditions. The solution (Fm An) of

dFn = 0 dx

00

8((

The case of (51) and (52a) is obtained for a 1 = a 2 = 0, while the case of (51) and (52b) is obtained for a 1 = 0, {3 2 = 0. A homogeneous linear equation (137) with homogeneous boundary conditions (138) is satisfied by the trivial solution Fn = 0. A nontrivial solution appears only for specific values of Am the eigenvalues of the linear differential operator

L = V2

55

To determine the arbitrary constants b~, the initial condition

The side conditions of the homogeneous second-order differential equation (13 7) are given at x = 0 and x = 1. The general form of these are

{3 2 dx

Solution of Linear Differential Equations

(136)

Assume that Gn = an exp (An(). In this case Fn is obtained as the solution of (137) with side conditions (138). 2 V Fn - [(1 - x 2)An - Pe- 2 A~]Fn = 0 (137)

dFn {3 1 dx

Sec. 1.7

(141)

= (At - Aj)

fb a

Wyiyj dx

r

py} 0 yp> dx

a

(147)

Mathematical Models from Chemical Engineering

56

Chap. 1

If the eigenfunctions satisfy the homogeneous boundary conditions (144), the left-hand side of (147) is zero; since A1 was assumed different from A1, for i ::P j one obtains

fb

(148)

W(x)yi(x)yj(x) dx = 0

a

Equation (148) is the fundamental relation for orthogonal functions. The solutions Fn of boundary value problems associated with linear partial differential equations satisfy homogeneous boundary conditions. If the boundary value problem is of the Sturm-Liouville type (143), then the eigenfunctions Fn are orthogonal over [a, b] with weight function W(x) and all eigenvalues are real. Equation (140) is a Sturm-Liouville problem and the eigenfunctions 2 are orthogonal with weight function x(1 - x ). Equation (137) is not a Sturm-Liouville problem. For such eigenvalue problems it is not possible to prove in general that the eigenvalues are real and the nonorthogonality of the eigenfunctions is a serious numerical problem when the Fourier coefficients bT of (141) are to be determined. The orthogonality property of the solutions to Sturm-Liouville problems is used as follows: Consider a function f(x) that is integrable in [a, b] but otherwise arbitrary. The Fourier series expansion for f(x) on a set of eigenfunctions {yi (x)} of a given Sturm-Liouville problem with positive weight function W(x) is n

I

f(x) ~ Sn =

(149)

bjyi(x)

!=1

The partial sum Sn of the Fourier series converges to f(x) in the following sense:

fb

W(x)[f(x)- Sn(x)] 2dx

~

0 for n

~

oo

(150)

a

The coefficient bj in the n-term Fourier expansion f(x) ~ Sn(x) is obtained when (149) is multiplied by W(x)y/x) and integrated.

r a

W(x)y/~;{f =

,t, bt

=

b'f

bfy;(x)] dx

r

tJ 1=1

r

W(x)y,(x)yi(x) dx

W(x)[yi(x)f dx

a

=

b'f~

=

r

W(x)y/x)f(x) dx

a

(151)

Sec. 1.7

Solution of Linear Differential Equations

57

.

E:very integral in the sum of n integrals is zero except that for which

z

= J. Formula (151) allows a one-by-one determination of the coeffi-

cients bj of (149). Furthermore, if f(x) is differentiable in [a, b ], the series (149) converges uni~ormly to f(x) in [a, b]; that is, ISn - f(x)l ~ 0 for any x in [a, b]. Om.te a few functions f(x) of practical interest (e.g., the square wave functiOn) lack this additional analytical property and oscillations of Sn - f(x) with non decreasing amplitude are observed also for n ~ oo ~ear points ~f discontinuity of f~x). For most purposes the "convergence m the mean property of (150) IS, however, what is desired. The eigenfunctions that correspond to the one-dimensional diffusion equation in the x interval [0, 1] are particularly important. The SturmLiouville problem is

d(

sd8 +

-X-

dx

dx

AxsF = 0

(152)

For s = 0, the eigenfunctions are harmonic functions sin x.JA or cos x-IA. The eigenvalues A depend on the coefficients ({3i, ai) of (144). .For s =:= 1 ?r 2, (152) has one solution that is finite for all x E [0, 1]. This solutiOn IS Jo(x.fA) when s = 1 and (sinx.fA)/x when s = 2. The other solution (e.g., cos x..fijx for s = 2) is infinite at x = 0. ~e const~nts of (151) can be found analytically for all three types o~ eigen~unctiOns m (152) and if f(x) is a simple function (e.g., 1, x, or x ), the mtegrals on the right-hand side of (151) can also be determined analytically. This is an exceptional situation. In most cases the solutions of the ~turm-Liouville problem are only available in numerical form-for ms~ance, as power ~eries .. Conseque~tly the determination of bj by (151), ~hich looks deceptively Simple, may m fact require a considerable numerIcal effort. . The two linear partial differential ~quations (82) and (83) are analyzed m c~ap!er 9.:. We shall see that y (or 0) can be eliminated when Bi = BiM (or If Y = 0 = 0 at x = 1) and when at the same time Le = 1. The resulting e~uation is a Sturm-Liouville problem. Ry and R 0 are complicated functions of x that are only available through numerical solution of the steady state equation. Even though the Sturm-Liouville problem can only be sol:ved n~mericall~, it is of great theoretical significance that (for L~ = .1, BI = BIM) the lmear model for pellet dynamics is a SturmLiou~Ille problem with real eigenvalues and mutally orthogonal eigenfunctions. In the remaining part of this section we shall define the so-called Jacobi polynomials Pn(x), the polynomial solutions of (154). They are orthogonal on [0, 1] with weight function W(x) = x 13 (l - xy:",

S

Mathematical Models from Chemical Engineering

58

~~a~/32

Chap. 1

> -1] p does not satisfy homogeneous boundary conditions (144) 1 and but (154) is still a Sturm-Liouville problem because

p(X) =

0

X

f3+1(

1

-

X

-0

)a+ 1

-

atx=Oandx=1

fora>-1,{3>-1 (153)

Differentiating the first term and dividing through by W(x) yields

2)x]~~ + Ay

+ [{3 + 1- ({3 +a+

= 0

(155)

solut(lh·o;;np~{y~~::~~) :r~o~~~m~:!e~~

Pn (x) = 'YnXn -

'Yn-1X

n-1

+ 'Yn-2X n-2-

...

+ (-1r

(156)

it is shown in chapter 3 [equation (3.9)] that 'Yk =

n-k+1n+a+f3+k k {3 + k 'Yk -1

k = 1, 2, ... , n

and

(157)

'Yo = 1

It is now easy to show that the eigenvalues of (155) are

+ a + f3 +

An = n(n

As an example, take a = f3 =

1)

(158)

-t in which case

x(1 - x)y;?) + (~ -o.x)y~) + n2Yn = 0

(159)

The three first polynomials that satisfy (159) are

Po= Yo= 1

for n = 0

p 1 = 2x- 1

for n = 1

p 2 = Sx 2

-

as good results as the "true Fourier" series. In this way a standardized method for solution of partial differential equations is obtained. The "true" eigenfunctions Fn that may be complicated transcendental functions never appear explicitly. A detailed description of this approximation process appears in section 4.3.

EXERCISES

It is easily seen that (1?5)fhas at. d in x The e1gen unc 1ons . . k ~~ee; t~ withi~ an arbitrary scalar factor, and if the polynommlls ta en

as

59

(154)

[p(x)yC 1)]( 1) + Ax 13 (1 - xty = 0

x(1- x)d2; dx

Exercises

(160)

Sx + 1 for n = 2

. f 1 · ls is named after Chebyshev. The resulting family o lpo yn~~l~hat are solutions to (154) are used in The offhogonal po ynomm s ainder of this text. Specifically' various circumstances th~oughout !he r~: t have been discussed in this . N-term series of for the partial differential equatiOns a . section, it will be s~own fth::c~n o~x~~:sl~ :g::functi~ns Fm n = orthogonal ?olynomi~ls o . (141) truncated after N terms yields just 1 , 2, ... , N, m a Founer senes

1. A tubular reactor for a homogeneous reaction has the following specifications: L = 2m, R = 0.1 m, T w = 600 °K. Inlet reactant concentration is c0 = 0.03 kmol/m 3 and inlet temperature, 700 °K. Heat of reaction -~ = 104 kJ/kmol, U = 70 J/m 2 s °K, cP = 1 kJ/kg °K, and p = 1.2 kg/m 2 • vz = 3m/s, k 0 = 5 s- 1 • Calculate Da, {3, Hw, and Ow of (1.32) The activation energy is assumed to be zero and the reaction is first order in the reactant. Calculate 0(?) and y(() and locate the hot spot. 2. The debate on the proper choice of boundary conditions for equations (1.35) and (1.36) has not yet been concluded. Deckwer (1974)a presents experimental results that seem to contradict the assumption of continuity of concentration across boundary surfaces. Give a comparison of the theoretical treatment in Deckwer's paper and that of section 1.4. Is there any clue in the experimental setup (figure 6 of the reference) that may explain the author's results? 3. Give a qualitative discussion of the early papers (ref. 19 and 20) on multiple steady states for model (1.35) and (1.36) compared to the newest papers (ref. 22). What are the major developments over the 10-year period between these papers? 4. Rony (1968, 1969)b was the first to give a theoretical model for transport restriction in a supported liquid phase (SLP) catalyst. Give an analysis of his single pore model and present a list of the required dimensionless numbers and boundary conditions. Calculate the effectiveness factor by solution of his model. Are all quantities in the model available from experiments?

5. Compare the SLP model of Rony with that of the much later paper by Livbjerg, et al. (1974Y. What is the major development? Are all quantities in the model now directly measurable? a Deckwer, W. D., and Maehlmann, E. A. Advances in Chemistry, Series 133 (1974):334-47. bRony, P.R. Chern. Eng. Set. 23 (1968):1021. J. Catalysis 14 (1969):142. c Livbjerg, H., S0rensen, B., and Villadsen, J. Advances in Chemistry, Series 133 (1974):242-58.

60

Mathematical Models from Chemical Engineering

Chap. 1

6. Yu and Douglasct discuss a model for oscillations in catalyst particles. Elnashaiee has criticized their model. Make your own "review" of Yu's and Douglas' model, and follow an eventual continuing "letters to the editor" debate. What is the numerical problem in the model? Give your own suggestion of how this should be treated. 7. Another way of obtaining oscillating reactions on catalyst pellets is suggested by Elnashaie and Cresswellf. Give an analysis of their model and explain why simultaneous adsorption and reaction is more likely to give oscillating behavior of the pellet than a chemical reaction alone.

8. Leonard and Bay J0rgenseng have recently given an up-to-date review of convection and diffusion in capillary beds. They treat the bed as composed of microcirculatory units-capillary tubes surrounded by tissue. A number of models is listed in their table 1. Make a careful comparison of models 1.2 to 1.5 for the capillary and 2.2 to 2.4 for the tissue. For each model a list of necessary side conditions should be given and a numerical treatment suggested. In their discussion of capillary penetration (p.334), one reference [to D. G. Levitt in Am. 1. Physiol. 220 (1971):250] seems to give an interesting modification of generally accepted boundary conditions between the microcirculatory units. Give your own summary of this discussion, and study Lightfoot's treatment of the same problem (ref. 10, p. 331ff).

References

61 (b) Next show that for an arbitrary scalar t exp (At)

A= UDBU- 1

where DB is diagonal except for a 2 x 2 block at position j and j + 1 The columns of U that correspond to row j and j + 1 of DB are 0 . and. ui+l·

1

Show that the kth component h(t) of the solution of dy dt = Ay

=

(

a -b

b) = a

!( 1 )(a +0 2 1 i

i

ib

0 a - ib

)( 1 -i

-i)1

\

ct Yu, K.

withy(t

=

O) =Yo

is Yk(t) = Uk1V1 exp (A1t)

uki

+ uk 2 v 2 exp (A 2 t) ... + (q cos bt + r sin bt) exp (at) + ... +

is the kth component of

q

= ukivi

+

ui

and

uk(j+1)vi+1

vi

ukNvN exp (ANt)

is the ith component of u-IYo·

and

r = ukivi+ 1 - uk(j+l)vi

As a trivial numerical example, you may use

A=(-~ -~ 1 -1

-;) -1

where th~ eigenvalues are a ± ib = - ~ ± i(J15/2) and A = -2. 3 The solutiOn of dy / dt = Ay with Yo = (1, O, O) is

+ 0.5 exp (-2t) 0.38729 sin bt) exp (at) + 0.5 exp (-2t)

Y1 = (0.5 cos bt- 1.67828 sin bt) exp (at) Yz

= (-0.5 cos bt -

Y3 = 0.51639 sin bt exp (at)

11. (a) Prove that A

cos bt sin bt) . exp (at) -sm bt cos bt

(c) When A has complex eigenvalues we are not able to diagonalize A Assume (~or simplicity) that A has one pair of complex eigenvalues; A-1 = a + zb and Ai+I = a - z"b wh"l · 1e all other eigenvalues A1 , Az, · · · 'Ai-1> Ai+z, · · · , AN are real. A can now be transformed to

9. Two papers by Davis, et al.h and Cooney, et al.' propose numerical solutions to the mass transfer problem from capillaries to tissue. Discuss their models and their proposals for a numerical solution in light of the review cited in Exercise 8.

10. Give your own summary of Lightfoot's discussion of the blood oxygenator (ref. 10, p. 380). What are the specific problems that have to be mathematically modeled? Make a list of references for a detailed quantitative study of this design problem.

(

=

~and Douglas, J. M.

Chern. Eng. Sci. 29 (1974):163. e Elnashaie, S. S. Chern. Eng. Sci. 29 (1974):2133. f Elnashaie, S. S., and Cresswell, D. L. Chern. Eng. Sci. 29 (1974):753. g Leonard, E. F., and Bay Jf<')rgensen, S. Annual Review of Biophysics and Bioengineering 3 (1974):293-339. h Davis, E. J., Cooney, D. 0., and Chang, R. Chern. Eng. Journal 7 (1974):213. • Cooney, D. 0., Kim, S. S., and Davis, E. J. Chern. Eng. Sci. 29 (1974):1731.

REFERENCES

Even 17 years after its publication, Transport Phenomena by Bird Stewart, and Lightfoot appears to be an outstanding reference for ~ ge~eral tr~atment of transport phenomena from a chemical engineer's pom~ of VIew..Bennet and Meyer's text (1962) may be somewhat easier to digest and It attempts to give a unified treatment of "theoretical" transpo~t phenomena and applications to unit operations. Slattery's text (1972) Is very similar to Transport Phenomena in scope as well as in

62

Mathematical Models from Chemical Engineering

Chap. 1

References

63 details. Development in this field runs parallel in the West and in Russia, as seen from Luikov's text (1965); however, this is considerably more mathematically oriented than any of the previous references. Many excellent books exist on fluid flow problems. Schlich~ing (196~) is a classic on boundary layer theory, as Batchelor (1967) IS on flmd dynamics. A good introductory text is Whitaker (1968), perhaps supplemented by, e.g., Skelland (1967) for non-New_toni~n flow probl~ms. Any serious quantitative treatment of hydrodynamics will refer to MllneThomson's classical book (1968). Biological applications of transport phenome~a-perhaps th~ mo~t promising field of further development i~ the ~ubject-~re descnbed m several recent texts. Lightfoot (1974) Is easily accessible to anyone familiar with Transport Phenomena. The cardiovascular system is treated in Middleman (1972) and the artificial kidney in Leonard, et al. ~196~), to mention two specific subjects. Sonrirajan (1970) and Lakshminarayanaiah (1969) may be used as further references to the reverse osmosis example of section 1.5. Most of what is said in sections 1.2 to 1.4 is touched upon by Froment whose reviews at the Reaction Engineering Symposia in 1970 and 1972 (and in several papers on the same subjects) are major contributions to a systematic treatment of various chemical reactor systems. Petersen (1965) and Aris (1969) are indispensable texts for the basic design problems of chemical reactors. Chapters 8 and 9 of Aris (1969) are especially useful for an understanding of the physical background of the reactor models. . . . Computations on the tubular reactor with axial dispersiOn was started in the early 1960s by Raymond and Amundson (1964) and further development is traced through a series of papers by Amundson and co-workers: Amundson (1965), Luss and Amundson (1967), Varma and Amundson (1972, 1973, and 1974); the 1973 papers show what astonishing computer results can be obtained from the fairly simple model (1.35) and (1.36). It is a good bet that even the detailed treatment that t~e problem has received in the last few years does not mark the en~ of t~Is research topic. Independent results on the tubular reactor With axtal dispersion are reported by Hlava~ek (1971) and McGowin and Perlmutter (1971). Taylor diffusion is treated by Wicke (1975) in a recent review of the physical pro~sses in reactor design. The original paper by Taylor (1953) is recommended as a masterpiece of scientific writing. There are numerous papers on the Graetz problem of subsection 1.2.4 and on the extended Graetz problem (i.e., with axial conduction). Some of these papers are referred to in chapter 9. . . The steady state and transient behavjor of solid catalyst particles IS the subject matter of Aris' monumental treatise (1975). Apart from the

almo~t overwhelming but always fascinating mathematical exposure that

con~tltutes the mai~ body of this book, two introductory chapters give an outhne of the physical problem and its historical background that with respect to clanty of pre~ent~tion _will be difficult to surpass. The many hundreds of references Cited mAns (1975) illustrate what possibilities for research work the contents of section 1.3 has offered over the last decade. Even thou~h Carberry (1974) wants to declare a moratorium on steady state ~ffe~hveness factor calculations, he will probably not succeed in enforcmg It-he may not even be willing to abstain from further publicatio~s in the field _himself [cf. Carberry (1975)]. A short but very readable rev~ew that also mcludes a number of references to experimental investigations on steady state and transient effectiveness factor problems is given by Schmitz (1975). The. tra~sient models for coupled fluid phase-solid phase systems may b~ studied ~n the text by Aris and Amundson (1973). Their object is to giVe a detailed treatment of first-order partial differential equations and their applications. A great number of models for adsorbers chemical reactors with poisoning of the catalyst, and other chemical e~gineering systems are set up and solved. . ~aede_ Hansen has contributed significantly to the subject of model Simphficatl~n for catalyst pellets and for pellets coupled to a fluid phase. Three of his papers are listed in the references. Reverse osmosis is treated in references 10, 13, and 14. The paper analyzed_ in section 1.5 (Gill and Bansal, 1973) has been followed up by an expenmental investigation (Dandavati, 1975). The paper ~y Zamodits and Pearson (1969) of section 1.6 refers only to a small portiOn of the work carried out by Pearson and co-workers in cooperation. with British industry. Several reports and computer programs ~or design of extruders have been published as a result of this work. Without further comment, references 38 to 51 are listed in their order of appearance in the text. 1. BIRD, R. B., STEWART, W. E., and LIGHTFOOT, E. N. Transport Phenomena New York: Wiley (1960). ' 2. BENNET, C. 0., and MEYERS, J. E. Momentum, Heat and Mass Transfer, New York: McGraw-Hall (1962).

3. SLAITERY, J. C. Momentum, Energy and Mass Transfer in Continua New York: McGraw-Hill (1972). ' 4. LUIKOV, A. V., and MIKHAILOV, Yu A. Theory of Heat and Mass Transfer. Translated from Russian by Israel Program for Scientific Translations Jerusalem (1965). ' 5. SCHLICHTING, H. Boundary Layer Theory, New York: McGraw-Hill (1968).

Mathematical Models from Chemical Engineering

64

Chap. 1

K An Introduction to Fluid Dynamics, Cambridge: Cam6. BATCHELOR, G · · bridge University Press (1967). . . to Fluid Mechanics, Englewood Cliffs, N.J .. 7 . WHITAKER, S. Introduction Prentice-Hall (1968). p Non-Newtonian Flow and Heat Transfer, New York: 8. SKELLAND, A · H · · Wiley (1967). M Theoretical Hydrodynamics, 5th ed. New York: 9. MILNE-THOMSON, L · · Macmillan (1968). Transport Phenomena and Living Systems, New York: 10. LIGHTFOOT, E. N. Wiley (1974). . 11. MIDDLEMAN, S. Transport Phenomena i~ the Cardiovascular System. BIOmedical Engineering Series. New York: Wtley (1972). d "The Artificial Kidney." Chern. Eng. Progress 12. LEONARD, E. F., ET AL., e . Symposium, Series 84, Vol. 64 (1968). . L d n· Logos Press (1970). 13. SONRIRAJAN, S. Reverse Osmosls, on o . Ti ort Phenomena in Membranes, New 14. LAKSHMINARAYANAIAH, N. ransp York: Academic Press (1969). Advances in Chemistry, Series 109 (1972):1-34. Proceed15. FROMENT, G · F · ings 1st ISCRE, Washington (1970). A5-1 to A5-20 in Proceedings 2nd ISCRE, Amsterdam 16. FROMENT, G · F · (1972). Elsevier (1972). . . Chemical Reaction Analysis, Englewood Chffs, N.J .. 17. PETERSEN, E · E · Prentice-Hall (1965). Chemical Reactor Analysis, Englewood Cliffs, N.J.: 18. ARIS, R. Elementary Prentice-Hall (1969). R c J. Chern. Eng. 42 . an. 19. RAYMOND, L. R., and AMUNDSON, N. (1964):173.

Can 1 Chern. Eng. 42 (1965):49. 20. AMUNDSON, N · R · · · N R Can 1 Chern. Eng. 45 (1967):341. · · 2 1. LUSS, D., and AMUNDSON, · · N R Can 1 Chern. Eng. 50 (1972):470. · . 22 . y ARMA, A., and AMUNDSON, · · Ibid., 51 (1973):206; and 52 (1974):580. . , H and KUBICEK, M. Chern. Eng. Scl. 26 23. HLAVt(:EK, V., HOFMANN, ., (1971):1639. TIER D D. Chern. Eng. Journal 2 ' · 24 _ McGOWIN, C. R., and PERLMU (1971):125. 25. WICKE, E. Advances in Chemistry, Series 148 (1975):82. 26. TAYLOR, G. Proc. Roy. Soc., London, Series A, 219 (1953):186.

References

65

27. ARIS, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Oxford: Clarendon Press (1975). 28. CARBERRY, J. J., and Burr, J. B. Catalysis Reviews-Sci. Eng. 10 (1974):221-42. 29. SMITH, T. G., ZAHRADNIK, J., and CARBERRY, J. J. Chern. Eng. Sci. 30 (1975):763. 30. ARIS, R., and AMUNDSON, N. R. Mathematical Methods in Chemical Engineering. Vol 2: First-Order Partial Differential Equations with Applications, Englewood Cliffs, N.J.: Prentice-Hall (1973). 31. GILL, W. N., and BANSAL, B. AIChE ]. 19 (1973):826. 32. DANDAVATI, M.S., DOSHI, M. R., and GILL, W. N. Chern. Eng. Sci. 30 (1975):877. 33. ZAMODITS, H., and PEARSON, J. R. A. Trans. Soc. Rheol. 13 (1969):357. 34. SCHMITZ, R. Advances in Chemistry, Series 148 (1975): 156. 35. WAEDE HANSEN, K. Chern. Eng. Sci. 26 (1971):1555. 36. WAEDE HANSEN, K. Chern. Eng. Sci. 28 (1973):723. 37. WAEDE HANSEN, K., and BAY J0RGENSEN, S. Advances in Chemistry, Series 133 (1974):505. 38. GUNN, D. J., and ENGLAND, R. Chern. Eng. Sci. 26 (1971):1413. 39. BISCHOFF, K. B. Chern. Eng. Sci. 16 (1961):731. 40. SATTERFIELD, C. N. Mass Transfer in Heterogeneous Catalysis. Cambridge, Mass: MIT Press (1970). 41. LIVBJERG, H., and VILLADSEN, J. Chern. Eng. Sci. 27 (1972):21. 42. VILLADSEN, J. Selected Approximation Methods for Chemical Engineering Problems, Lyngby, Denmark: Instituttet for Kemiteknik (1970). 43. HOlBERG, J. A., LYCKE, B. C., and Foss, A. S. AIChE J. 17 (1971):1434. 44. S0RENSEN, J. P. Chern. Eng. Sci. 31 (1976):719. 45. FROMENT, G. F. Ind. Eng. Chern. 59 (1967): 18. 46. KJAER, J. Computer Methods in Catalytic Reactor Calculations, rev. ed. Vedbrek, Denmark: Haldor Tops0e (1972). 47. TROWBRIDGE, E. A., and KARRAN, J. H. Int. f. Heat Mass Transfer 16 (1973): 1833. 48. WILKINSON, J. H. The Algebraic Eigenvalue Problem, Oxford: Clarendon Press (1965). 49. AMUNDSON, N. R. Mathematical Methods in Chemical Engineering. Vol. 1: Matrices and Their Application. Englewood Cliffs, N.J.: Prentice-Hall (1966).

Mathematical Models from Chemical Engineering

66 50

Chap. 1

. CoURANT, R., and HILBERT, D. Methods of Mathematical Physics, New York: Interscience Publishers (1953).

51. TITCHMARSH, E. (1946).

c. Eigenfunction Expansions,

Oxford: Clarendon Press

Polynomial ApproximationA First View

2

of Construction Principles

Introduction In this chapter (sections 2.2 to 2.4), the methods of weighted residuals (MWR) are introduced. We use a fairly simple example-that of an isothermal, irreversible nth-order reaction on cylindrical catalyst pellets-to illustrate differences in quality between various approximations by polynomials of degree N. It is seen that the approximation is improved when N is increased and also that quite different accuracy is obtained with different MWR. The example is sufficiently simple to allow an analytical solution y to be obtained when the reaction order is n = 1. In section 2.1 a Taylor series solution of the analytical solution is discussed. It is shown that the analytical solution is not necessary to obtain a Taylor series approximation of order N Differences between the Nth-order polynomial approximation YN by Taylor series and by MWR are studied by means of the distance function EN = y - YN and it is noted that the properties of the Taylor series approximation are less desirable than the properties of the approximations obtained by MWR. Section 2.2 treats the first approximation by MWR; in section 2.3 a generalization to the Nth order approximation is given. The linear case n = 1 is used in both sections. A nonlinear case (n = 2) is studied in section 2.4 and an important analogy between two MWR-one which gives very accurate results and one which is very easy to apply-is discussed. 67

Polynomial Approximat1on-Construct1on Pnnc1ples

68

Sec. 2.1

Chap. 2

~:~::: expressed by the following series in ascending powers of the

No new approximation methods are introduced in section 2.5; however, this section contains some important material. In the previous sections approximations in ascending powers of the independent variable are always used. Here it is shown that these finite series can be reformulated either into a finite series in polynomials of ascending degree or into a sum of Nth-degree polynomials-the Lagrange interpolation polyno-

I .

Iv(z) = (!z)v (iz2)i 2 i=O z!f(v + i + 1) 2 2 v = O: 1o(z) = .'-~ (iz = 1 .!. 2 +1(1-z 2) +1 (1 )3 ("1)2 + z ,=o z. 4 4 4 36 -4z 2

r

mial for the approximated function. Finally it is shown that one particular MWR-the Galerkin methodgives the optimal set of N approximation constants ai when applied to the solution of the linear isothermal reaction-with-diffusion problem.

v

1

'Y1

with boundary conditions

=0

and y

=

00

1

1 at x = 1

= 1 + (<1>2x214) ~ i(<1>2x214)2 + :k(<1>2x214)3 + ... 4 1 + (<1>12) + i(l2) + :k(l2)6 + ...

(7)

= ~ ~<1>(1 + ~(<1>12)2 + f2(l2)4 + !h(I2)6 + ... ) 1 + (<1>12)2 + i(l2)4 + :k(I2)6 + .. . = 1 + ~(<1> 214) + f2(<1>2 14)2 + _1 144(<1>214)3 + ... 1 + ( I 4) + i< <1>2 I 4) 2 + :k<<1>2 I 4) 3 + . . .

(8)

y(x)

(1)

at x

(2)

dx

In particular we wish to calculate the effectiveness factor T/, which is given by (1.72) (3)

or, by integration of (1),

2

Both series (7) (cf>l2 or <1>xl2); for and z < (8) 1 are rapi"dly convergent for small values of z Y1(x) = 1 + (xl2)2 1 + (<1>12)2

and . = 1 + ~(<1>12)2 <1>2 T/1 1 + (<1>12)2 = 1 - 8

Io(2)

An analytical solution of (1) and (2) is possible for n = 0 and 1. For the case n ~lone obtains 10 (<1>x)

= Io()

2 11(<1>)

and

(5)

0

1

X

'Y1 = Io()

Where 1 and 1 are modified Bessel functions of order zero and one.

(9)

· · fare reasonably good app roximatwns to the profile and to the eff f actor. Note that the same number of terms should be ec I~eness numerator and denominator of (7) . used m the x = 1 It is f h to satisfy the boundary condition at . seen rom t e form of (7) and (8) that approximations YN(x) I 0 (2x) y= --

(4)

y(x)

+ ...

= dl0 (z) dz

In this chapter we shall treat the model for diffusion and nth-order irreversible, isothermal reaction in the radial direction of a cylindrical catalyst pellet. The dimensionless concentration profile y(x) is obtained from equation (1.69), letting y = 0 or (} = 1 for all x:

=0

c 2)'

(6)

= 1: I 1(z) = - z L . ~z = _ z( 1 + .!.(.!. 2) 1 ( 1 2) 2 ) 2 i=O z !(z + 1)! 2 2 4z + 2. 2. 3 4:Z + ...

2.1 A Taylor Series Approximation

dy

69

A Taylor Series Approximation

Figure 2-1. Solution of equation (2.1) for n = 1.

70

Polynomial Approx1mat1on-Construction Principles

Sec. 2.1

Chap. 2

and YJN with N + 1 terms in the numerator and denominator are always above the exact values y (x) and YJ for any and x < 1. Figure 2-1 shows y(x) as computed by (7) for = 2. The shape of the curve indicates that the approximation (9) by a parabola is quite satisfactory. This is substantiated by table 2.1, which gives values of the so-called linear distance function: EN(x)

=

A Taylor Series Approximation

71

equat~on. The differential equation (1) and its boundary conditions (2) contam exactly t~e same information regarding y(x) as the exact solution (5). When we Wish to study the Taylor series expansion of (5) truncated aft~r ~ + 1 terms, we may just as well start from (1) and (2) The ?envatwn of the ;aylor series follows most conveniently if (1) is re~ritten m terms of u = x for 0 :::; x :::; 1:

y(x) - YN(x)

dy dy du dy dx = du . dx = du · 2-J;,

and by table 2.2, which shows the rapid convergence of YJN to YJ. EN(x) is of one sign throughout the interval as explained above; for N = 1, the error level is approximately 0.065 except close to x = 1. EN(l) = 0 by definition of the method for obtaining YN(x).

2 d y d 2 x

=

!!._ ( 1 dy) d 2vu x

du

for u

2:

0

_ d ( I dy) du dy - 2~ vu- - - 2 du du dx du

(10)

+4

d 2y u du 2

Equation (1) is reformulated to TABLE

2.1

ERROR IN TAYLOR SERIES APPROXIMATIONS FOR X

- E 1 (x) 102

- E 2 (x) 103

- E 3 (x) 10 4

y(x)

0 0.2 0.4 0.6 0.8 1.0

6.1 6.4 6.8 6.9 5.2 0.0

5.7 6.0 6.7 7.4 6.7 0.0

3.3 3.5 3.9 4.6 4.5 0.0

0.439 0.456 0.512 0.611 0.768 1

TABLE

2.2

EFFECTIVENESS FACTORS FROM TAYLOR SERIES APPROXIMATIONS FOR

171 '172 '173

d2y u du 2

= 2

= 2

( 11)

(~)e ~ounda:y condition dyI dx =

0 at x = 0 is automatically satisfied smce Y IS only a function of u = x 2 • The boundary conditions of (11) are consequently y(u = 1) = 1 and dky/duk = (k) fi · _ Y mte at u - 0 for k = 0, 1, . . . . Differentiating (11) k times with respect to u yields .

+

uy(k+2)

+

(k

1)y(k+l)

=

py(k)

(12)

Introduce y 0 = y(u = 0) = y(k) (k = o, u = 0) . Eq ua t'Ion (12) can now be used to construct derivatives of any order at u = 0 by the followin recurrence formula: g Y6k+l)

= y(k+I)(u = 0) = _P_y(k) k

0.6978 (k+l)

Yo

A serious objection to the procedure that leads to the polynomial approximation YN(x) by (7) truncated after the (N + 1) term in the numerator via the exact solution (5) is that this procedure assumes a knowledge of the function I 0 (z) and published series representations of this function Only a very limited number of differential equations have solutions tha~ are named and discussed in literature, and in our example (1) any value of n except n = 0 and 1 will bring the problem outside the haven of these "known" exact solutions. Furthermore the detour via (5) on the route from (1) to (7) is completely unnecessary since an algorithm exists for calculation of the coefficients in the Taylor series expansion of y (x) for any differential

4

In

= 0.75 = 0.704 = 0.6982

1'/ex =

+

dy <1>2 du = y = py

p

=

(k

2

(k-1) -

+ 1)k Yo

The Taylor series for y(u) from u

+1 -

°

p

(13) k+l

- · · · - -(k-=--+--)! Yo 1

(14)

= 0 is defined by (15)

;q~at~on (14) for Y6k) inserted into (15) gives the following expression for b

-

0, 1, 2, ....

Polynomial Approx1mat1on-Construction Principles

72

and y is finally determined by the requirement y 0

N

YN(u)

= ~ bku

k- L~[pk/(k!)z]uk - L~ [pk /(k !)z]

N

Chap. 2

= 1 at u = 1: = 1, 2, ... , oo

(16)

Equation (16) is identical to (7). Exactly the same procedure may be followed for n :f:. 1 although the computational work is heavier since no simple recurrence formula (13) exists for the derivatives y and N Immedmtely signifies that a better approximation can be found. . . Take approximation (9) by a second-degree polynomtal Y~· E1 IS negative throughout (0, 1) and has a minimum valu~ so~ewhere m (0, 1). Hence addition of a . (1 - x 2 ) with a < 0 to y 1 will give a new seconddegree polynomial approximation to y with a s~aller maximum value of - E or in general a smaller value of the functiOnal 1

F

=

r

W(x)\y -

y,J' dx

(17)

for any nonnegative function W(x) and s > 0. . This improvement can be obtained when ?ne of the c~nstruct10n principles of section 2.2 is used to find n coefficients ai of YN m place of the N Taylor coefficients bk of (16). All these meth?ds. construct ~he coefficients ai by operations on the residual RN(x ), which IS t~e f:unctton that appears when YN is substituted into the diffe~ential equatiOn m ~lace of y. Hence it may be interesting to make a bnef study of the residual RN(u) for the Taylor series method. The manner in which the recurrence formula (13) was used to construct the Taylor coefficients bk of (15) immediately shows that R(k)[y (u)] = R(u) = 0 N

N

N

at u

= 0 for

k = 0, 1, ... , N- 1

(18)

i.e., that the residual RN(u) and its first N - 1 derivatives with respect to u are zero at u = 0. Conversely the Taylor series method may be defined (somewhat artificially) as a method in which theN constants bi of (16) are found as tlfe solution of theN equations (18). The methods in section 2.2 apply information on RN(u) throughout the interval 0 ~ u ~ 1 to construct the coefficients. Intuitively such procedures would appear to give "better" ap?roximations b! Y in the given u-interval (0, 1) than the extremely one-s1ded Taylor senes approximation (16 ).

Sec. 2.2

The Lowest-Order MWR Approximations y1 (x)

73

These remarks concerning the quality of Taylor series approximations are suggested by the peculiar fashion in which RN(u) is treated in this method. In general, such evidence is at best circumstantial: A small residual at some u-value does not always imply that EN = y - YN is small at this point or that any other relevant measure of approximation accuracy such as '17 - '11N in (3) is small. The constant sign of EN that shows that a better approximation of the same order can be constructed is much stronger evidence against the Taylor series. Even though quantitative information on important measures of the approximation accuracy [e.g., the maximum value of jEN(u)IJ is difficult to extract from a study of the residual RN(u ), this function is of great importance in approximation theory. In the absence of an analytical solution y(u), the distance function EN = y(u) - YN(u) is not available but the residual RN(u) is a known function of u once the coefficients a. are given. Some results on bounds for jEN(u)j based on the residual hav~ been reported by Ferguson and Finlayson (1972) for linear problems. It may be proved that some of the methods in section 2.2 will guarantee that RN(u) converges uniformly to zero for N ~ oo in the range of the independent variable. From this it may be proved that EN(u) converges uniformly to zero for N ~ oo also when y(x) does not admit to a convergent Taylor series throughout the interval. At present it is sufficient to note that Taylor series approximations are unsatisfactory from a practical viewpoint because more rapidly converge~t approximations can be constructed. Also in some problems they may give completely erroneous results when the radius of convergence of the series is less than the interval length.

2.2 The Lowest-Order MWR Approximations Y1(x) We shall now study different methods of obtaining the coefficient a 1 in the first approximation y 1 (x) to y(x) of equation (1): Yt(x)

= 1 + at(1 - x 2 )

(19)

Equation (19) is a slight reformulation of the expression that was used in the previous section. Here the term a 1 (1 - x 2 ) appears as a perturbation fu~ction to the known boundary value y = 1 at x = 1. Equation (19) satisfies both boundary conditions (2), and for the Taylor series one obtains (20)

by rewriting (9) in the form (19).

74

Polynomial Approximation-Construction Principles

Chap.2

Sec. 2.2

Inserting y 1(x) from (19) into the differential equation (1) with n = 1, one obtains the residual R 1(x) or R 1 (ab x) to accentuate that the function R 1(x) depends on the choice of a 1: R1(y1(x)] = R1(x) = R1(ab x) = - 4a 1 - <1> 2[1 + a 1(1- x 2)] (21)

The Lowest-Order MWR Approximations y, (x)

or - 4a1 - <1>2(1 + ~a1.) = 0 - 2<1>2 a - - - -2 1 - 8 + <1> (26) For = 2 a 1 = _ l y _ 1 + 2 2 d 2 , 3, I - 3 3X ' an 'T/t = 3 = 0.667. 3 Ch · ( )~se W(x) = aytfaal = the perturbation function (1 _ x2) of

No choice of a 1 will make R 1(x) zero throughout (0, 1) since this would mean that (1) was satisfied by a second-degree polynomial. One adjustable constant is, however, enough to obtain a mean value of R 1 in (0, 1) or the mean value of a weight function multiplied by R 1 equal to zero. This looks like an entirely logical procedure when we wish to avoid giving undue preference to any particular part of the interval (0, 1). Thus we determine a 1 by the following equation: 1 R 1(ab x)W(x) dx 2 = 0 (22)

19

t

Rt(l - x2) dV =

- 3<1>2 a - - - - -2 1 - 12 + 2<1> For = 2, a1 = - ~' Y1 = ~ + ~x2, and 4. Choose W(x) = aR 1ja a1.

1. Define W(x) by

= {:

for x

:f:.

2

or

W(x)

x1

= 5(x - x1)

(23)

L\-

(24)

a 1 is determined such that the residual R 1 of (21) is zero at one interior point x 1 in (0, 1). As an example, x 1 = !-is chosen: - 4a 1 - <1>2[1 + a 1(1 - ~)] = 0 - 4<1>2 a1 = 16 + 3<1>2 ~

For = 2, one obtains a 1 = - ~, y1 = ~ + ~x 2 , and 11 1 = 0.714. 2. Let the mean value of R 1 be zero; i.e., W(x) = 1:

Jv

R1 W dV =

f

Jo

1

{ - 4a 1 - <1>2[1 + a 1(1 - x 2)]} dx 2 = 0

(25)

t=

f

v

(27)

= .l 'TJI

-

10 -

0.7.

f.

R1 . aR1 dV = _j_ Ri dV = 0 aa1 aa1 v

~e see that this method implies a choi'ce of a t 1 f 1 such that the m egra o the square of the residual is minimum.

in which case (22) degenerates to R1(ab x1) = 0

+ at(l - x2)]

- 2a1 - <1>2(~ + ~a1) = 0

or, in general, a 1 is determined such that the weighted residual W · R has a mean value of zero in the volume of the system. Each of the various methods of weighted residuals (MWR) is characterized by a specific choice of the weight function W(x); we now shall compare four choices of MWR:

W(x)

4at - cp2[1

x (1 - x2) dx2 = 0

0

= X1

r{-

or

J.

for x

75

4at - cf>2[1 + at(l - x2)]}[- 4- cf>2(1 - x2)] dx2 = 0 at=

- <1>2[ 1 + 1<1>2] s 4 + <1>2 + fi4

(28)

For = 2 a = _JL _ s 9 2 ' 1 14, Y1 - I4 + I4X , and TJ 1 = ~ = 0 679 Table 2.3 compares the four approxi . . . . approximation (9) (which we hav ~ations. With the Taylor series method 1 with e seen m section 2.1 corresponds to x1 - 0). All values of E1 are multiplied by 100 . Note that each MWR method 1 to 4 ·ves one n : .

anbd ~~lt~ough this shoul~ not be asc~i~:dp:~~::~~= Pd~i~;: ~nc:ho~hE_l, Ic IS a m t-m feature of methods 1 4 b

=~~;~;;::::~:sf!rt:!i:g~t h(xa)v~ISbetter properties t~~n ~h:e;:;l~~c:e:~:: negative for any x and N Th

•

N

Imum error of approximations 2 and 4 is larger than them : e maxof the Taylor series approximation, but methods 1 to 4 g~~~~u~e~~~;

76

Polynomial Approximatlon-Cvnstruction Principles

Chap.2

2.3 y - y 1(x) FOR DIFFERENT MWR SOLUTION OF Eo. (1) FOR n = 1 AND l/J = 2) TABLE

DISTANCE FUNCTION

(y

IS THE

E 1 (x)

Sec. 2.3

0 0.2 0.4 0.6 0.8 0.9 1 '11 - '111

1.0 0.4 -0.9 -2.3 -2.7 - 1.8 0 - 0.016

2

3

4

Taylor

10.6 9.6 7.2 3.8 0.8 -0.3 0

3.9 3.2 1.6 -0.5 - 1.6 - 1.3 0

8.2 7.3 5.2 3.7 -0.1 -0.5 0

-6.1 -6.4 -6.8 -6.9 -5.2 - 3.2 0

0.031

- 0.002

0.019

77

or, after considerable manipulation,

=

2

)(.! __!_) + .!J _<1>2( 4 2) 1T 2 -

a 1[(1T + <1> 2 4 4

Method X

Higher-Order MWR Approximations VN(x)

2

2

1T

- -

1T

(30)

For t:P = 2, al = - 0.6331 and 711 = 0.6834. The accuracy of approximation (29) w'th . 1 ble with that obtained by the pol . a1 : ~ 0.6331 IS compara1 the computational effort is consid;~~: approximatiOns in table 2.3, but

2.3 Higher-Order MWR Approximations YN(x)

- 0.052

2.3.1 Definition of the methods

error, J~ E 1 dx 2 in T/b the reason being that positive and negative errors partly cancel in the integration. In particular we note the very small error of method 3. Throughout this text we shall restrict our attention to polynomial approximations. Approximations with other functions-especially harmonic functions in analogy with the classical Fourier series-and also with functions in which the parameters enter nonlinearly are well studied in literature [see, e.g., Rice (1964) and (1969)]. We shall briefly illustrate the use of a trigonometric approximation that is analogous to (19) and thereafter leave this subject, noting that the computational work in obtaining the coefficients ai and the subsequent application of YN for approximation purposes is in general heavier than for a polynomial approximation with an equal number of adjustable parameters. (29) The perturbation function cos [(1T/2)x] is again zero at x satisfies both boundary conditions.

1, and y 1

Having been acquainted with several MWR . to obtam the ~rst approximation Yt for equation (1) with n = 1 we approximations. Our general approxi~atio~roc~eddtofi study htgher-order YN Is e ned by N

YN = To

+ I aiT;

(31)

t=l

To satisfies the boundary conditions of the . d'ff . Th . 1f . gtven I erenttal equation e tna unctiOns (or perturbation fun . ) . . . N-term sum Each tr' If . . ctwns T; are contamed m the . · ta unction satisfies homo b geneous oundary conditlons, i.e., for the problem ( 1), (2), dT dx' (x = O) = T;(x = 1) = 0 i > 0 (32) The choice of trial functions T is in ener I . satisfy conditions (32) and are' . gl . da qmte free as long as functions~ 1mear Y m ependent . We restnct our attention to ol . 1 . . . study (1) (2) T. p ynomm ~nal functiOns and for our case . ' , o - 1 and T; = (1 _ x2) 21-2 1. _ . tlons (32). x ' - 1, 2, · · · satisfy condiYN = 1

N

+ (1 - x2) I aix2i-2

(33)

1

It is more convenient to work with (11):

= ayifaa 1 = cos 1T/2x]:

We choose method 3 [W(x) l

J.

0

1TX

d 2y dy u-2+--py=O du du

2

R 1 cos2dx = 0

y(u = 1) = 1,

Y(k)(u

= 0)

finite for k ~ 0

Polynomial Approximation-Construction Pnnc1ples

78

Chap. 2

Here an expansion that includes all positive integer powers of the independent variable u can be used. N

YN(x 2 )

=

=

YN(u)

1

+

L

(1 - u)

aiu'-

1

(34)

=

=

~2 :~ L~

1

= ;

.~

1

a,[(i - 1)2 u'- 2

-

J

uj

= 0,

uN

= 1) and the subinterval method can be

RN(a, u) du = 0,

= 1, 2, ... ,N

j

(40)

3a. The method of moments Here VV;(u) in (38) is chosen as u1- \ j = 1, 2, ... , N. (35)

YNdu

:~I ~I

i 2 u'- 1] -

1 1

RN(a, u)ui- 1 du = 0,

j = 1, 2, ... , N

(41)

0

(36)

Other functiOJ:?-S cpi(u), j = 1, 2, ... , N could be used instead of the 1 in (40), e.g., cos [(j - 1)1Tx/2] or an arbitrary polynomml of degree ( j - 1 ). The name method of moments is, however, cutomarily given to the method based on (41 ). 3b. Galerkin's method Here lo/f(u) = ~ = ayNjaai or in our case (42) monomi~ls u'-

u

The residual RN(a, u) is formed as before by substitution of YN into the differential equation (11): 2 d yN dyN RN(a, u) = u du2 + du - PYN =

u3,. · · ·, uN-1 (uo wntten:

1=1

r r YNdxz

79

Higher-Order MWR Approximations VN(x)

Uj-1

We shall determine the coefficients ai of (34) and finally compute TlN =

Sec. 2.3

r( 1 + (1- u) Ja,u,_,]

rRN(a, u)(1 - u)u 1- 1 du

1

=

j = 1, 2, ... , N

0,

(43)

(37)

4. The least squares method The following N relations determine the coefficients ai: (44)

Jv RN(a, u) VV;(u) dV = 0, or

j

=

1, 2, ... , N

(38)

J RN(a, u) VVj(u) du = 0, 1

j = 1, 2, ... , N or, in an alternative formulation, determine a such that

0

Each MWR is characterized by a different choice of the sequence weight functions VV;(u) in (38). In methods 1, 3a, 3b, and 4, we discuss the Nth-order analog of each of the four MWR that introduced in section 2.2; by addition of a fifth MWR-method comprehensive catalog of the most popular MWR is established. 1. The collocation method Choose

j = 1, 2, ... ,N

of N shall were 2-a

(39)

whe'e ui are N points in (0, 1). This is equivalent to RN(a, ui)

= 0,

j = 1,2, ... ,N

j

= 1, 2, ... , N

2. The subdomain method Divide the region V into N subregions Vj. Choose VVj(u) = 1 inside T;j and 0 outside Vj. In our case study we divide the range 0 ::::: u ::::: 1 into N subintervals by the interior points u 1 , u 2 ,

(45)

The method owes its name to this formulation. Note that aRN! aai is usually a much more complicated expression than ayNjaai, the. weight function in Galerkin's method (42), and that whereas. VV; 1s al~ays a polynomial of degree j in (43), this is ~ot necess~nly true m (44). If the differential equation is nonhnear, VV; 1s also a nonlinear function of ui. "W_e shall now mak~ a brief numerical study of the approximate solutiOn YN of (1), (2) w1th n = 1 for the following four MWR: 1. ~he collocation method with ui = j/(N tion). 2. The subdomain method with ui = j/ N.

+ 1) (equidistant colloca-

Polynomial Approximation-Construction Pnnciples

80

Chap. 2

. 1

Sec. 2.3

3b.

3a. The method of moments ( lVj = u'- ). 1 3b. Galerkin's method ["'f = (1 - u)uj- ].

1

In each of these four cases one obtains a set of N linear algebraic equations of the form (46) (A+ p · B)a = p · c

Aji

1

[(z. - 1)2 u i-2 - z·2 u i-1 ]u' -1 (1 - u) du

=

0

!I

i

(i - 1)' - i2 N ~ 1] . (N ~ 1) Bj, = (N ~ 1) •-1 . (N ~ 1- 1)

H

Bji

J.j/N

Aji =

[(i - 1) 2 u

(J-1)/N

=

1 2 -

2

i u'-

1 ]

=

0

+

=

1

1)

\u ui- )(1 1

=

1

-

u)uj- 1 du

f1

.1

(50)

1)

1

Jo (1 - u)u'- du

= j(j + l)

The approximate effectiveness factor may be found from

du

e

i

-2 (i + j - l)(i + j)(i + j +

(47)

c1

2.

i ¥= 1

- 1

j(j

Aj, = [

+ j - 2ij

= (i + j - 2)(i + j - l)(i + j)

The elements of the (N x N) matrices A and Band theN-vector care 1.

81

Higher-Order MWR Approximations YN(x)

e

i

1

(i - 1)[(~f1 - ~ 1r1J - H~r - ~ 1rJ

TJN =

o

YNdu

1

=Ill+ (1- u) I o

aiui- ]du

= 1 +I . . ai

1=1

t=1

z(z + 1) (51)

(48) 2.3.2 A sample calculation of y2 (u)

Let us consider a simple numerical example, e.g., method 3b, N We find that 1

j/N

c

=

du = -

J

(j-1)/N

J

3a. Ai, =

-

N

r[(i - 1)2ul-2 - i2u'-1]uj-1 i2

(i - 1)2

i

du ~

For

i = 1

r

i.e., p

'

= <1> 2 /4 =

=

(u' - u 1- 1 )ui- 1 du

I

1

cj =

o

. 1

1

u'- du =-:1

=

_(i_+_j-)(-~-:-1-.-_-1-)

'

and

c=m

1, we obtain

-!-~

(49)

j

Bi,

1 -30

( -t--lz

1

=\i+j-2-i+j-l - 1

= 2,

- -lz}

~}

-61

- ~a1

- ia2

=!

- ia1

- !a2

= ~

a1 = -

¥s,

a2

= - fs

Y2

= 1 - ¥s(l -

TJ2

= 1 + !a1 + ~a 2 = i~~· = 0.6977778

u) - fs(l - u) · u

=

2.

82

Polynomial Approximation-Construction Principles

Chap. 2

A general parametric representation in p = <1> I 4 may also be obtained: 2

- 2p 2 3p 2

-

40p

+ 32p + 40

00

~

00 "'
+

+s

l:l..,"'
~ +

+

N

Nl:l.., 1/")

l:l..,\0

::::: + <'ll:l...

2.3.3 Tabular results for N

= 2, 3, 4

The form of equation (46) shows that the solution for the ai and for 'Y1 may be written as the ratio of two Nth-degree polynomials in the parameter p, the denominator polynomial being common for all the ai and 'YI· The simple dependence on the parameter p makes it possible to derive parametric expressions as demonstrated in the example above but, except when the influence of p is studied in general, it is easier to 2 substitute the appropriate value of p = <1> I 4 in (46) before solving the equations. In table 2.4 parametric representations of a 1 and 'Y1 are given for the four methods described above and N = 2, 3, 4. Next we compare the approximate solutions obtained using N = 3 and = 2 with the exact profile, y(x) = I 0 (2x)II0 (2). The following approximations are found: 1. Equidistant Collocation y3(u) = - R(u)

3

+ 982u + 240u 2 + 32u ) 2 · 2233 = 32u 3 - 48u + 22u - 3

1 22

33(979

(52)

2. Subdomain Method with Equally Sized Subdomains 3 2 yiu) = lisi550 + 551u + 135u + 18u ) 2 - R(u) · 1254 = 18u 3 - 27u + 11u - 1

(53) 83

Sec. 2.3

Higher-Order MWR Approximations YN(x)

85

3a. Method of Moments with W; = ui- 1

+ 612u + 150u 2 + 20u 3 ) - R(u) · 1393 = 20u 3 - 30u 2 + 12u - 1

y3(u) = 13~ 3 (611

(54)

3b. Galerkin 's Method \0

r--- r---

0

lr) r-f r-f r---

0l ('r)

~ 0

0l

-.::t

0

+

('r)

-.::t

r---

~

0l 0l

00~

lr)~

0

0l

0

r---

r---

+

0l

+ ~ 0

-.::t

~

-.::t

+

0l

0l

+

l::l..

00

0l

~

~ +

-.::t

\0

+

+

Nl::l,. 0l N l::l,.f'-0l

+

~

N

~

v)

+

('r)

('r)

\0

\0

+ +

+ ~ 0

~ ~

-.::t

-.::t -.::t

+ + 0~

+ +

~

<')

l::l.. 0l

0

lr)

+ +

~

lr)

..,; \0

0l

+

+ N

~ 00

N

~

tr;_

0l

lr)

Nl::l,.

l::l..

.-<J<'\

l::l..

('r)

+ +

=~ ~

0l <')

<')

+

'
.-<~ '
~

0l

('r) 0~

-.::t

~ + ('r) 0~ ~

0l

I

'"";:s

00 -.::t 0l I v)

II

~--

0

~ r---

0l

"' E

0l

s s

"'d

0

..s::: Q)

~ ~

~

I

0

0

+

r---

I])

~ \0 + I + 0

-.::t

\0

0l

~

0l

I

~ ~

+

N

N

~ I

;} r---

('r)

N

~ +

-.::t"N

0

~

lr)

0

I

-.::t

<'l<'l 0l

~

<')

l::l..

+

d

I])

~

l::l..

(") ~

\0

I N

0

~

l::l,."
0

+

-.::t I

;}

N

l::l..

('r)

I

00

~

).

0

-.::t

lr)

+ N

l::l..

lr)

l::l,.f'-lr)

N

l::l..

('r)

I

+

+

\0

I

\0

0

r---

<')

..,;

00

\0

('r)

+ N + «>

0l

I

~

0

lr)

l::l..

0l

N

l::l..

~

-.::t

lr) ~

0

+

lr)

I

«>

l::l.. 0 l::l..OO

'
-.::t -.::t I + '
l::l..

lr)

(")

~

"1::S

..::

~ ('r)

~--d' 0 ;9

~ ......

'
l::l..

II

"'

~

~ 0

'"";:s

l::l..

0l

I

I

0

0l

+

r---

~

~s::

+

r-f

I

s

~ + I

lr)

I])

lr)

N

r---

<'l~

r--- r---

"i

0l

('r)

-.::t

..::

~

~

84

0l

('r)

y3(u)

= iz(36 + 36u + 9u 2 + u 3 )

- R(u) · 82 = u 3

(56)

~

+ +

lr)

0

4. Taylor Series

0

~

+ N+

+

'
l::l..

-.::t

0l 0l

l::l..

(55)

0l

N N ~ ~ ;} ;} ~ lr)l::l.. \0 00 ('r) r--+ -.::t

~<')

0l

0

0

lr)

0l

<')

l::l..

0

~N

~ 0

0

('r)

+ 547.5u + 135u 2 + 17.5u 3) - R(u) · 2494 = 35u 3 - 45u 2 + 15u - 1

y3(u) = 12~7(547

+

+

~ + 00 0l ~

0l

lr)

\0

-.::t

Figure 2-2 shows computed values of - E 3 (u) = y3(u) - y(u) and of R 3 (u) for the four approximations (52) to (55). From the plots of y3 (x) - y(x) we notice that all these MWR approximations are superior to the corresponding Taylor series approximation. The Galerkin approximation is noteworthy for its very even error distribution (notice in particular the three node points) and a maximum error of only 5% of that of the Taylor series, followed closely by the method of moments with a maximum error of 10% of the Taylor series error. The subdomain method yields an approximation with a rather large error at x = 0, and the result obtained from the collocation method is rather disappointing, the error being comparable to that of the Taylor series. As mentioned earlier, the evenly distributed errors for the method of moments and Galerkin's method indicate that very precise estimates of the effectiveness factor (J~ yNdu) can be obtained due to error cancellation in the integration. We further notice that all the MWR approximations (1, 2, 3a, and 3b) have their maximum error at x = 0, this error being equal to the error on the first expansion coefficient a 1 [yN(O) = 1 + a 1]. We may therefore judge the accuracy of the profile by the accuracy of a 1 {the exact value of a 1 is - 1 + [1/ / 0 (<1>)]}. The effect of the parameter on the error of TJ and a 1 is shown in figures 2-3 and 2-4, respectively, for yiu) obtained by the different MWR. It appears that for any given value of the Galerkin method yields the best results, followed by the method of moments, the subdomain method, and the collocation method. We notice that the difference

Log l77 - 17N I

-E · 105

Log¢

R · 103

100

10

-1 50

Eq. (2.56)

5

Eq. (2.52) -3

Eq. (2.53) Eq. (2.54)

-50

-5

Eq. (2.55)

-5 -100

-E · 10 5

Taylor series

-7

R · 103

3

-E · 105

R · 103

Figure 2-3. Error off~ YNdV = 'YIN for various YN(x); N = 3.

3

2 0.5

-1

0.5

-1

-0.5

-0.5

-2 -3

-1

-1 -3

Eq. (2.55) Galerkins method

Eq. (2.54)

-0.2

0

0.2

0.4

Method of moments

-1

-E · 105

R · 103

R • 103

3

-3 10 0.5

0.5

-5

-1

-0.5

-10

-1

-3 -20 Eq. (2.53)

Eq. (2.52)

Subdomain method

Equidistant collocation

Figure 2-4. Error of first expansion coefficient; N = 3. Figure 2-2. Distance function and residuum for solution of equation (2.1); n = 1 by various methods; N = 3.

86

87

0.6

Log¢

Polynomial Approximation-Construction Principles

88

Chap. 2

between the various methods is much more pronounced with respect to their ability to determine the value of 'YI accurately than with respect to that of finding a 1 . This is due to the well-balanced error curves (figure 2-2) for the first two methods that allow cancellation of errors in J~ YNdu. When increases beyond 3 or 4, the behavior of y (x) = 112 10 (<1>x)/I0 (<1>) is well approximated by x- exp [- <1>(1 - x)] for x > 0.5, and it is apparent that we need more than a third-degree polynomial in u to imitate the rapid decrease of y (x) near x = 1. Finally, figure 2-5 shows the effect of N on accuracy for a fixed <1> = 4. The error of 'YI decreases rapidly with N for Galerkin's method and the method of moments. All four MWR give comparable results for

Sec. 2.3

89

Higher-Order MWR Approximations YN(x)

2.3.4 Expansion of 'YI and of a, in powers of the parameter ql

The quantities 'YI and a 1 can be expressed by power series in the = <1> 2/ 4) by a Taylor series expansiOn of the exact solution or of any of the approximate solutions. We have param~ter (or more conveniently in p

() = 'YI

~r/1(<1>) fo()

1

(57)

= 1 - 1iP + 13P 2

-

11

3

4

473

5

48P + 12oP - 432oP + ... __12_

a1.

and Log I 77 - 17N I

2

3

4

(58)

N

5

The Galerkin approximations for N = 1 are

a =

-3

1

-5

-<1>2 4

-

-3p

+ ~<1>2 - 3 + 2p an d 'Y/1

= 1

+ !a 1 -- 3 + ~P 2 3 + 2p

and expansion of these quantities in powers of p yields

Eq. (2.52)

'Y/1 = 1 - ~p + iP2- ~p3 + ...

Eq. (2.53)

-7

a1 = -p

+ ~p 2

-

(59) (60)

•••

Similar expansions for N = 2 are found in the example in subsection

2.3.2: 2

Log la 1

-

iP + 12p + 40 1 712 -- 3p 2 + 32p + 40 = 1 - 2p

a 1,N I

2

3

4

N

5

-

a

Eq. (2.56)

/

Eq. (2.52) Eq. (2.53) Eq. (2.54) Eq. (2.55)

Figure 2-5. Error of

'TIN

and a 1,N by various methods;

cp

=

4.

1 -

+!

3p

2

-!!

48p

3

+ 19

4 1051 5 120p - 9600p

+ ...

(61)

2 - 2p - 40p 3 2 21 3 2 3p + 32p + 40 = -p + 4P - 40P + · · ·

(62)

A comparison of (59) and (61) with (57) shows that the series are identical up to and including the term p 2 N, and comparison of the expansions for a1 shows that the terms are identical up to the term pN. Table 2.5 gives the results of similar expansions for the other methods. Addition of one extra term in methods 3a and 3b gives two extra correct terms in the expansion of 'YI in powers of p, compared to one term (in the average) for the other methods.

Chap.2

Polynomial Approximation-Construction Principles

90

TABLE

Sec. 2.3

Higher-Order MWR Approximations VN(x)

manner and from the integral, the reason being that both of these methods require the relationship

2.5

POWERS OF p CORRECTLY REPRESENTED IN TAYLOR SERIES

1 1

EXPANSION OF NTH-ORDER APPROXIMATION

RNdu

al

'YJ

Collocation (1)

N

N

Subdomain (2)

N

Moments (3a) Galerkin (3b)

N N

91

=0

0

to be satisfied; i.e., { N

Nfor Nodd } N even 2N-l 2N

+ 1 for

or dyNI1 u-d U

The numerical factor of the first incorrect term is furthermore very close to its exact value as seen from the example with N = 2: Exact:

473 5 - 4320p

Galer kin:

1051 5 - 9600p

Difference:

1

5

86,400p

One final point is worth noting. We remember that the effectiveness factor may also be calculated as lJ

~ ~2 (2) x~t ~ %•(~~) u~t

Approximate values 'Y/N calculated from this relation with YN instead of y may differ from those found by integration. We obtain 'Y/N

d d [ ( = -1 -(yN)u=1 = -1 -d 1+ 1p du

=

.!_(p

p

u

~f...

)

U

;= 1

a;U

I a;) i=1

i-1]

40 + 12p 1 - 1- -p 2 'Y/N - 40 + 32p + 3p 2

= (dyN) -d U

u=1

=P·

I 0

1

YNdu

2.3.5 Estimates of approximation accuracy

In general it is impossible to ascertain when a given approximation accuracy has been reached since the exact solution is unknown. It is, however, important to establish empirical rules for discontinuing further computation at a certain N and every loose estimate of approximation accuracy is valuable for this purpose. The simplest approach is to compare results obtained with different approximation order N. As a rule of thumb, the error in the Nth-order approximation will normally be much smaller than the difference between the Nth-order and the (N- 1st)-order approximation; e.g., j'Y/ 4 - 'Y/ 3 1 » I'Yiex - 'Y/ 4 1, cf. figure 2.5. Some a priori guidelines may, however, be given. Returning to our original equation (1) for n = 1, d2; dx

+ .!_ dy = <1>2 x dx

y(l) = 1,

y,

y'(O) = 0

we notice that u

=1

(63)

max (dd2; + X

.!_ dy) = max (<1>2y) = z X

dx

(65)

Now, if a trial solution of the type

For the Galerkin method and N = 2, the result is -

0

+ -13p 2 40

YN = 1

+ (1

- x 2)

N

L

21 2

a 1x -

i=l

(64)

that is, the Taylor series expansion is only correct for terms of powers :=:;N - 1. The method of moments (3a) and the subdomain method, however, give identical results for effectiveness factors calculated in this

is used and we require (by physical arguments) that YN is positive and has positive derivatives in (0, 1), we obtain 2

d yN 1 dyN) __ N 2 ma 4 ( x--+-dx2 x dx

92

Polynomial Approximation-Construction Pnnciples

Chap. 2

(the steepest possible function satisfying these conditions being .YN = x 2N); i.e., the approximation order should at least satisfy 4N2 >

Sec. 2.4

93

Nonlinear Problems

is again used as a first approximation. The residual is R1(ab u) = -a 1

-

p[1

+ a 1(1

(68)

- u)]n

The formulation of the equation for a 1 is just as simple as in the linear case when the collocation method is used:

or N > 2

Thus the results for N = 3 are only expected to be reliable up to

In the given example, the "best" approximations are definitely obtained with Galerkin's method. The method of moments gives comparable but slightly less accurate results, whereas the subdomain method and the collocation method both fall rather far behind. One should, however, bear in mind that for the last two methods we have quite arbitrarily chosen equal-sized subdomains and equidistant collocation points. It might well be that better results are obtainable with different choices. With regard to the computational effort for this linear problem, all the MWR methods require the solution of a set of N-linear algebraic equations. These equations are almost directly available in the collocation method, whereas all the other methods require integrations. This presents no problems here, but in nonlinear problems the formulation of the equations for the a; may actually be more cumbersome than solving the equations.

R1(ab u1)

= 0

or (69)

The equation is now nonlinear and has to be solved numerically. In general, a parametric solution in p cannot be given. Using Galerkin's method, we obtain 1

R 1 (ab u)(1- u) du

{

=0

0

or 1 {

{-a~ -

p[1

+ a1(1

- u)r}(1 - u) du

= 0

(70)

0

If n is noninteger, this integral normally has to be evaluated numerically and an explicit equation in a 1 cannot be obtained. Even for integer values of n, say n = 2 or n = 3, considerable algebraic manipulation is necessary to evaluate the integral. It appears that we have to choose between the easily applicable but in general rather inaccurate collocation method and the trustworthy but cumbersome Galerkin method. A compromise is, however, possible. The integral in (70) may be approximated by a numerical quadrature. Quadrature formulas are treated in section 3.3 and here we shall only state some results:

2.4 Nonlinear Problems (71) 2.4.1 The first approximation y1 (x)

or, in a more general formulation,

We now consider (11) with n ¥- 1: d2y u du 2

+

dy du

=

q,2

(72) n

4 . y = pyn

y = 1 at

u = 1

(66)

(67}

The one-term approximation (19) Y1 = 1

+ a1(1

- u)

In case a weight function W(u) can be extracted from the integrand as in (72), it is possible to build this weight function into the quadrature by means of a different choice of weights wi and quadrature abscissas ui in (71) and (72}-thereby improving the accuracy of the quadrature. In our present case (70), we shall use W(u)

= 1-

u

and

F(u)

=

-a~ -

p[1

+ a1(1

- u)r

94

Polynomial Approximation-Construction Principles

Chap. 2

Sec. 2.4

Nonlinear Problems

95

u 13 (l

For a weight function W(u) = - ut' there exists an optimal choice of quadrature points in the sense that the highest possible power of u in a power series expansion of F(u) is correctly integrated when these uvalues are used as quadrature points. The optimal quadrature points are zeros of PXf13 )(u) where PXf 13 )(u) is a polynomial of degree M in u that satisfies the following relation:

r

uP(1 - u)"u1P\:iPl(u) du

=

0 for j

=

0, 1, ... , M- 1

(73)

The quadrature is exact when F(u) is a polynomial of degree ::; 2M -1 in u and in general the terms in a power series expansion of F(u) are correctly integrated up to and including u 2M- 1. A discussion of the properties of orthogonal polynomials PXf13 )(u) and the proof of the optimality of the proposed quadrature are deferred to chapter 3. Here it suffices to give the result 1

1

R 1(u){1- u)du

0

M

~I

t

We have now shown .that the specific choice u 1 = is optimal in the sense t?at the collocatiOn method becomes identical to the approximate Galerk~n m~th~d .(74) forM= 1, and we have a feeling that the Galerkin approximatiOn Is m general better than other MWR. . w_e further notice that the linear case n = 1 requires M ?. 1 for Identt~y betwe~n (74) and (70). Hence a collocation method (75) with U1 =:= 3 would giVe the same result as Galerkin's method in the example of sectwn 2.2. For other values of n, (75) and (70) give different results-but since the outcome of the calculation.s in (70) is an approximation for y, we cannot conclude that a 1 determmed from (70) is always "better" than a from (75). 1 As an example, n = 2, = 3 (p = ~) will be chosen. The optimal collocation method gives -a1 - ~[1 + a 1(1 - ~)] 2 = 0

M

wjR1(ui)

1

=I wi{-a1- p[1 + a1(1-

ui)r}

1

a 1 - { - 0.6771 - (- 3.32)

(74)

where {ui} are theM zeros of P~ 0 )(u). All zeros of PXf 13 )(u) are real and distinct and 0 < ui < 1. All weights wi are real and positive. Using (74) instead of (70) relieves us of the task of evaluating an analytical expression for the integral without really sacrificing anything: The values computed from (74) with a given value of a 1 and increasing M form a rapidly convergent sequence. Also the original Galerkin expression (70) only provides us with a first approximation (19) to the solution of the differential equation (1), and a reasonable approximation in the determination of a 1 cannot conceivably give a final result {19), which is in principle worse than that obtained by means of (70). If n is an integer, the residual RN(u) of equation (1) is a polynomial of degree nN in u. Exact integration of {70) and a quadrature with M quadrature points consequently give the same result when M?. (n + 1)/2. An even simpler version of this method is obtained when M = N, i.e., when one quadrature point is used for the first approximation. Equation (74) can,now be written 1 R1(u)(l- u) du = 0 w1R1(u1) ~ 0 (75) -a1 - p[1 + a1{1 - u1)r = 0 u 1 is the zero of Pi1'0 )(u) = 3u - 1 or U1 = ~. The collocation method (69) contains u 1 as an adjustable parameter.

J

(76)

The solution a1 = -3.32 is discarded since it leads to a physically unreasonable approximation y 1 = 1 - 3.32(1 - u), which is negative for

u < ~.

The Galerkin method gives

(77)

a

1

= { -0.7005 (-2.86)

The solution a1 = -2.86 is also discarded. The effectiveness factor is found from 2

l

YJ 1 =

I 0

[1 + a 1(1 - u) f du = 1 + a 1 + a 1 3

that is, Optimal collocation: Galerkin: (Exact result:

0.476 0.463 0.44621)

Polynomial Approximation-Construction Principles

96 2.4.2 Optimal

11

Chap. 2

Sec. 2.4

97

Nonlinear Problems

Galerkin with quadrature"

method 2.4.3 Computer results for n = 2

The trial solution N

YN = 1 + (1 - u) L aiut-

(78)

1

i=1

yields

!'fth-order approximations for n = 2 and = 3 are found optimal collocation method and by the G a1er k"m method (70)by The the f h ahccurac! o t e effectiveness factor TiN [computed from (3)] and.f . s own m figure 2 - 6 a s a function . o f the approximation order N. oInathis I IS Log 10 e N

The Galerkin method gives the following N relations: (80)

j = 1, 2, ... ,N

r

or

J'i(u)(l- u) du = 0, 1

j

(81)

= 1,2, ... ,N

1

where Fj(u) = RN(a, u)u - • Each of the N integrals in (81) is evaluated by quadrature: 1

J

M

Fj(u)(1 - u) du =

0

L

wkFj(uk)

=

k=1

1

M

L

wkRN(a, uk)u{;·

(82)

k=1

where p<;/\uk) = 0, k = 1, 2, ... , M. We shall again choose M = N, and the following N equations are obtained for the N unknown coefficients ai: N

L

1

wkRN(a, uk)ut- = 0,

j

= 1, 2, ... ,N

(83)

k=1

The N equations can be satisfied for all when (84) 0

where uk are the zeros of P~' )(u). Again we notice that the best approximate Galerkin method is a collocat"n method, provided we choose as collocation points the roots of P~' 0 )(u)

= 0. In the case n = 1, Fj is a polynomial in u of degree N + j - 1, or at most of degree 2N - 1, which means that for the linear problem the optimal collocation method and the Galerkin method will give identical results. In the general nonlinear case we see that the optimal collocation

F_ig~re 2-6. Approxi~ation error by Galerkins method (line with closed c(lu/c e)s)(l)and ~y optimal collocation (line with open circles)· y + X Y - 9y = 0. .

Polynomial Approximation-Construction Principles

98

Sec. 2.5

Chap. 2

example the optimal collocation method yields results for 'Y1 that_ for the same N are about 2.5 times less accurate than results obtamed by Galerkin's method. But more important, the same dependence of. the accuracy on N is found by the two methods. Furthermore, the optimal collocation method gives a more accurate value of a1. . In comparison with the tremendous increase in accuracy that ~s obtained with an increase of N, the difference between the two method~ IS insignificant, making the optimal collocation m~thod our preferred chmce of MWR due to the ease with which the equatiOns (84) for a are set up.

The coefficients '}'; and '}'~ of (87) and (88) are all positive. If we specifically wish to stress that (-1y-i-v. is the coefficient of u 1 in P." a • I] double mdexing 'Yii is used. Recurrence formulas for 'Yii are given in chapter 3 but a few examples are shown in table 2.6. TABLE

2

0

0,0 1, 0 2,0

Approximation YN(x)

2.6

EXAMPLES OF ORTHOGONAL POLYNOMIALS

a,{3

2.5 Reformulations of the Mh-Order

99

Reformulations of the Mh-Order Approximation YN(x)

2u- 1 3u- 1 4u- 1

1 1

6u 2 10u 2 15u 2

-

6u + 1 8u + 1 lOu + 1

2.5.1 An optimal Fourier-type expansion It is easily seen that the polynomials of table 2.6 satisfy the orthogonality relation:

Approximation (34) to y is of the following form: N

YN(u) = 1

+ (1

- u)

L a;ut-

f u 13

1

1

i=l

YN is a polynomial of degree N in u, but we have used the bounda~y condition y(1) = 1 to eliminate one of the N + 1 param~ters a; that m general are required to determine an Nth-degree polynomml. The following (N- 1)-degree polynomial can be extracted from (34): ~

YN - 1

= 1 - u = t::-1 a;u

gN-l(u)

t-1

(85)

Equation (85) can be rewritten into an N-term expansion in any set of (i - 1)-degree polynomials P1-1 (i = 1, 2, ... , N). ·· gN_ 1 =

N

N

t=l

t=1

L a;u 1- 1 = L

b;P;-I(u)

(86)

One purpose of this reformulation may be to obtain a ~ore rapidly convergent series, i.e., a series where b; decreases more rapidly than a;. We shall often choose PJu) as the orthogonal polynomials defined by (73). These polynomials can be written t-1 t-2 + (-1)i (87) ( {3) Pt' (u) = 'Yiu - '}';-1u + '}';-2u - · · ·

(1 - utP;(u)Ij(u) du

= Cz S;i

(89)

0

To illustrate the equivalence of (85) and (86) we shall express one series in terms of the other for the specific case P?· 0 )(u) and N = 3.

U2

=

1p(1,0) 6 0

+ .±.p(l,O) + _l_p(I,O) 15 1 10 2

or the following values for bi in (86)

= a1 + ~a2 + ~a 3 b2 = ~az + 1~a3 b1

b3 =

(90)

ioa3

A systematic procedure for obtaining b; from a; follows from the orthogonality property (73) of the polynomials:

I

or normalized differently such that the leading coefficient is 1: (-1)' (ot,{3) -

p· I

- u

i

-

'Yi-1 ~

u

t-1

+ ... + - ~

= ut - 'Y:-1ut-1 + ... +

(-1)~'}'~

(91)

(88)

N

N

= L ai L wkPt-1 (uk)u~j=

I

k=I

1

100

Polynomial Approximat1on-Construct1on Principles

Chap. 2

The quadrature points in (91) are theN zeros of P~,f3)(u). uj- 1 • P;_ 1(u) is at most a polynomial of degree (N- 1) + (N- 1) = 2N- 2 in u and the quadrature is exact. Similarly the aj can be derived from b; of (86) by differentiating both sides of the equation and setting u = 0. 1

aj

N

= (. _ 1), "L b; ] • i=1

dl-1 p~a,{:J)'

d ~~~

N

1

=·

U

u=O

..

."L. (-1r-1Yi-l,j-lbi z=J

1

1 2 =(35u + 305u + 1400) 2494

5 (280, 61, 7) 2494

5

1

2494

10

= - - . -(3015 222 7) '

YN

= 1 + (1 -

f

1

0

'

rapidly than the a; coefficients, an obvious advantage for approximation purposes since a rapid convergence of the series for increasing N is expected when addition of an extra term contributes very little to the approximation. Table 2.7 illustrates that the low-order coefficients b; of an N-term expansion in orthogonal polynomials P~ 1 ' 0 ) are very rapidly stabilizedcompared to the coefficients ai in the expansion in monomials, which for every N is obtained by reformulation of the polynomial expansion.

1 2

3.86. 10-4.45. 10- 3

2.7 (85)

AND

(86)

FOR

p

=

b;P~~~)(u)

(94)

1

(1 - u) du = _ 2

and TJ

1

2

3

4

af,b1

1.3 . 10- 3 -3.8 · 10-6

2.3 · 10-s -2.4 · 10-9

-2.6 · 10-7 -3.6 ·10- 13

-0.56132 -0.60445

1.07 . 10-2 -6.2. 10- 5

-3.5 · 10- 4 -2.4. 10-8

6.3. 10-6 -3.6 · 10- 12

-0.12265 -0.044507

= 1 + bt = 1 _ 3015 = 6961 2

9976

9976

which is the same value that can be obtained from table 2.4, Galerkin's method, N = 3 for p = 1. We observed in table 2.5 that terms up to and including p 2 N in a series ~xpansion of TJ in powers of p were represented exactly when the Galerkm MWR was used to determine YN(u ). It has now been shown that b 1 of expansion (94) has the same accuracy as TJ, i.e., that terms up to and including p 2 N in a series expansion in powers of p of b! obtained from the exact solution are correctly represented.

b* = - -1 I

It is seen that the sequence of coefficients b; is decreasing much more

TABLE

"L

since all higher terms in the sum are integrated to zero due to the orthogonality relation (89). The value of

1

ExPANSION COEFFICIENTS OF

N

u)

i=1

(93)

Correspondingly the coefficients bin an expansion of y 2 (u) in P~ 1 ' 0 \u) are determined from (90). b

101

One further advantage of expansion (86) with P; = pp,o) is that the mean value of y is determined from b 1 alone.

C0 =

i.e., a = -

Reformulations of the Mh-Order Approximation YN(x)

(92)

where (-1) 1- 1')';- 1,1 _ 1 is the coefficient of u 1- 1 in P~~f) as defined in (87). In (55), a second-degree polynomial approximation for g(u) is obtained using Galerkin's method. Y3 - 1 _ u g2(u) =

Sec. 2.5

1

[

1-

Co o

I (2JP;t] du 0

= f(p)

Io(2vP)

(95)

by the formula for determination of Fourier series coefficients in the infinite orthogonal polynomial expansion: g

y- 1 1- u

00

= -- = L

i=l

bfP}~~)(u)

(96)

This high accuracy of b 1 is not exceptional and it may in fact be shown that an arbit~ary coefficient in (94) is accurate up to and including the t 2N+l-z · . erm f! * m a power senes representation of the corresponding coeffict~nt bi of (96). In contrast the coefficients a; of expansion (34) in monomtals are accurate up to and including pN in a power series representation of a f as obtained in the infinite Taylor series solutitm of

g(u).

. Cons~quently th~ full strength of Galerkin's method as an approximation toolts first reahzed when the trial functions are chosen in the optimal way-in t~is e~ample as_ (1 ~ u)P~~~)(u), i = 1, 2, ... , N, or if the original expanswn m monoquals ts converted into the optimal expansion using (91).

Polynomial Approximation-Construction Pnnciples

102

To prove the high accuracy of the bi obtained by Galerkin's method with trial functions (1 - u)P~~~)(u), we return to (11):

!!_(u dy) du du and

y(1) = 1

=

PY

103

Reformulations of the Mh-Order Approximation YN(x)

or, in matrix notation, (A+ pB)b = pc where A, B, and c are given by expressions similar to (50). AI, =

r {u,:U[(lc~

u)P,_IJ})(l-

u)~_ 1 du

(99)

y(O) is finite (100)

We now introduce an approximation that is entirely different from the expansions in increasing powers of u: y(u, p) ~

(101)

M

L

fk(u)pk lnt~e~r:!lticm

k=O

This is a perturbation solution to (11). The increasing powers of p = <1> 2 /4 are multiplied by unknown perturbation functions fk(u). We choose f 0 (u) to satisfy the boundary conditions of (11) while fk(u) (k > O) satisfy homogeneous boundary conditions: / 0 (1)

=

1

while fk(1) = 0,

k

=

d (

dA) {o,A-b =

Aii = u

d~ [(1 -

u)Pi-1](1 -

u)P;- 1 1~

1, 2, ... , M

Equation (97) is inserted into (11) and coefficients of equal powers of p on both sides of the equation are identified: du u du

by parts in (99) gives

The first term is zero and the integral is symmetric in i and j. Hence A is symmetric, and we can show furthermore that A is a diagonal matrix.

k=O k>O

The first differential equation yields fo(u) = 1 The second is integrated to / 1 (u)

= u +A = u - 1

when f 1 (1) = 0 is inserted. The next two perturbation functions found in the same way and generally it is noted that fk(u) is a polynomial of degree k in u.

in (99) is a polynomial of degree i - 1. Using the orthogonality property of the expansion polynomials P = p(l,o)(u), it follows that all A 1i with i < j must be zero. Since A is symmetric, it must be diagonal. Equation (100) shows that B is also symmetric. (1 - u)Pi-I is a polynomial of degree i and all B1i with i < j - 1 are zero. Hence B is a svn[lffi1etn'tc tridiagonal matrix. Finally, c1 = 0 for j > 1, while

cl

'

= r(l-u)Podu =1

0

Next contider expansion (94) and Galerkin's method to obtain bi:

0

A22

B23

+p

A23

I

B23

1

=

p

0

(1- u)Pi-1 du

j

B33 · . .BN-l,N

= 1, 2, ... ,N ANN

0

BN-l,NBNN

b=p

Polynomial Approximation-Construction Principles

104

Chap. 2

In a discrete analog of (97) we now expand b in a power series in p: b

=

r M

k

(103)

tkp

k=O

where f are perturbation vectors that are determined one by one when equal p~wers of p are collected in (102). A(fo

M

f

2

+ flp + fzpz + ... + fMP ) + B(foP + flp + ... + MP

or

M+l) -

- p

C

Af0 = 0

c

Afk

+ Bfk-1 = { O

k = 1 k > 1

(104)

The solution of (104) is fl =A-le= (-1-, 0, ... 2A11

f 0 = 0,

fk = -A- 1 Bfk-l

'o)

fork > 1

A -1 B is a tridiagonal matrix that is obtained from B by division of row i by Au. Hence fk from (105) has k components different from 0: fz = __ 1_.(B 11 , B12' O, ...) 2A 11 Au Azz

1 (Bi 1

f3 - - --z A 11 - 2A 11

Bi 2

B12Bu

+ AuAzz'A u A 22

+ B12B22 2

Bz3B12, O,

A 22 'A 22 A 33

...)

Let us mark the components of fk by an index (J) that is obtained as the highest index from Bii or Au that occurs in the component:

Sec. 2.5

Reformulations of the Mh-Order Approximation YN(x)

105

The double indexing lik is used to indicate the ith component of vector fk. A comparison of (107) and (97) with the appropriate expressions for the polynomials /;(u) shows that all polynomials l;(u) of degree i s N can be exactly represented by the linear combination of polynomials (1 .;. . u)Pi-b i = 1, 2, ... , N in (107). For these terms (107) is simply a reformulation of (97) and the coefficients bi found by a Galerkin approximation of order N will all contain the correct coefficients /;k in a power series expansion of the true Fourier series coefficients bf up to and including the coefficient of pN. The low-order coefficients bb b 2 , ••• are, however, considerably more accurate. Consider a 2Nth-order Galerkin approximation similar to (107). Now all coefficients bi are correct up to and including the coefficient hzN of 2 p N. The schematic solution (106) for fk does, however, show that only the first N rows in (1 02) enter into the determination of the first element Ilk for k :S 2N, i.e., that Ilk are identical fork :::; 2N whether they are determined by an Nth-order or a 2Nth-order Galerkin's method. In the same way (106) shows that lik are identical whether they are determined by an Nth-order or a (2N - i + 1)-order Galerkin approximation. Hence for an Nth-order Galerkin approximation (94) in orthogonal polynomials P~~~)(u), the same coefficients /;k appear in a power series expansion b; = L~ l;kPk up to and including /i,ZN-i+l as those that would have been obtained from a similar power series expansion of bf in (96). This completes the proof of the optimal character of Galerkin's method using expansion (94) for the linear differential equation {11 ). The result explains the accurate value of 'YJ in table 2.5 for Galerkin's method (TJ is given by b 1) and also the rapid stabilization of bi in table 2.7. The method of moments with the same set of trial functions has an accuracy of 2N- i (i = 1, 2, ... , N) forb; in the sense defined above.

f1 = [(1), 0, 0, ...] 2.5.2 Lagrange interpolation polynomials

f 2 = [(1), (2), 0, 0, . · .]

f3 = [(2), (2), (3), 0, 0, ... ]

f4 = [(2), (3), (3), (4), 0, 0, ... ] + 1), (k + 1), ... , (2k - 2), (2k - 2), (2k - 1), 0, 0, ..

fzk-1 = [(k), (k), (k f2k

= [(kjil(k + 1), (k + 1), ...· '(2k

- .1), (2k

~

Two formulations of the Nth-degree polynomial approximation YN have hitherto been discussed: N

YN

= L a;xt

YN

= L

1), (2k ), 0, 0, " .] (106)

i=O

(108)

N

The fk determined in (105) are mserted mto (94).

YN

= 1

+ (1 - u)

I

i=l

b.P_ 1 = 1 + (1 - u) I

I

I ( ~ /;kPk)P;-1(u)

i=l

k=O

i=O

biP;(x)

(109)

A third approximation operator will, however, be frequently used in this text: N+l

YN

= L y;l;(x) i=l

(110)

Polynomial Approximation-Construction Pnnciples

106

. · 1 The "building blocks" are N· This is the Lagrange interpolation polynomla. now N + 1 polynomials l; (x) that are all of degree .

x - Xz) ... (x - xN+1) i~ a polynomial o~ degr~e N + PN+~(x) (~ 1)( . 1 It is called the node polynomial while l;(x) 1 with leadmg coefficient . . . N + 1 interpolation points and Y; of

=

- x

are the Lagrange polynomials. xl are . h N + 1 ordinates at these pomts. h (110) are t e . 1 0 f degree N passing through t e To verify that (110) is a polynomia ) it is sufficient to note that N + 1 points (xb ~1), (xz, Yz), · · · '(xNdb~~+l(~) = 0 for i "# j and 1 for l; are all polynomials of degree N an t a I ' i = j. . . Another formulation of l; can be giVen. N+1 X - X (112)

z.

n

=

I

_____L X]

j=1,1 XI -

. . 1 . that the product consists of all factors where the notation 1 = , z means ._ . (x - xJ)j(xt - xj), j = 1, 2, ... 'N + 1, except 1 - z. N+1

n (X -

) x,

=

PN+1(x) X - Xt

x)

=

dpN+1(X) dx

j=1,1

N+1

n (xi -

Equation (114) is completely analogous to the distance function for the Taylor series approximation from x = x 0 •

(111)

PN+1(x) l; = (x - x, )PN+1 (1) ( Xt )

f

or X =

XI

,=1,i

and the identity o~ (112) and ~1111 is( p)r~;:!terpolative approximation of To obtain the distance functiOn N x . . . . f . ( ) define s (x) by the followmg relation. a giVen unction Y x , ( ) 113 s(x) = EN(x) - KPN+1(x)

N + 1 interpolation points, s (x) is obviously For x equal to any of the F other choice of x in the approxizero whatever the value of K. or an~ h that s(x) = 0 Hence s(x) mation interval [a, b] we can choose sue x a~d the current has at least N + 2 zeros E[a, b ], namely Xb x2,h .. ' fu+~ (N+l)(x \ has at A repeated application of Rolles theore~ s ows a s J x. ~~... [ b] This (unknown) zero IS at 0. least one zem .E a, · cN+1>(0) = 0 = £<:+1)(0) - K(N + 1)! 8 (N+l)( ) - (N+1)(lJ)· Now YN(x) is a polynomial of degree N in x and. EN 0 - y ' since EN(x) = KPN+1(x) at the current x, we obtam (N+1)(0) (114) EN(x) = y ) 1PN+1(x) (N+ 1 .

=

EN(x)

y
Ef a, b ].

(115)

The two values 0 and 0 1 in (114) and (115) are, of course, different but the real difference between the two distance functions lies in the last factor, as easily seen by comparison of (114) and (115) for y(x) = xN+ 1 and x 0 = 0: (114):

(116)

(115):

EN(x)=xN+

1

(117)

In both cases EN(x) is.an (N + 1)-degree polynomial with leading coefficient 1 but the N + 1 zeros of PN+ 1 (x) that all lie in the interpolation interval, e.g., [0, 1], will render the maximum value of jEN(x)j in [0, 1] in (116) much smaller than 1, the value obtained in (117). This is one of the major entries into the field of general approximation theory and a "best polynomial" approximation of degree N can be obtained by minimization of IPN(x )j in [0, 1] with a suitable choice of the N + 1 interpolation points. It is natural to close the discussion of construction principles for Nth-order polynomial approximations by noting that once YN(x) has been constructed in any one of the three approximation modes (108), (109), (110), it is a simple matter to compute the coefficients of the other two approximations. As an example, take the Fourier-type approximation (109) with P1 = P~a,f3) and the interpolation approximation (11 0): N

YN(x)

= L

b;P~a,f3)(x)

N+1

= I

t=O

y;l;(x)

i=1

The coefficients b are determined in the usual way for Fourier series:

where W(x) = x 13 (1 - x)a

and

G

=

i

1

WP7 dx

0

The integrand YN(x )P~a,f3\x) is at most a polynomial of degree N + N = 2N. Hence b; can be calculated exactly by a quadrature formula that accurately integrates a 2N-degree polynomial using N + 1 quadrature points. N+1 Gb; =

I

k=1

for anv Q:iven x-value

107

Reformulations of the Mh-Order Approximation VN(x)

Chap. 2

wkP~a,f3)(xk)YN(xk)

(118)

108

Polynom1a I Appro ximation-Construction Pnnciples

References

Chap.2

109

. node polynomial PN+l(x) that permi~s In subsection 3.3.4 we denve a 1 . 1 when one of the nodes IS . f 2N degree po ynomta f exact integration o a . · N nodes are zeros o an . - 0 or x = 1 and the remammg etther x . (a,f3)(x) orthogonal polynomtal PN : d' t (x ) the exact values of hi Thus with N + 1 interpol~tiOn ~~s)ns~;~l~~e~ts (91) and (92), which in (109) can be found. Equation ( 'd ntical expansions of the Nthare other conversion formulas between 1 e degree polynomial YN(x).

2

EXERCISES

A tan A = Bi

Show by perturbation analysis that the eigenvalue of smallest magnitude is approximately

= ko + a(T- To) A1 =

Show that the heat balance takes the following form:

!i.[(1 + bO) dO]

=

0

0(0)

=

1

dx

dx

=

0

0(1)

with one parameter a1. . - 1 = 1 x = 0. . b _ 1 by collocation at x - ' x 2• Fmd at« for . of a1 m . Powers of b is rapidly convergent. 1 an -expansion For b

= qo + q1b + q2b 2

. and q 1 using collocation at x = !. Determme qo ·d· t lloca (1) by eqm tstan co. 2 Write a computer program for so lution .of equation (34) n and N are program mput • tion and trial functwns · · by equation · = 0 5 1 and 2. gtven df 3 8 should be use or n · ' ' _ 1 data and N - ' ' · · · ' · 1 2 ' ,' . h Y< )(X -_ O) = 0 in a power• senes 1-, 1 · f y(2) - y = 0 Wlt 12) 3. a. Expand t.l\' so utwn o ( to and including the term m x from x = '0. Find the first seven terms up

= O) of p art a is

6

360

REFERENCES

What is the solution for b = 0? below the solution for b = 0? . d t te the form of the approximatiOn Is the solution for b > 0 above ~r Select a proper set of trial functions an s a

of the series. _ b. Assume that the parameter Yo - y(x

.112 ( 1 - Bi .) - + -11 B I2

BI

for small values of Bi.

in the dimensionless variab~es x and _o. 1 arameters? What is b in terms of the given phystca p

a1

2

4. The eigenvalues A for the heat conduction problem in slab geometry and with film transport resistance are given by

nduction through a slab of thickness L. The two 1. Consider steady state he~t c? ( = 0) = To and T(s = L) = T1. The f the slab are mamtamed at T s . . faces o d t' 't of the solid is a linear function of T. heat con uc IVI Y k(T)

2

e. Consider y< ) - y = 0. What is the largest value of Yo for which the series developed in part a is convergent for x = 1? f. Find a perturbation series for the solution of y< 2 ) - 2 y 2 = 0, y = 1. If the result is disappointing, then use continued differencing on the sequence for Yo that you obtain from the partial sum of series with N = 1, 2, ... terms. h. Reflect on the range of applicability of the perturbation series. Can it be used for the same values as the Taylor series? i. Repeat for the trivial equation y<2 ) - 2 y = 0. Show that the radius of convergence for the perturbation series is = 7T/2.

1. Solve the

differential equation in closed form. f the power series in part a with c. Prove that the radius of convergence or = 1 is smaller than or equal to 2.9745. Yo Yo in part a wh en y (x -- 1) = 1 . d. Find

Weighted residual methods have been used to solve mathematical models of physical science for the last 50 years, and naturally a vast number of references to applications exist. A substantial part of these references is listed in Finlayson (1972). Over the years, numerous papers have appeared on weighted residual methods from research groups at the University of Minnesota (Sparrow, Scriven, Amundson, etc.), and Finlayson's review (1966) with Scriven introduced a number of chemical engineers to the subject. Finlayson's authoritative textbook from 1972 is today undoubtedly the best reference to applications of MWR in the various fields of transport phenomena. Rice (1964, 1969) is a highly readable introduction to approximation theory and Natanson (1955) cites much of the older literature on the subject. Morse and Feshback (1953) has a long chapter on perturbation methods but it is difficult to read and lacks newer references. The best text is probably Nayfeh (1973), which contains numerous simple examples. Other texts are Van Dyke (1964), which exclusively treats problems from fluid mechanics, and finally Cole (1968). 1.

FINLAYSON, B. A. The Method of Weighted Residuals and Variational Principles, New York: Academic Press (1972).

110

Polynomial ApproxJmatJon-A First View of Construction Principles

Chap. 2

2. FINLAYSON, B. A., and SCRIVEN, L. E. Appl. Mech. Rev. 19 (1966): 735.

3. RICE, J. R. The Approximation of Functions, Vols. 1-2. Reading, Mass.: Addison-Wesley (1964, 1969). 4. NATANSON, I. P. Konstruktive Funktionentheorie. Berlin: Akademie Verlag (1955).

Some Important Properties

5. MORSE, P. M., and FESHBACK, H. Methods of Theoretical Physics. Chapter 9, pp. 1001-105. New York: McGraw-Hill (1953).

of Orthogonal Polynomials-

6. NAYFEH, A. H. Perturbation Methods. New York: Wiley (1973):

3

Formulation of a Standard

7. VAN DYKE, M. Perturbation Methods in Fluid Mechanics, Academic Press Series in Applied Mathematics and Mechanics. New York: Academic Press (1964).

Collocation Procedure

8. COLE, J.D. Perturbation Methods in Applied Mathematics, New York: Blaisdell (1968).

Introduction Chapter 2 shows that a very efficient collocation method results when the collocation points are chosen as zeros of certain orthogonal polynomials, the so-called Jacobi polynomials defined by equation (2.73). In section 3.1 methods of constructing Jacobi polynomials and of determining their zeros will be developed. The MWR of chapter 2 are based on either power series or Fouriertype approximations of the unknown function. The object of the MWR is to determine the coefficients of these expansions. In section 2.5 the Lagrange interpolation polynomial is introduced as a reformulation of the primary expansions of YN in increasing powers of x. A computer-oriented collocation algorithm is much easier to set up when the N ordinates at the collocation points are used as unknowns rather than the coefficients ai or bi of expansions .in terms of increasing powers of x or polynomials of increasing degree. Consequently formulas for differentiation of an arbitrary Lagrange polynomial l;(x) are set up in subsection 3.3.2 and quadratures based on integration of certain Lagrange polynomials are developed in subsection 3.3.3.

The results of sections 3.1 to 3.3 are utilized in section 3.4 where computer programs for calculation of zeros of Jacobi polynomials, derivatives of YN at the collocation points, and quadratures based on the collocation ordinates are described. 111

112

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

Finally in section 3.5 a rigorous formulation of our collocation procedure is given. This section is the basis on which the standard problem types of chapter 4 are treated.

Sec. 3. 1

(t1 -- t .!)('Yo) ~ 'Y1 = 0 .!.

or

'Yz

p~~f3J(x)

since Yo = 1. The solution is

(1) t=O

13 (1

xt~(x)PN(x) dx

-

= 0

j = 0, 1, ... , N- 1

(2)

p~,f3)(x)(1

- xt'xf3

1

x 13 (1 - xtxiPN(x) dx = 0

j

= 0, 1, ... , N - 1

(3)

= 6x 2

-

+1

6x

= (-1)Nf({3 + 1) dN [(1 f(N + {3 + 1) dxN

(6)

)N+a N+f3] X

(7)

X

For a = {3 = 0 and N = 2,

0

or from any of a number of other relations that are all in one way or the other derived from the fundamental relation (2). The orthogonal polynomials defined by (2) are called Jacobi polynomials. A more convenient form of (2) is

P~o,o)(x)

Another method is based on Rodrigues' formula, which may be derived from (2) [see, e.g., Villadsen (1970), p. 37, or Szego {1967), p. 67]:

Yo is taken to be 1 and the remaining N coefficients can be found either directly from the orthogonality property 1

= 'Yz = 6 and

'Y1

In section 2.5, equation (2.87), P~,f3)(x) was written in the following form

I

113

For N = 2,

3.1 The Power Series Representation of

Ix

The Power Series Representation of ~o:.m(x)

(O 0)( ) P 2' x

=

1 d

2

2 dxz[(l

2

2

- x) x ]

=

2

6x - 6x

+1

Computing the Nth derivative in (7) and comparing coefficients of equal powers of x on the two sides of the equation, one may-after considerable al~ebra-arrive at the following explicit expression for 'Y; [Villadsen (1970,, p. 40]:

0

xi can be expressed as a linear combination of Pk (k

Y·

= 0, 1, ... , j)-see

(2.90)-and (3) is obviously equivalent to (2). The following set of linear equations for 'Yi is derived from the integrals (3).

'

= (N)f(N + i + a + {3 + 1)f({3 + 1) i f(N + a + {3 + 1)f(i + {3 + 1)

where ( N)

M'Y = 0

i

M .. = f({3 + 1 + i + j)f(a + 1)(- )N-i 1 Jl r(a + {3 + 2 + i + j)

(4)

(8)

N!

= i!(N- i)!

An even simpler recursive computation of 'Y; is obtained from (8) by taking the ratio of 'Y;- 1 and 'Yi:

i = 0, 1, ... ,N, j = 0, 1, ... ,N- 1 _N-i+1N+i+a+{3

since

f

Jo

1

xm(l- xr dx = f{m + 1)f(n + 1)

f(m

+ n + 2)

M .. Jl

= f{i + j + 1)f(1)(-1)N-i = (-1)N-i f{i + j + 2) i +j + 1

'Yi -1

(5) Yo

For a = {3 = 0, one obtains the Legendre polynomials:

i + {3

i

'Yi -

=1

and

i = 1, 2, ... , N

For N = 3 and a = {3 = 0, (9) immediately gives Yo -

2

= 1, 5

'Y2 - 2 . 2 • 'Y1

'Y1

=3·4

= 30,

-

· Yo 1

= 12 6

'Y3 - 3 · 3 · 'Yz = 20

(9)

114

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

The simplicity with which 'Yi can be calculated from (9) makes it the recommended formula whenever an explicit expression for P<;,'f3) is desired. Similar formulas can be derived from (7) for the coefficients 8i in N

XN

= L 8iP~a,f3)(x)

Sec. 3.2

Zeros of Orthogonal Polynomials

h1 = 0, hN =

h2 =

115

(a + 1)({3 + 1) (a + {3 + 2) 2 (a + {3 + 3)

(N - 1)(N + a - 1)(N + {3 -· 1)(N + a + f3 - 1) (2N +a + {3 - 1)(2N +a + {3 - 2) 2 (2N +a + {3 - 3)

(10)

for N > 2

i=O

The final result is an analog of (9): i(2i + a + {3 - l)(N + i + a + {3 + 1) 8 8 1 i- = (N - i + 1)(2i + a + {3 + 1)(i + a + {3) i

(11)

with 8N = 'YN

and

i = N, N- 1, ... , 1

(12) where FN, GN, and HN are functions of N, a, and {3. A three-term recurrence relation (12) can be shown to hold for any family of orthogonal polynomials and the functions FN, GN, and HN can be calculated by an application of the fundamental definition of the orthogonal family-in our case, equation (2) [Villadsen (1970), p. 55]. The coefficients gN and hN of a recurrence formula for the rescaled polynomials PN = PNI'YNN are given in equations (13) to (15):

~

(13)

PN = [x - gN(N, a, {3)]PN-1 - hN(N, a, f3)PN-2

{3+1

+ {3 + 2

gN =

1[

21-

az - {32 (2N + a + {3 - 1)2

=

1

2

(N

_

For a = 1, {3 = 0, and N = 2, one obtains y 1 = 8 and y 2 = 10 from (9) and 8 2 = -fo, 8 1 = 1~, and 8 0 = ~ from (11) in accordance with the results of table 2.6 and (2.90). Whereas (9) is excellently suited for a calculation of the coefficients in the explicit expression (1) for PN, it is not at all suited for machine computation of PN at a specific x-value. 'YNi are rapidly increasing with N and since the terms in (1) occur with alternating sign, a significant loss of digits is unavoidable. As an example, the tenth-degree Legendre polynomial may be considered. Here y 10,7 = 2,333,760, while the maximum value of Pi~o) is 1. Hence another computational scheme is used to obtain PN(xk). The following recurrence relation is recommended:

g1 = a

The recursive evaluation of PN(xk) is started with N = 1, p_ 1(xk) arbitrary, and p0 (xk) = 1. The following computation on a 10-digit machine of pco,o)(O 98) = (0 O)( 10 • YNNPH) 0.98) by (1) and by (13) with gN

1

] -

1

(14)

for N > 1

(15)

1'10,10 -

2:

1),

h N

-

(N- 1)2 4(2N- 1)(2N - 3)

(N

2:

2)

f(2N + 1) 20! f(N + 1)f(N + 1) = [(10)!Y = 184,756

shows that (13) is far superior to (1);

Pi~0)(0.98) = -0.2552433289

= -0.2552433288 = -0.255252

(exact) (by recurrence)

(by power series)

Th~ advantage of (13) over (1) is even more obvious when an expansion YN m orthogonal polynomials is used to calculate YN(xk): N+l

YN(xk)

= L

biPi-1(xk)

i=1

(16)

In this case every term Pi(xk) calculated in the sequence (13) is used.

3.2 Zeros of Orthogonal Polynomials In the example in chapter 2 it was shown that an "optimal" collocation inetho~ ';ith accuracy comp~rable to-or even equal to-the accuracy of

Galerkm s method was obtamed when the collocation points were chosen as the zeros of an orthogonal polynomial, pJ!r,o). We shall henceforth call a collocation method with collocation points equal to the zeros of any Jacobi polynomial an orthogonal collocation method. In this section, various methods for an accurate computation of the zeros of P<;,f3)(x) [or p<;'13 )(x)] are discussed.

Properties of Orthogonal Polynomials-Collocation Procedure

116

Chap. 3

It is easily proved from the fundamental relation (2) that PN(x) has N distinct, real-valued zeros xk and that 0 < xk < 1:

I

1.

I

1

1

x13 (1

- xtPoPNdx =

0

Sec. 3.2

sign changes in this sequence is equal to the number of real-valued zeros of.PN larger than x~). Hence if K sign changes are observed for a given xg), there are K zeros larger than x~):

W(x)PNdx = 0

C)

0 < X1 < X2 < · · · < XN-K < X~ < XN-K+ 1 < ... < XN < 1 (18)

0

Since W(x) is positive for x E (0, 1), there is at least one realvalued zero of PN(x) in the open interval (0, 1). 2. Let Xt, x 2 , ••• , xk (k ::; N) be the total number of real-valued, distinct zeros in (0, 1). The remaining N- k zeros are either complex, of multiplicity > 1, or outside [0, 1]. PN(x) changes sign at each xi? j = 1, 2, ... , k, but the polynomial (x - x 1)(x - x 2) ... (x - xk)PN(x)

=

Qk(x)PN(x)

N-K

K

In this manner an interval 8k in which each zero xk is located can be determined. The final computation of the location of xk within 8k is made by Newton iteration since a parallel development of p< 1 )[x~)] and p[x~)] is easily performed. Pi

= [x~)

- gJPi-1 - hiPi-2

is of constant sign throughout [0, 1]. Consequently,

I

(1) -

Pi

1

I=

-

[

(i)

]

(1)

xk - gi Pi-1 -

W(x)Qk(x)PN(x) dx ::/: 0 Po= 1,

0

On the other hand if Qk(x) is a polynomial of degree < N, the orthogonality relation (3) shows that I = 0. Hence k = N, and it is proved that PN has exactly N distinct, real-valued zeros in (0, 1). These properties of the zeros of PN considerably facilitate their numerical determination, and they are used in all the algorithms discussed in the following: 1. Algorithms based on the explicit power series coefficients 'Yi· 2. Algorithms with PN(x) and pW(x) computed by (13). a. Bisection followed by Newton's method. b. Newton's method with suppression of previously determined zeros. 3. QD algorithm on an equivalent tridiagonal matrix. The classical methods for computation of the N distinct zeros of .a polynomial PN(x) [e.g., Henrici (1964)] compute the value o~ ~N(~g)) where x ~) is the current estimate of the zero xk by nested multlphcatlon (Horner's scheme) using the coefficients of the polynomial. In our case, PN(x)

= [(yNx - 'YN-t)x + 'YN-2]x -...

(17)

We have alr~dy seen that PN(x~)) is inaccurately determined by this method, and serious errors in the zeros close tO X = 1 may result for large N. . It is just as easy to compute PN[xg)] by (13) and the accuracy of the computation is usually of the order of the machine accuracy. The numbers p 0 [x~)] 1, p 1 [x~)], . .. , PN[x~)] computed from (13) form a so-called Sturm sequence and it may be shown that the number of

=

117

Zeros of Orthogonal Polynomials

h

(1)

jPj-2

p~l)

+ Pi-1

= 0,

P-t

j

=

1, 2, ... ,N

(19)

= p~~ arbitrary

If the Newton method fails to converge, 8k can be reduced by bisection, but the final iteration is always done by Newton's method, which is strongly convergent locally. The following routine avoids the initial Sturm sequence-bisection calculations to obtain intervals 8k small enough for the Newton method to be convergent. A Newton iteration starting from x = 0 will produce a sequence m h . xro 1 , x 1 , ••• t at converges to x 1 from below-the reason bemg that IP~\x)l is monotonously decreasing in (-oo, xt). In the normal Newton-Horner scheme based on the coefficients of the power series representation of PM one automatically obtains the coefficients of GN-t = PN/(x - x 1) when the smallest zero x 1 has been determined. Hence the iteration may be repeated on GN-b producing a sequence x~l)' x~2 ), ... that converges to x 2 from below when x~o) = x 1 since IG.2~t(x)l is monotonously decreasing in (-oo,x 2 ). By this deflation process the zeros of PN are determined one by one and there is no danger that the iterations will converge to one of the previously determined (smaller) zeros. We shall use (19) with x~o) = 0 to compute x 1 • Even though the explicit coefficients of GN-k never occur when this procedure is used to determine xk+t. it is, however, still possible to suppress the previously determined zeros x 11 x 2, ... , xk. (20)

118

Properties of Orthogonal Polynomials-Collocation Procedure

G~~k - p~)

d[ln GN-k(x)]

=

dx

G~~k = p~)-

f_1_

GN-k

i=1X - XI

PN

_

x
~[x
••• ,

Sec. 3.3

Differentiation and Integration Formulas N

L ai(l

YN(x) =

- x)xi- 1 or

i=1

119

N

L bi(1 i=1

- x)Pi_ 1(x)

(25)

(21) N+1

YN(x) =

L

y(xJli(x)

(26)

i=1

(pNIPW)

= G~~k = 1 - (pN/P~)) L~=1 1 /(x -

The Newton iteration for xk+l with xb x 2 , is (i)

n:=l ;(x

-xi) n:=l(x - xJ

GN-k - PN -

GN-k

o(x)

L~=l

Chap. 3

xJ

(22)

xk previously determined

In all three cases, YN can be represented as the scalar product of two vectors, 0 a coefficient vector a, b, or y and a vector of expansion functions fr = (x , x\ ... , xN), gr = (P0 , P,, ... , PN), or hr = [/ 1(x), lz(x), ... , lN(x), lN+1(x)].

(23)

Linear operations are easily performed on the power series expression in (24):

where e is a small positive quantity, e.g., 10-4 • • It is seen that the iteration for xk+ 1 proceeds on GN-k JUSt as would be the case if the coefficients of this polynomial had been known, but in the actual calculations the recursive relations (19) are all that is needed. An entirely different method may be set up by noticing that the recursive calculation of PN[x~)] by (19) is exactly what is done when the determinant of a tridiagona~, symmetric matrix w~h ~~ -diag~nal elements Tj,,-1 = T-,-1,]. = .JJ;J and diagonal elements T..JJ - x •k - gj Is calcu_lated. (i) If x~> happens to be an eigenvalue ofT, the determmant = PN[xk] = 0, and x ~) is one of the zeros, xb of PN· . . Very efficient methods exist for computatiOn of the eigenvalues .of a tridiagonal matrix-e.g., the QD algorithm. In a sequence of operatiOns, T is transformed into a diagonal matrix with the eigenvalues equal to the desired zeros of PN(x) on the diagonal. In our experience this is the fastest method available, but the zeros are not so accurately determined as by the Newton metho~ based on (22) and (23). The Newton algorithm has been tested for different parameters (a, {3) of the Jacobi polynomial P
(27)

Xk+1 -

k+1

for-i = 1, 2, ...

and 0

x~ l1 = xk

+e

3.3 Differentiation and Integration of Lagra~ge Interpolation Polynomials

r

W(x)yNdx

=I

a,[f W(x)xj-] dx]

=[f

W(x)fdxra (28)

Similar linear operations on the other expansions in (24) and (25) are obtained with almost equal ease but the Lagrange interpolation polynomial (26) presents some problems since li (x) is not differentiated and 1 1 integrated as easily as X - or Pi_ 1 • In the present section we shall develop methods of obtaining expressions similar to (27) and (28) for the Lagrange interpolation polynomial dk(YN) = [ dk I(x)Jr dxk dxk y

(29)

(30) 3.3.2

Differentiat~of a Lagrange interpolation polynomial

Differentiation of (26) yields d [yN(x)] dX

3.3.1 Linear operations on YN(x)

= N£1 dfdJx) y(x,) = [J(1))Ty z=1

X

(31)

or in general

In chapter 2 several approximation operators were discussed: (24)

(32)

120

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

In particular, we are interested in obtaining the derivatives (dyNfdx)x=xi at the interpolation points x1, j = 1, 2, ... , N + 1.

Sec. 3.3

l~2 )(x1 ) = ~[p~~ 1 (xi) xi

PN+1(x) (1) ( ) PN+1 xi

=

(34)

(x - x-)1-(x) I

I

(1)

(

(2)

(

)

(k)

(

=

(x -

x-)l(2 )(x) I

(35)

I

+ 2l(l)(x)

3

(1) _- [(dy) -

dx x=x 1

I

'(dy) , ... ,(dy) dx x=xz dx x=x

d -(y) dx

(36)

JT

(44)

= Ay

(45)

+

kl~k-1)(x)

= l~l)(x1 )

(37)

dx2(y) =By

(46)

and similarly d2

I

yields

. p~~1(xi)

p~~1(xJ

(38)

where B1i = l~ 2 )(x1 ). A simple algorithm exists for numerical calculation of the derivatives p~~1(x,) of the node polynomial PN+ 1(x) at the interpolation points x1 : PN+l(x)

=

x.]

may be expressed

and, for x = xi ¥- xi,

z~k)(xi)

X· :

!"!

where A 1i

z
-

I

)

X;

Z.(x)

I

PN+1 x = (x - x-)l~k)(x) (1) ( ) I I PN+1 Xj

Equation (37) with x =

+

(43)

x1 - xi

All the coefficients l~l)(xi) and t?)(xi), i = 1, 2, ... , N + 1 and j = _+ 1 may thus be calculated from p~~ 1 (x1 ), p~~ 1 (x1 ), and PN+1(x1), 1 - 1, 2, ... , N + 1. The vector of derivatives

1( ~, •.• ,

y

)

PN+ 1 x (1) ( -) PN+1 X;

2lP)(x)[ lji)(x1 )

2-1-] I

Equation (34) is differentiated: PN+1 x = (x - x-)l(1)(x) (1) ( ) I I PN+1 Xj

-

PN+1(x,)

=

2l?)(xi)]

xi PN+l(xi)

l~l)(xi)[p~~1(xi)

= and similarly for higher-order derivatives. We now wish to obtain expressions for or method of calculating l~k)(xi), i = 1, 2, ... , N + 1, j = 1, 2, ... , N + 1: From the definition (2 .111) of 1; (x ), one obtams

121

Differentiation and Integration Formulas

_1_[P~~1(xi) - kl~k-1)(xi)]

N+1

n (x -

xi)

i=l

(39)

xi - xi PN+1(xJ

may be defined by the recurrence formula

Normally only zp)(x) and t?)(x) are of interest. For these derivatives, we

Po(x) = 1 \

obtain

Pi(x) = (x

(40)

~1 )Pi-l(x)

(41)

v?)(x) = (x - x)p}:?l(x)

pf)(x) = (x - xi)pj~ 1 (x) with (42)

j

= 1, 2, ... , N + 1

(47)

Differentiate (4 7) to obtain P?)(x) = (x - xi)PJ~I(x)

[noting that li(x) = 0]

=

+ P1-1 + 2pj~1(x) + 3p}:':\(x)

(48)

(49) (50)

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

Sec. 3.3

Values of p~~ 1 (x,) may thus be found by simultaneous evaluation of (47) to (50) with xi inserted for x. A further simplification is obtained when the interpolation points are reordered such that the xi value at which p~l 1 (x) is to be evaluated is always placed first:

where

122

Starting from p 0 (x 1 ) = 1, p&k)(x,) = 0, k = 1, 2, 3, we obtain PI(xJ = (xi - Xt)Po(xi) = 0 p~l)(xi)

= (x, - xJp61)(xi) + Po(x,) = 1

p~2 )(xi)

= =

p~3 )(x,)

2

(xi - x,)p6 )(xi) (xi - xi)p63 )(x,)

pj(x,) remains zero for all j

2:

+ 2p&1\xi) = 0 + 3p&2 )(x,) = 0

Differentiation and Integration Formulas

wi

=

J

\(x) W(x) dx

123

(57)

0

In subsection 3.3.2, simple expressions for derivatives of the Lagrange polynomials li at the interpolation points have been developed for any choice of interpolation points. Similar expressions for the quadrature weights w; cannot be found for an arbitrary weight function W(x) and arbitrary interpolation points. We shall, however, only be interested in the specific choice W(x) = x 13 (l - xt, (a, {3) > -1, and PN(x) = a Jacobi polynomial, in which case the resulting formulas are called Gauss-Jacobi quadratures. In the following, simple expressions for the weights w of Gauss-Jacobi quadratures will be developed. The node polynomials p~,/3) satisfy the orthogonality relationship

1, and (48) to (50) become

J p~a,f3)(x)pj"''13 )(x)(1 1

pji)(x,) = (xi - x,)pD~l)(xi) P?)(x,) P?)(xi)

(51)

- xtx 13 dx = da,f3)8ij

(58)

0

= (xi - x,)p~~~l)(xi) + 2p~}~l)(xJ = (xz - xj)pu~I)(x,) + 3p~~~l)(xJ

(52)

where

(53) 1

with j = 2, 3, ... , i - 1, i + 1, ... , N + 1 and, starting from p~ )(xi) = 1, p~2 )(x,) = p~3 )(X 1 ) = 0. We finally notice that (40) to (43) and (51) to (53) are valid for any choice of distinct interpolation points xi.

We evaluate wi

=

J.

1

li(x)(1 - xtx 13 dx

(59)

0

where

3.3.3 Gauss-Jacobi quadrature

p~,/3)

.

Consider YN- 1 (x), the (N- 1)-degree Lagrangian interpolation polynomial given by theN interpolation abscissas xi, i = 1, 2, ... , N, and the corresponding y values y(xi). N

YN-l(x)

= I

N

li(x)y(xi)

= I

i=l

(54)

li(x)yi

li(x) = ( _ ) (l)(a,{3)( ) X xi PN x,

(60)

~N(x)

The short notation PN will be used for p');·M(x). Certain properties of the expansion functions li are first proved:

i=l

1. Any set of Lagrange polynomials li of degree N - 1 satisfies where N

PN(x)

=

n (x -

Xj)

(55)

j=l

J 0

N

0

li(x) = 1

(61)

i=l

For a given weight function W(x ), we obtain

l W(x)YN-I(x) dx = i~/i f.l l;(x)W(x) dx =wry

N

I

(56)

Proof: The only polynomial of degree < N having the value y (xi) = 1 at N distinct points x 1 is the polynomial y = 1. Inserting y and Yi in (54), equation (61) is immediately obtained.

124

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

2. The expansion functions l;(x) defined by (60) are mutually orthogonal with weight function W(x)

(1 - xt'x 13

=

Sec. 3.3

125

Differentiation and Integration Formulas

The last integral is integrated by parts using d(x - x,)

I q~(x)(1 1

1 1 xt'+ x 13 +

=

dx:

1 dx

•

0

J

1

= (x - Xt)qNx)(1 - xt'+ 1x 13 +1 16

-r r

l;(xg(x)(1 - xt'x 13 dx

0

1 p~x) a f3 (1) (1) (1 - x) x dx o PN (xt)PN (xj) (x - xt)(x - x)

f

=

1

= (1)(

1 )

(1\

PN X; PN Xj

)

f

1

PN(x)[

0

TI. (x -

k=1,1,]

(X

xk)](l - x)axf3 dx

= 0-

=0

since the polynomial

=

n

k=1,i,j

(x - xk)

I

J l;(x)LI

1

+ ({3 + 1)(1-

x)q;(x)] dx

W;

x

[

2(x

x tx

13

dq;(x) - x)~ +(a

2

+ 1)xq;(x) + ({3 + 1)(x -

J

l)q;(x) dx

1

l;(x)(1- xt'x 13 dx

=

0

0

J /~(x)(1

This result is inserted into the expression for s;:

lj(x)](l- xt'x 13 dx

,=1

(62)

1

=

1)xq;(x)

I

With the preliminary results of (1) and (2), we next proceed to find from (59):

=

x)2d~;x)- (a+

LPN(x )(1 -

is of degree N - 2.

W;

x)"+lXP+lj dx

(x - x,)q,(x)(l - xrx•

x [x(1-

N

ON-2(x)

-X,)~ [ql{x)(l -

13

- xt'x dx

0

Let x

Then S;

=

J

I

{ 2(x

2

dq;(X) - x)~

= -cc;,/3) +

2 [q;(x)] x;(l - x;)(1 - xt'x 13 dx

I

+

q;(x)[2x- 1 +(a+ 1)x

+ ({3 + 1)(x- 1)] } dx

1

PN(x)(1 - xt'x 13 GN(x) dx

0

0

GN(x) is a polynomial of degree N with leading coefficient

Substitute x,

=

x - (x - x,)

and

1 -X;

= (1 -

x)

+

(x - x;)

2(N - 1)

Hence

-I p~x)(1 1

S;

=

r f r 0

+

q~,x)(l

= -cc;,/3) +

-

xt'x 13

dx

+

f

~

+ 2 + (a + 1) + f3 + 1 = 2N + a + {3 + 2

We may write I (x

- xi)q;(x)(2x - 1)(1 -

xtx 13

dx

0

- x)"+lXP+l dx

where GN_ 1 is some polynomial of degree N- 1. S;

= -cc;,/3) + (2N +a + f3 + 2)cc;,/3) = (2N +a + {3 +

pN(x)[(2x - 1)q;(x)](1 - xtx 13 dx

or

0

+

1)c~,f3)

1

q~(x)(l

- x)•+lXP+l dx

W;

_ (2N + a + {3 + 1)cc;,m x;(1 - x 1 )[p}.Y(xz)]2

(63 )

126

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

In the development of (63) we have only considered quadrature points in the interior of (0, 1)-i.e., the zeros of a Jacobi polynomial. When YN(x) is obtained as the solution of a differential equation, the side conditions contain information on y at the interval end points (e.g., y = 1 at u = 1 in the example of chapter 2). It is thus of interest to include the interval end points as additional interpolation points and to develop quadrature formulas that include the ordinates at these points. As an example the following Nth-degree polynomial approximation is considered: N+1

I

YN(x) =

(64)

li(x)y(xz)

i=1

with the node polynomial PN+1(x) = p~' 13 \x)(x - 1)

(65)

Sec. 3.3

If both x 0

127

Differentiation and Integration Formulas

= 0 and

xN+1

_ xi(l - xJ ( wi - [ (1) ( )] 2 2N PN+2 xi

= 1 are included, )

(a,f3)

+ a + {3 + 1 c N

,

i

= 0, 1, .. . ,N + 1 (71)

The quadrature weights (63) were developed by integration of an (N- 1)-degree polynomial (54). With N free parameters wi, any function YN- 1 (x) can be integrated correctly by (56) whatever the choice of quadrature points xi. The remarkable property of Gauss-Jacobi quadrature formulas is that they integrate a polynomial of degree 2N - 1

correctly. We might say that N extra parameters-the quadrature abscissas-appear beside the N weights, and this makes these formulas particularly accurate. The disarmingly simple proof follows: Let YN+j be an arbitrary polynomial of degree N + j. This may be written (72)

The quadrature weights are obtained as before: Wi = fo1!Jx )( 1 -

X

)a

X

f3

f1 ( dx =

_PN+1(x) ·) (1) ( ·) (1 o x xi p N+ 1 xi

Take xN+ 1 = 1 and i ¥- N

_f

Wi -

X

X

in which the coefficients of the jth-degree polynomial Gi are chosen such that the identity (72) is satisfied. 1

f 0

W(x)yN+J dx =

f

1

+

W(x)YN- 1 dx

0

f

1

W(x)Gi(x)pN(x) dx

0

p~~ 1 (xz) = (x, - 1)pW(xz) 1

_I

1

0

X

(66)

+ 1:

PN(x )(x - 1) (1)

)

o (x - xz)pN (xz)(xi - 1

-

-

)a {3d

(1 _

X

PN(x)(x - Xt + Xt - 1) ( _ 1 (1) (x - xi)PN (x, ) (xi - 1)

)a

X

f3

)a X

d

13 X

X

d

_ X -

I

1

0

(67) 13

PN(x)(1 - xtx dx ( (1)( ) x - xi ) PN xi

when j
The integrand is identical to that of (59), and the wi are unchanged. Fori = N + 1, we obtain

I

1

WN+1 =

0

/N+1(X)(l -

xtx 13 dx =

f 0

1

p~,{3) (1)

(

PN+1 1

) (1 - xtx 13 dx = 0

(74) (68)

If an expression for the wi containing the derivative of the actual node polynomial PN+ 1(x) is desired, we may substitute p~~1(xi)= (x, - 1)pW(xz) in (63):

_ ,~1 - xz)(2N + a + {3 + 1)c~,f3) Wi - . ( (1) ( )]2 ' · xi PN+1 xi

i = 1, 2, ... , N + 1

(69)

for j = 1, 2, ... , N, RN--a-polynomial of degree N in u and {ud zeros of the Jacobi polynomial with weight function W(u).

= theN

3.3.4 Radau and Lobatto quadrature

In chapter 2 it is shown that the functional 1

Y'f

Similarly, if x 0 = 0 (but not xN+ 1 = 1) is included, the result is xi(2N + a + {3 + 1)c~,f3) wi = (1 - xz)[p~~1 (xJ] 2

(73)

i = 0, 1, 2, ... , N

=

f YN( u) du

(75)

0

(70)

is calculated with an exceptionally high accuracy-2N in a power series expansion of Y'f in the parameter p of the differential equation-when the

128

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

Sec. 3.3

N coefficients ai of (2.34) were determined by a collocation procedure based on the zeros of p~· 0 >(u). Expansion (2.34) is identical to

= L

(76)

li(u)yi

First we note without proof that (62) also holds when 1-(x) are given 1 by (77). si is defined by

i=1

with _ PN+l(u) l .( u ) ) (1) ( ) ' ( u - ui p N+l ui

=

PN+1(u)

si

(77)

(u - 1)p~·o>(u)

1

si =

f. = f.

Multiply by W(x)

f.

0

l Y2N ~~ WdX --

=

'N+1

~k=l

YN

+ GN-1(1

N

rr (x -

where Yb y 2 , ••• , YN are the interior ordinates and YN+l is the ordinate at x=l. Having proved that the best node polynomial for the quadrature (78) is PN+ 1 (x) = (1 - x)p~+t,~>(x), we shall next determine the weights wk of

xiq;(x)(1 -

0

xtx~ dx

(79)

x) 2(1 - xt+ 1 x~ dx N

l

(1 - x)[x - (x - xJ]. fi (x - x) 2 (1 - x)a+Ix~ dx

(80)

J=1,z

The first i~tegral in (80) is completely similar to the third integral

m the expressiOn for si for Gaussian quadrature (a being changed to a + 1), and its value is [2(N- 1) + a + 2 + (3 + 1)c~+1,~) = (2N +a + (3 + 1)c~+ 1 ,m.

The second integral is simply c~+ 1 ,~) since the product is an Nthdegree polynomial with leading coefficient 1 which may be exchanged with p~+ 1 ,~) in the integral. Thus si = (2N~+ {3

+

2)c~+t,~)

- x)p~+ ,~)

0

1

J=1,z

1

+1 f lGN-1 x~(1 - x)a+1p(a+1,~) dx = wTy N

i

(x - x,?(l - xtx~ dx

xi(1 - x)

Also

x~(l - xt and integrate:

wkyk

fi

0

.

=

=

j=1,i

0

m

Y2N

xi[p~~1(xJ] 2 wi

N+1

l

=

The development in (64) to (71) shows that an extra quadrature pomt at x = 0, x = 1, or at both locations does not at all improve the accuracy of the quadrature (59) and (60) since the quadrature weights at these points are zero. In fact (73) implies that N extra interpolation points besides the zeros of p~·~>(x) have no effect on the quadrature. The reason is that theN ordinates y(xk) at the zeros of p~·~>(x) are sufficient to integrate a polynomial of degree 2N- 1 exactly. We shall prove that the correct choice of N interior quadrature points

are the zeros of p~+ 1 ,~)(x). When this has been proved, it immediately follows that the collocation ordinates y(uk) at the zeros of p~,Q)(u) are indeed the best we could wish for when (75) is evaluated by (78). Let YN be represented by (76) and (77) (with x in place of u). Any polynomial of degree 2N can be written in a form similar to (72):

1

xt

0

.

(78)

=

First, consider i ::P N + 1:

The high accuracy of 'YJ indicates that a quadrature formula based on (76) with one quadrature point at u = 1 has the highest possible accuracy 2N, when the remaining N quadrature points are chosen as zeros of P~,o>(u).

129

the formula. The result is given by (83). Formulas with x = 0 or with both x = 0 and x = 1 as extra quadrature points are (84) and (85) which can be derived similarly to (83). '

N+1

YN

Differentiation and Integration Formulas

i [p~·~>]2 (1 1

sN+1

=

0

-

x)ax~ dx

i = 1, 2, ... , N

(81)

Properties of Orthogonal Polynomials-Collocation Procedure

130

Sec. 3.4

Chap.3

used as a node polynomial. As shown in (69) to (71), the interior weights and wN+l are the same as in (83), while w0 , the weight of the ordinate at x = 0, is zero. A Radau formula based on N interior points and one end point will integrate any polynomial YM exactly, provided M ~ 2N Similarly, a Lobatto formula (N interior points and two end points) will give exact results for polynomials of degree ~2N + 1. In general the quadratures of this section may be used for exact evaluation of integrals of the type

. 1 . (a+L/3) The second integral is integrated by parts: The first mtegra 1s c N ·

r

(1 -

x)"x~+lpf.,dx = --=i-(1 - xr+lxf3+lp~~ a+1

e

_1_ (1 +a + 1 Jo

= _1_

xr+l_E_(xf3+1p~) dx dx

1

1 = --(2N a+1 -

SN+l-

I

r (1- x)"+lpNxP[ 2X ~; + {jJ + 1)pN] dx

a+ 1 Jo

) (a+l,/3) 1 CN

2N + a + {3 + 2 (a+l,f3) CN a +1

Hence the following weights are obtained: (2N + a + (3 +

wi =

2)c~+l,f3) .

(1) ( )]2 xi [ PN+l Xt

A similar formula using the va1ue

Yo

a

t

(3

t~

ii:N+1

(83)

i = N+ 1

a +-1

- 0 (but not YN+l at XN+l = 1)

Xo -

(3

2

(1 -

Wi

YM(x) W(x)f(x) dx =

(2N +a

Xo

1

1

i=O i ,c 0

(84)

= 0 and Xl = 1,

(a+1,{3+1)

+ {3 + 3 )CN

wJ(xJyi

(86)

where M = (N- 1) (Gaussian), N (Radau), or (N + 1) (Lobatto) for an arbitrary polynomial f(x) of degree less than or equal to N

In Section 3.4, computer programs for obtaining zeros of Jacobi polynomials, derivatives of node polynomials, interpolation weights, differentiation weights, and quadrature weights are described. The programs are all in double precision and in FORTRAN IV. They are listed in the Appendix. 3.4.1 Zeros and derivatives of Jacobi polynomials

This subroutine calculates the zeros of p~'13 >(x) and also the three first derivatives of the node polynomial PNT(x)

Finally, using both end points

I

3.4 Program Description

may constru::: +a + + l)c~·P+ll . j ~ = Xt)[p~~l(xz)] 1 be

1

0

(82)

+ {3 +

131

Program Description

1 (3+1 1

(87)

at the interpolation points. Each of the parameters NO and N1 may be given the value 0 and 1. N~ +NO+ Nl.

i = 0 i ,c 0, N + 1

= (x)N°pN(x)(x - l)N 1

(85)

The subroutine call is CALL JCOBI (ND, N, NO, N1, ALFA, BETA, DIF1, DIF2, DIF3, ROOT)

1 a+1

i = N+ 1

I

. ~ f (83) d (84) are called Radau quadrature The weight factors wi o an . h weights and the wi of (85) ~~e Lo~a~to qu:~~~u~ew:;yts~ish to include In the solution of ~ dl e~entm equ = 1 because boundary 1 b d - 0 as an interpolatiOn pomt as well as XN+l Xo . E f n (83) may a so e use - x(x - 1)pN(x) is conditions are given at both these pomts. qua 10 to calculate an integral derived from YN+l when PN+2 -

Input parameters: INTEGER ND:

The DIMENSION of vectors DIFl, DIF2, DIF3, ROOT. In all subroutines described in this text an automatic dimensioning of vectors and matrices is made via one or two input parameters: ND for vectors and ND, NCOL for matrices. The values of 'these parameters are given in the DIMENSION statement of the main program. In the description of the following subroutines, ND (and NCOL) will not be commented on further.

132

Properties of Orthogonal Polynomials-Collocation Procedure

Chap.3

INTEGER N:

The degree of the Jacobi polynomial, i.e., the number of interior interpolation points.

INTEGER NO:

Decides whether x = 0 is included as an interpolation point. NO must be set equal to 1 (including x = 0) or 0 ·(excluding this point).

INTEGER N1:

As for NO, but for the point x = 1.

REAL ALFA, BETA:

The values of a and {3.

Sec. 3.4

Program Description

133

Hence, if the interpolation points x1 (ROOT) and the first derivative of the no~e polynomial p)J«xi) (DIFl) are known, the interpolation weights are easily calculated. The subroutine call is CALL INTRP (NO, NT, X, ROOT, DIF1, XINTP)

With input data: The output is

NT:

REAL ARRAY ROOT:

One-dimensional vector containing on exit the N + NO + Nl zeros of the node polynomial used in the interpolation routine.

REAL ARRAY DIF1, DIF2, DIF3:

Three one-dimensional vectors containing on exit the first, second, and third derivatives of the node polynomial at the zeros.

+ NO + Nl) for which the

REAL X: The abscissa x where y(x) is desired. REAL ARRAY ROOT, DIFI:

The vectors ROOT, DIFl, DIF2, and DIF3 may be declared in the main program with any dimension greater than or equal to the actual value of N +NO+ Nl. The program consists essentially of three parts. First the values of the coefficients gi and hi, i = 1, 2, ... , N of the recurrence formula (13) are computed according to formulas (14) and (15). The 2N coefficients are temporarily stored in DIFl and DIF2. Next the zeros of PN are determined as described in section 3.2 by the Newton method with root suppression. The smallest zero x 1 is stored in ROOT(l), x 2 in ROOT(2), and xN in ROOT(N). If x 0 = 0 is included as an interpolation point (NO = 1), the N zeros are shifted one position so that ROOT(l) = 0, ROOT(2) = X1 and ROOT(N + 1) = xN. Finally an extra element ROOT(NT) = 1 is added to ROOT if Nl = 1. ROOT now contains the interpolation points in ascending order. Finally the derivatives of the node polynomial are evaluated at the interpolation points using equations (51) to (53).

The total number of interpolation points ( = N value of the dependent variable y is known.

Interpolation points and derivatives of node polynomial, derived in JCOBI.

The output data is REAL ARRAY XINTP:

The vector of interpolation weights /i(x)

y(x) is then found from NT

y(x) =

L

XINTP(I) · Y(I)

1=1

(90)

INTRP can, of cour~e, be used for any choice of NT interpolation points, but DIFl and the mterpolation points ROOT have to be specified in other applications. In the main program the dimension of XINTP must be equal to or larger than NT. 3.4.3 Differentiation weights and Gaussian quadrature weights

3.4.2 Lagrange interpolation (91)

The value of y at any desired point x = xA may be found from NT

y(xA) =

L

i=l

where

li(xA)Yi

(88)

(92)

Integrals of the type 1

I y(x)(l -

x)""x 13 dx

0

·are determined by Gaussian quadrature.

(93)

134

Properties of Orthogonal Polynomials-Collocation Procedure

The subroutine call is

Discretization of Differential Equations Using Ordinates

135

Input data:

CALL DFOPR (ND, N, NO, N1, I, ID, DIF1, DIF2, DIF3, ROOT, VEC)

INTEGER N, NO, N1:

As in JCOBI.

As in JCOBI. ID is an indicator. ID = 1 gives Radau quadrature weights including x = 1. ID = 2 gives Radau quadrature weights including x = 0. ID = 3 gives Lobatto weights including both.

INTEGER 1:

The index of the node for which the weights in (91) or (92) are to be calculated.

The exponents a and f3 of the weight function that appears in the integral (95).

INTEGER ID:

Indicator. ID = 1 gives the weights for dy / dx, ID = 2 for d 2 yj dx 2 , and ID = 3 gives the Gaussian weights. The value of I is irrelevant in this last case.

The node polynomial is given by

Input data:

REAL ARRAY ROOT, DIF1, DIF2, DIF3:

PNT(x) = xN°pX;',f3')(x)(x - 1)N1

The arguments a' and /3' to be used in JCOBI for calculation of ROOT and DIF1 depend on whether x o;= 1, x = 0, or both are used as extra quadrature points. Thus:

The computed vector of weights

ID = 1:

a' = a

ID = 2:

a'= a,

ID = 3:

a' = a

ID = 1:

N1 = 1 and NO = 0 or 1

ID = 2:

NO = 1 and N1

Output data: REAL ARRAY VEC:

[l~l)(xJ, /~2 )(xj),

or

wb

k = 1, 2, ... , NT]

This dual-purpose algorithm applies the expressions developed in section 3.3. The Gaussian weights are normalized, however, such that their sum equals 1. If the true Gaussian weights of (51) are desired, the calculated weights should be multiplied by

f.

(a,{3) _

I

(96)

The one-dimensional vectors computed in JCOBI.

-

0

1 _

(1

a

{3

x) x dx

ID

NO

=

+ 1,

W= W=

1 and N1

(N1 = 1 and NO = 0 or 1)

~

+1

(NO= 1 and N1 = 0 or 1)

~

+1

(NO = N1 = 1)

=

0 or 1

=

1

ROOT is the vector of zeros of PNT(x) in (96) and DIF1 is p}J~ (ROOT). Both are determined in JCOBI.

= f(a + l)f(/3 + 1) f(a + f3 + 2 )

The computation of 1~1) and 1~2 ) by (91) and (92) is valid for any choice of interpolation points and may be used whenever the appropriate values of ROOT, DIFl, DIF2, and DIF3 are available. The computation of quadrature weights is restricted to node polynomials of the form (87) with PN(x) a Jacobi polynomial.

= 3:

+ 1, /3' = f3

REAL ARRAY V:

Th:! NT computed quadrature weights. These are also normalized such that NT

I

wk = 1

1

To obtain the true weights of (95), each wk must be multiplied by I(a,f3) in (94).

3.4.4 Radau or Lobatto quadrature

The weights of a quadrature

J.

1

y(x )(1 - x tx 13 dx

0

are determined. The interior interpolation points are zeros of where a and f3 are defined below. 1

3.5 Discretization of Differential Equations in Terms of Ordinates

1

3.5.1 The dependent variable known at the boundary

An approximate solution to the problem of chapter 2 The subroutine call is CALL RADAU (ND, N, NO, N1, ID, ALFA, BETA, ROOT, DIF1, V)

d2 y u du2

dy

+ du = py,

Yu=1

=1

(97)

136

Properties of Orthogonal Polynomials-Collocation Procedure

Discretization of Differential Equations Using Ordinates

137

was obtained in the following form: YN = 1

N

+ (1

- u)

(B*- pl)y = c

L a;ut-l t=l

Bfi

YN was inserted into the differential eq~atio? and ~n a collocation m~thod the residual was equated to zero at N mtenor pomts U;. The resultmg N

equations were solved for a. . . YN of (98) is a polynomial of degree N m ~ and It rna~ be reformulated into an Nth-degree Lagrangian interpolatiOn polynomial: N+l

YN

= L

l;(u)y(u;)

i=l

where YN+l

=

Yu=l

c1

(!)PI = 2(:~) u~l ••• ,

uN are selected,

2

N

d yN du

dyN du

= u-2 + -

-

_L 1=l

AN+l,i . Yt = -2 (1 -

(102)

YN+l)

= K(1 -

YN+l)

(103)

N

YN+l(K

j

+ An+l,N+l) = - L AN+l,iYi + K i=l

or

= 1, 2, ... ,N

or

=

f.

-AN+l,i

L..

i=lK

+ AN+l,N+l

•

K

Y· + - - - - - I

K

+ AN+l,N+l

(104)

Equation (104) is inserted into each of theN collocation equations (100) and a system of N equations for theN interior ordinates is obtained:

(B*- pl)y = c

(105)

The elements of B* and c are slightly different from the corresponding elements in (101):

N

L BitYi + i=l L Ai;Y;

BiM

N+l

PYN

YN+l

ui

1)]

Equation (103) and the N collocation equations (100) supply N + 1 equations for theN+ 1 unknown y-values. For a linear boundary condition it is usually simpler to eliminate the unknown boundary ordinate from the system of equations:

is evaluated and equated to zero at the N interior ui.

N

=

An expression for (dyN/du)u=l may be obtained from (91) applied for = 1:

(u - ui)

in which uN+l = 1. The residual R

BiM [1 - y(u

=

UNT

dyN) ( -d - = U u-1

j=l

(101)

Collocation at N interior points supplied the exact number of equations that were required to find theN unknown ordinates of (99). In a similar problem with mass transfer resistance at the pellet surface, YN+l = y(u = 1) is given in terms of mass transfer coefficient hM, pellet radius R, and pellet diffusivity D by the following relation:

N+l

L

+ Ait = -(uiBiN+l + AiN+l) uiBJt

3.5.2 Unknown boundary ordinates

= 1

Equation (99) contains N unknown ordinates instead of the N unknown coefficients a. Hence, N interior collocation points u 11 u 2 , u 3 , yielding the node polynomial PN+ 1(u) =

=

- PYi = -uiBiN+l - AiN+l

j

= 1,2, ... ,N

i=l

B~ ]I

The elements of B and A are evaluated by the algorithms of section 3.4 and N linear equations (101) in the unknown Y; are immediately

= u-(B .. _ 1

]I

BJ,N+tAN+l,t) +(A·· _

K

+ AN+l,N+l

]I

Ai,N+lAN+I,i)

K

+ AN+l,N+l

(106)

139

Properties of Orthogonal Polynomials-Collocation Procedure

138

2. Show by comparison of the power series for the two orthogonal polynomials that

The most general type of linear boundary conditions are

(dxdy) (dxdy)

+ a 1 y(x = O) =

az

dP~·13 l - = N(N + a + {3 + 1) p
x=O

+ b 1y(x =

x=l

1)

= bz 3. Let y 0 ) = f(x) and approximate f(x) by

We choose an interpolation polynomial PNr(x) with NT= N + 2, - 0 x x x = zeros of PN(x) and xN+l = 1. In the computer Xo - , b z, · · · ' N ' .. programs, the index of xi is shifted one posttlon, but t~e nomenclat~re (x , y ) for [x = 0, y(x = 0)] is easier to use here and m the followmg 0

0

chapter. The boundary conditions are N+l

L

Ao,iYi

+ a1Yo

L

AN+l,iYi

= az

+ b1YN+l =

bz

i=O N

M( YN+l Yo)

(A 00

L

b;P~o,o)(x)

•=0

How would you find b; from N ordinates f(x;) at the zeros of P~· 0 l(x)? Write an explicit expression for YN(x) in terms of a given y(x = 0) = Yo and the coefficients b;. Show that the accuracy of /(1) is determined solely by b0 • 4. In the Graetz problem a Newtonian fluid in laminar flow is being cooled from the tube wall, which is at temperature T = 0, in appropriate units. The fluid enters the tube at z = 0 with uniform temperature 1 and, during the flow through the tube T(x ), the radial temperature profile gradually falls to 0 for all x. The following integral is an expression for temperature T(x) · v(x) averaged over the tube cross section.

i=O

N+l

N-1

f(x) =

+ al)Yo + Ao,N+IYN+l = - i=l L AoiYi + az N

AN+l oYo ,

+ (AN+l,N+l + bl)YN+l = - t=l L AN+l,iYi +

bz

J

1

x(1 - x 2 )T(x) dx

=I

0

- M -l[(az) - ( A01 ) Y1- ( Ao2 ) Y2 _ b2 AN+l,l AN+1,2

)yN]

Ao,N - ( AN+l,N

In each of theN collocation equations, y 0 and YN+l are inserted ftom the solution (108) of the boundary equations and again N equations of the form (101) or (105) appear for the interior ordinates Yb Yz, · · ·: YN· In subsection 7 .2.1 the problem of eliminating a number of vanabl~s that are given through linear equations is described in a general matnx notation. In the present context the explicit elimination of (yo, YN+l) given by (108) is, however, sufficient. EXERCISES

1. Show that f(N + a + 1)f(f3 + 1)

p~,f3l( 1 ) = f(N + {3 + 1)f(a + 1) for the polynomials defined by equations (7) or (8).

We wish to approximate I by I 1 = w1 T 1 = w1 T(x 1) with the purpose of finding the best measuring point x 1 for this average temperature, and we wish to compute the corresponding "best" weight, such that a temperature profile of the highest possible degree in x is integrated correctly by I 1 • a. The following proposals are first studied: i. Take F(x) = x(1 - x 2 )T(x) and find the optimal quadrature point x 1 and the corresponding weight. ii. Take F(x) = (1 - x 2 )T(x), W(x) = x and repeat. iii. Take F(x) = (1 + x)T(x), W(x) = x(1 - x) and repeat. iv. Use your knowledge that F(x) = (1 - x 2 )T(x) = 0 at x = 1. Thus take F(x) = (1 - x 2 )T(x ), W(x) = x and two quadrature points with x 2 = 1. [You will still obtain a quadrature of the form I 1 = w 1F(x 1 ) since F(x 2 ) is zero.] Apply (i) to (iv) to T = constant (say 1) and to T = x. Explain your results by comparison with the exact value of I and your knowledge of the accuracy of the formulas for I 1 • b. We know that T(x) is symmetric in x. Thus introduce u = x 2 and convert the integral to I = ! J~ (1 - u)T(u) du. i. What is now the optimal quadrature formula? Check the accuracy by integrating T = 1, u, and u 2 •

140

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

c. We also know that T(u = 1) = 0. This can be applied to find an even. better average temperature based on measurement of only one value of T inside the tube. i. Find the optimal one-interior-point formula. Check the accuracy by integrating T = (1 - u) and T = (u - u 2 ). (Now T has to be zero at u = 1, therefore the slight alteration of "trial" functions.) d. The previous formulas can be used to find the smallest eigenvalue of

1 d (xd~ - -A(l-x 2) Y=O x dx

dx

(1)

which is the eigenproblem (1.140) that was discussed in section 1.7. Write Y 1 (x), the first eigenfunction of equation (1), as a first-degree interpolation polynomial in u = x 2 with u 1 and 1 as interpolation points. Discretize the equation to obtain A 1 as a function of u 1 • When u 1 is the optimal quadrature point of question c, one obtains A1 = -7.11. This is close to the true value of At. which is -7.31. When u 1 is the optimal point in question b, one obtains A1 = -9. For larger N the best results are, however, found by the latter choice of u 1 .) 5. Consider the polynomials P)::; 1 / 2 ,- 1 / 2 >(x ). a. Construct P 1 (x), Pz(x), and Pix) using formula (3.9). b. Show that PN(x) = cos NO where x = (1 + cos 0)/2. Determine (exact) values for the zeros of P 3 (x). c. Find the recurrence formula for PN(x) [or PN(x)]. d. Determine J~ W(x)[PN(x)] 2 dx. e. Determine weights in J~ W(x)F(x) dx = L~ wkF(xk). f. Find an interpolation polynomial of degree 2 for ex based on the zeros of P 3 (x). g. Find the expression for ai in YN-l = L~ ap,_1(x). h. Find ex ~ eapp(x) = L~ aiP,-1 (x ). What is the general algorithm for finding ai in an expansion based on P~=Y 2 ,- 1 / 2 )? i. Make a sketch of ex - eapp(x) in [0, 1] and note that the maximum error is approximately evenly distributed. 6. Solve the equation

for the quantity Yo· The integrand is singular at g = 1 (which does present a problem): but since Yo ~ lo-s, there will be a large part of the interval where a simple Gauss formula Will work well. Thus it is suggested that the interval [1, l/y 0 ] is split up in sevetih parts. In the first subinterval [1, e] a suitable weight function is separated out, and for g » 1 the integrand may be simplified by neglecting 1 in comparison with f Make up your own computer program following these guidelines and show that

Yo

= 0.8804 ·

This result is applied in subsection 7.1.3.

lo-s

References

141

7. Go through the statements of JCOBI and make a careful analysis of how the algorithm in equations (19) to (23) is applied.

8. Use JCOBI and RADAU to make a list of Radau formulas for integration of I

1

= -

2

I

1

u- 112/(u) du

0

with N = 1(1)10 quadrature points. The list should contain weights and quadrature abscissas. 9. a. Compute exp (-x) at the zeros of P~· 0)(x) (N = 1, 2, ... , 8) to machine accuracy, and use these ordinates and INTRP to compute a table of exp (-x) at x = 0(0.05)1. Compare with results obtained directly at these abscissas. b. Repeat for interpolation points chosen as zeros of P)::; 112 ,- 1f 2 >(x) and note whether the maximum error for a given N is smaller here than in a. c. Finally use collocation at the zeros of P~ 12 • 112 )(x) to solve the equation

d2y -y =0 2 dx

y(O)

=

1,

y(l)

= l/e

Note that the maximum error is comparable to that obtained in a or b.

REFERENCES Man~ results in. section 3.1 are derived in Villadsen (ref. 42 of chapter 1). Szego (1967) ts the standard reference to properties of orthogonal polynomials. Zeros of polynomials are treated in all textbooks on numerical a~alysis, e.g:, He~rici (1964). Modern Computing Methods (1961) contams some dtsturbmg examples of how difficult it may be to determine the zeros. The implicit deflation process equations (20) to (23) is also described in Wilkinson (ref. 48 of chapter 1). The method of section 3.3 to differentiate Lagrange polynomials was first proposed b~ Michelse~ and Villadsen (1972). It is substantially better than previOusly published methods, e.g., Villadsen and Stewart (1967). Quadrature formulas are treated in Kopal (1961) and in Davis and Rabinowitz (1967). ~illadsen (ref. 42 of chapter 1) makes a summary of convergence properties of quadratures based on orthogonal polynomials, but far more complete presentations of this subject are given in Szego (1967) and in Natanson (ref. 4 of chapter 2). 1. SZEG6, G. "Orthogonal Polynomials." American Math. Soc. Colloquium Publications 23 (1959).

142

Properties of Orthogonal Polynomials-Collocation Procedure

Chap. 3

2. HENRICI, P. Elements of Numerical Analysis. Wiley (1964). • "A-.f h d Published as No. 16 in National Physical 3 Modern Computing met 0 s. · Office . Laboratory's Series "Notes on Applied Science," H. M. Stationery

Solution of Linear

(London) 2nd ed. (1961). and VILLADSEN J. Chern. Eng. Journal 4 (1972): 64. 4. MICHELSEN, M · L ., ' ART W E Chern. Eng. Sci. 22 (1967): 1483. STEW ' · . 5. VILLADSEN, J ., an d . l A alysz·s 2nd ed London: Chapman and Hall (1961). , · 6. KoPAL, z. N umerzca n · l lntegra tz" on. New York·· BlaisNumenca WITZ p 7. DAVIS, p .J. , and RABINO ' . dell (1967).

Differential Equations

4

by Collocation

Introduction The examples and programs that are contained in the present chapter are intended to form the main body of our text and a large number of linear models can be solved approximately by the methods described here. 1. Solution of two-point boundary value problems. 2. Solution of one or several coupled ordinary differential equations. 3. Solution of parabolic partial differential equations. Two-point boundary value problems are discussed in chapters 2 and 3. In section 4.1 the collocation solution for the example of chapter 2 is briefly recapitulated on the basis of a computer program. It is shown that the problem may also be solved without using the known symmetry of the solution and different choices of collocation points are discussed. Initial value problems are integrated using a marching technique with collocation within each step. Three efficient collocation algorithms are developed in section 4.2 for a single first-order equation, and it is shown that higher-order initial value problems can be solved by the same techniques without increasing the dimensionality of the problem. Parabolic partial differential equations (PPDE) are solved in section 4.3 by a collocation treatment of the underlying eigenvalue problem combined with a standard method for solving linear differential equations with constant coefficients. A general program EISYS for diagonalization of a matrix is described. 143

144

Solution of Linear Differential Equations by Collocation

In section 4.4 the collocation solution is compared with the infinite Fourier series solutions. The two solutions are very similar and both break down in the entry region. A penetration solution is developed that gives a very accurate solution for this region. The eigenfunctions of the PDE can be found with an arbitrary accuracy by an application of the methods of section 4.3. A general algorithm for solution of boundary value problems by an initial value technique is discussed in section 4.5. Finally a PDE with an ordinary differential equation as one of the boundary conditions is treated in section 4.6. Models of this type are shown to be of considerable importance for chemical engineering systems.

4.1 Solution of a Linear Boundary Value Problem

Our desire to compute 17 as accurately as possible using N interior quadrature points and the boundary point u = 1 thus leads us to a collocation process in which the collocation points are chosen as the zeros of P~,(s-1)/2l(u ). The interpolation polynomial PN+ 1 is (u - 1)p~,(s- 1 ) 121 and the collocation equations (3.100) take the following form for equation (2): N+1

Uj

.L z=l

BjiYi

2

d y 2 dx

= y = R (y)

y(1)

=

2

S

2

y(1)

=

<1>

w· = w' '

4Y = py

= =

V;(1)

1 1

f

v R(y) dV

+1 y(x)dxs+ 1 = _s__

11

(3) y(u)u(s- 1 )12 du

2 0 In the prese1, section the details of a collocation solution by the procedure of section 3.5 will be described for this problem. It is first noted that the weight function W(u) in (3) is of the form 13 u (1 - u t with a = 0 and {3 = (s - 1)/2. Furthermore, y(u = 1) is known, and if we wish to include this ordinate in a Radau quadrature representation of (3), the interior quadrature points should be chosen as the zeros of p~+ 1 , 13 )(u )-i.e., as the zeros of p~,(s- 1 )/ 2 l(u ). 0

1

2

1

'o

W(u)du = w~

Consequently when the weights

y(O) finite

is treated in detail in chapter 2 for s = 1. Analogous equations for first-order, isothermal reaction in slab geometry and in spherical geometry appear when s = 0 and s = 2. A major object of the calculation has been to find the catalyst effectiveness 17 as a function of the Thiele modulus . 11

(4)

(5)

WiYi

i=l

w:

2

1

=0

In section 3.4 it is noted that program RADAU determines quadrature weights with a fixed sum of 1 and that the proper weights of (5) are obtained from w ~ by

1

s + 1 dy + -2- du =

pyj

+ 1 N+l

= -- L

or d y u du 2

-

with YN+I = y(u = 1) = 1 and j = 1, 2, ... , N. The N linear equations for the interior ordinates y~, Y2, ... , YN are solved by Gauss elimination and the concentration profile is next found by Lagrangian interpolation using the program INTRP. Finally the effectiveness factor is evaluated by Radau quadrature: 'Y/ ~ 'Y/N

s dy x dx

+- -

S + 1 N+1 + -2- L AjiYi z=1

The linear boundary value problem

-

145

Solution of a Linear Boundary Value Problem

11

'o

2 u(s- 1 )12 du = - - w '

s+1'

w: are used, (5) should read (6)

The complete computer formulation of the problem applies five subprograms: JCOBI, DFOPR, GAUSL, INTRP, and RADAU. With the exception of GAUSL these are all described in section 3.4. Even though a program for solution of linear equations is probably available anywhere, for the sake of completeness we shall also describe our Gauss elimination routine, GAUSL: Let A be an [N x (N + NS)] matrix that contains the square matrix B in its first N columns and an (N x NS) matrix C in columns N + 1 to N + NS. The call of GAUSL will replace C by B- 1 C. The B part of N

matrix A is also destroyed, but the determinant of B is obtained as

TI

Aii.

i=1

The program is thus seen to solve N linear algebraic equations with a fixed coefficient matrix B and NS different right-hand sides C. Specifically, GAUSL can be used to invert B, in which case C is an (N x N) unity matrix.

146

Solution of Linear Differential Equations by Collocation

The subroutine call is

Finally a profile is obtained at a fixed grid of interpolation points

= 0, 0.1, ... , 0.9, 1 by means of the interpolation program INTRP.

CALL GAUSL (ND, NCOL, N, NS, A)

that the Lagrange polynomials are

The input parameters are

(u -

INTEGER ND, NCOL: Explained in input parameter list of JCOBI.

li (u) =

INTEGER N:

Current size of the system of equations to be solved N

INTEGER NS:

Number of right-hand sides for which the system of equations is to solved.

ARRAY A:

The coefficient matrix augmented by the NS right-hand sides in columns N + 1 toN+ NS.

:5

(u

1)p~,(s-1)/2J(u)

)

(1)

(

- ui PN+l ui

)

i

'

= 1, 2, ... , N + 1

(8)

ND.

The output is ARRAY A:

147

Solut1on of a Linear Boundary Value Problem

The last NS columns of A contain the solution of the system of equations for the NS right-hand sides. The determinant of the coefficient matrix 1 Au.

I1:

The main program for solution of (2) and (3) using the five subroutines mentioned above is shown in the Appendix, p. A12. The input data are N, s, and (N = approximation order, s = geometry factor, = Thiele modulus). The zeros of p~,(s-l)/ 21 (u) are found by JCOBI using NO = 0, Nl = 1. The derivatives DIF1, DIF2, and DIF3 of PN+l(u) = (u - 1)p~,(s-l)/ 21 (u) at the interpolation points ROOT (u = Ui =the collocation points and uN+l = 1) are used in DFOPR. This routine is called N times and with ID = 1 and 2. In this manner the discretization matrices for the first and the second derivative of y with respect to u are found row by row starting with the collocation point nearest to u = 0. The rows of A and of B are next combined to give the matrix left-hand sides of equation (2). Finally the right-hand side of (2) is subtracted in the diagonal. The last column of the [N x (N + 1)] matrix BMAT contains the known boundary terms

that u = x 2 should be used in the argument of INTRP. The substitution u = x 2 was introduced because it was known that the series contains only even powers of x and it was assumed that for same N a more accurate approximation could be obtained when the functions (here orthogonal polynomials) contain only "correct" of x. In other situations it might not be immediately obvious that the n has certain symmetry properties and the collocation procedure of course, also work when both even and odd powers of x are din YN(x). Consider (1) for s = 1 and use collocation at the zeros of P~,f3)(x ). There are several attractive choices of a and {3. The weight function W(x) of (3) is x\1 - x) 0 and this suggests a collocation procedure at the of P~'l)(x) when a Radau quadrature with y(x = 1) as the extra ture point is used to evaluate 11· One might also collocate at the of P~' 1 )(x) and subsequently evaluate 17 by a Gauss quadrature. the factor x of (3) might be included in the integrand z = · y(x) and (3) could be evaluated either by Gauss quadrature [collocaat the zeros of P}3' 0 )(x)] or by a Radau quadrature [collocation at the of P~' 0 \x)]. In all these examples, x = 0 is included as an interpolation pointBI is called with NO = N1 = 1. The boundary condition y< 1 ) = 0 at = 0 is used to eliminate the ordinate y 0 as described in section 3.5, the known ordinate y(x = 1) = 1 appears as shown in (7). The error of 1JN is given in table 4.1 for = 2 and for each of the of (a, {3) mentioned above. TABLE

s+1 ) s+1 ( UjBj,N+l + -2-A],N+l YN+l = UjBj,N+l + -2-A],N+l in the jth row~ef BMAT. Thus wheri BMAT is used in the argument of GAUSL (NS = 1), the vector of N collocation ordinates is obtained as the last column of the output matrix from GAUSL after a sign change. RADAU is called with values a = 0, {3 = (s - 1)/2 of the weight function in (3). The NT= N + 1 quadrature weights are used to compute 17 = RADWT · Y.

I7J - 7JNI FOR

=

2,

4.1

AND CYLINDER SYMMETRY

a,{3

Optimal polynomials N

(1, 1)

2 3 4

4.0 · w- 4 6.1 · w- 6 5.3 . 10- 8

(0, 1) 2.4 · 4.4 · 5.7 ·

w- 3 w- 5 w- 7

(1, 0) 2.0. 4.6 · 6.0 ·

w- 3 w- 5 w- 7

(0, 0) 9.0. 2.5 · 4.2 ·

w- 3 w- 4 w- 6

pCl,Ol(u)

u = x2

3.0. w- 6 1.2 . w- 9 <10- 12

148

Integration of Initial Value Problems

Solution of Linear Differential Equat1ons by Collocation

The choice (a, {3) = (1, 1), which applies the known structure of weight function and the known boundary value in the optimal way, give the best value of TJN, as expected. At the same time it appears the difference in quality between the methods is less than between successive N values. In practice all four methods are applicable and increase of N by one corresponds to a decrease in the error of TJ by factor of 50-100. The polynomials in u are obviously better than the polynomials x-we have seen in chapter 2 that a polynomial approximation based these polynomials yields the best possible value of TJ for a given N they are not necessarily the best when other measures of quality are Table 4.2 shows that approximation by means of P~· 1 )(x ), which was selected as probably the best in table 4.1, does not fall far behind the maximum error of YN(x) in [0, 1] is compared. TABLE MAXIMUM VALUE OF

N

4.2

IY - YNI

FOR X E

[0, 1] ( = 2)

Collocation at zeros of P~· 0 )(u)

Collocation at zeros of p~·l)(x)

3.3 . 10-3 1.3 . 10-4 1.5 . 10- 5

2

1.3 . 10- 3

3

2.4 . 10- 5

4

2.6. 10-7

4.2 Integration of Initial Value Problems In all examples of the previous sections MWR have been used to integrate boundary value problems. They are, however, just as applicable for integration of initial value problems. In the present section several very attractive collocation procedures are derived following the arguments of section 2.4 and especially of 2.5 where the optimality of a Galerkin procedure and a corresponding orthogonal collocation MWR was proved. The general initial value problem for one first-order equation is dy

Integrate dx

=

f(y, x)

from x

=

0 to X

with initial value element (0, y0 )

(9)

4.2.1 Truncation error and stability in forward integration

Usually the interval [0, X] is divided into M subintervals of length h and (9) is integrated through each subinterval, using the right-hand end

149

ordinate of one interval as initial value in the next interval. It is to tabulate y at x = n · h, n · 2h, ... , X; as far as y is accurately .t............ at these x-values, it does not matter if intermediate y-values inaccurate. The repetitive application of the right-hand interval end point ordias initial value for the next interval will lead to an accumulation of . This may not only give inaccurate y-values at the entries of the x = n · h, . . . but the whole integration process may be ruined instability of the method. The most common instability phenomenon appears when h is larger a certain critical value. Any explicit method, of which the Rungemethods are the best known, exhibits this steplength-induced }instab:thty. It may be remedied by taking small integration steps h, but at the cost of a large and unproductive computational work between the tabular points. Stability properties of various integration methods are discussed in 8. In this chapter we only study the error within each integration step-the truncation error. The truncation error at the end of each step x = h, 2h, ... is usually reD,res,en1tea as some functional of y multiplied by h k. If k is large (and h -su.mcienltlV small), this seems to indicate that the integration method is well constructed. One way of obtaining a high value of k is to use a high-order explicit method: Euler's method applies y and / 0 at x 0 to integrate y from x 0 to x 0 + h, while Runge-Kutta's fourth-order method applies the first four derivatives of y in an indirect way. The less complicated Euler method has a truncation error proportional to h 2 , while Runge-Kutta's fourth-order method, which demands four evaluations of f(y, x ), has a truncation error proportional to h 5 • In these explicit "look ahead" methods a high-order truncation term is indicative of a small error per integration step, but the critical h value, above which instability occurs, does not increase significantly with increasing order of the integration method. Another way of obtaining a high-order truncation term is to use previously determined ordinates y(x 0 - h), y(x 0 - 2h) ... and f[(x 0 - h), y(x 0 - h)] ... besides y (x 0 ) to determine y (x 0 + h). Both explicit and implicit variants of these "backward interpolative" methods exist and they are widely used in practice often in a predictor-corrector scheme where the predictor formula is explicit and the corrector formula is implicit. For a purely explicit method the critical value of h decreases when several previously determined y-values are included (i.e., when the truncation error is diminished) and these formulas may be quite dangerous when used improperly. It is characteristic for the interpolative methods mentioned above that at most one y-value is implicitly given [y(x 0 + h) is found from an '\Prt

150

Solution of Linear Differential Equations by Collocation

algebraic equation in which f(x 0 + h) also appears]. The high order of the method is obtained by filling in with previously determined y- or /-values. A truly "forward interpolative" method determines y(x 0 +h) and y at certain intermediate abscissas between x 0 and x 0 + h by simultaneous solution of a corresponding number of algebraic equations. Only y (x 0 ) is assumed to be known during this computation. An interpolation polynomial YN(x) of degree N can be constructed on the basis of y(x 0) and N ordinates taken at x-values between x 0 and x 0 + h and this polynomial is used to represent y (x) in the interval. In subsection 4.2.2, three interpolative methods are developed, and it is shown in general that optimal interpolative methods may be constructed for any number N of interpolation points. The truncation error for these methods is much smaller than for an explicit method of the same order N and this may compensate for the heavier computational work per step. The residual R (yN) = y W- f(yN, x) can be interpolated to zero at any set of N x-values in (x 0 , x 0 + h) and the N + 1 constants of YN can be determined from the N collocation equations and the known ordinate at x 0 • Our main objective will be to choose the N points such that y (x 0 + h) is especially accurate. The preferred interior interpolation-or collocation-points are again zeros of an orthogonal polynomial p ~· 13 )(x ). Here we shall prove this in detail for the extremely simple differential equation yc 1) - Ky = 0

y(O) = 1

[y(x)

= exp (Kx)]

but the proof also holds for an equation of the general form (9) as shown in chapter 6 for a one-point collocation method.

Integration of Initial Value Problems

r

151

The N constants b are determined by the method of moments. R(yN)Wj(X) dx

the weight functions chapter 2. Our first choice is

wi

=

o

j

= 1. 2•...• N

are not chosen as the monomials ui- 1 used in

N

RN

(13)

= I bi[P~~1 + P~ 1 )]

N

I

- K - K

z=1

bi(Pi-1

+ Pi)

(15)

z=1

r

RN is inserted into (13): RN(PJ-l- Pj) dx

=

j = 1,2, ... ,N

0

(16)

in which the integrand is a polynomial of degree N + j. Equation (16) is written in the same way as (2.99) to (2.101):

Al, =

(A- KB)b = Kc

r

[Pill.

r

+ Pi1>](P1- 1

-

(17)

Pj) dx

(18)

(P,_. + P,)(Pj-t- Pj) dx

Bjt =

1

ci =

i

(PI- 1

-

PI) dx

= { 1 for j = 1 0

0

r

(19)

(20)

for j > 1

In (20) we have used 4.2.2 Development of three attractive collocation methods

p~O,O)(X) dx

By a transformation of the independent variable the interval [0, h] is first transformed into [0, 1]. As trial functions for YN in [0, 1] we use T 0 = 1 and

YN = 1

N

+ i..J '\'

b.[p(O,O) 1

1-1

=0

Next we evaluate the elements of A and B: O) + pO) . a po 1ynomta . 1 o f d egree 1. - 1• If 1. <. J,. 1t . may b e P 1-1 , Is 0 0 reformulated into a sum of Legendre polynomials P~ • )(x )(k < j - 1) and each of these is orthogonal on PI_ 1 and PI. Hence Aii = 0 for i < j. Integration by parts yields

+ pCO,O)] 1

z=1

The value of P~a,f3)(x = 0) is (-1)' and all trial functions Ji(i > 0) satisfy a homogeneous boundary condition Ji(x = 0) = 0. Equation (12) is a polynomial of degree N and it may if desired be converted into the monomial expansions of chapter 2 by means of the explicit power series for P~o,o).

for any n > 0

Aj, = (PH - Pl)(P,-t

+ P,)J5-

r

(P,_.

+ P,)[Pjll.

- pji>] dx (21)

The first term is zero since by a result from (Exercise 3.1) P~'M(x

=

1) =

f(N + a + 1)f(/3 + 1) = 1 for a = f3 f(N + f3 + 1)f(a + 1)

152

Solution of Linear Differential Equations by Collocation

Integration of lnit1al Value Problems

153

and 13

Pc;· )(x

= 0) =

-Pc;:!{(x

= 0) for any (a, {3)

The integral in (21) is zero for j < i since the last factor is a polynomial of degree j - 1. Consequently A is diagonal. The diagonal elements are easily calculated:

f [P~l!,

+ P~'l](P,_, -

P,) dx =

=

f

P,P,_,J~-

=

r

P;P~I!, dx

YN(X = 1) = 1 +

Bji

= 0.

(P,_, + P,)(P,_, - P,) dx

-

YN(1) = 1 + K

(P?_,- P?) dx

8

1+ 1 ,1

=

1

=

f

r

(30)

y(x) dx

1+

f

b,(Pi-1

+ P.) dx

1

= 1

+ K + Kb,

=

P? dx =

2

i

~1

6 + 4K + K 1 1 1 4 = 1 + K + 2K 2 + 6K 3 + K + ... 6 _ 2K 18 3

1

j

1

4

IlK + 18K + ... i=O}. 2 3 (1) = 60 + 36K + 9K + K 2 y 60- 24K + 3K

1

+ P,)(P,- Pi+,) dx

rr

= 1+K

2

y 1 (1)

f~P, + Pi+ )(P,_,- P,) dx = -f P? dx = - 2i ~ 1 (P,_,

(29)

(31)

The total accuracy of YN by (31) is K 2 N+ 1 in a power series representation of the exact solution at x = 1 in powers of K. A few examples are given:

[f(/3 + 1)]2 N!f(N + a + 1) f(N + {3 + 1)f(N + a + {3 + 1)(2N + a + {3 + 1)

r

1

In this formulation the accuracy of yN(l) is seen to depend only on the accuracy of b~, which is much higher than the accuracy of the series in (29).

The side diagonals Bi,i+ 1 and Bi+1,;, i = 1, 2, ... , N- 1, are found from B,,,+l =

N

bi[P;-1(1) + Pi(l)] = 1 + 2 I bi

Tl},e integral is evaluated by (12):

in which the result of Villadsen (1970), p. 55, has been used for a = {3 = 0:

CN

N

I

term in (29) is bN, which is accurate up to and The least accurate 1 Iinc:ludi'r1_g KN+1 , but the first missing term bN+l in (29) is itself of the of KN+ and the error term in (29) is thus eJ(KN+ 1). A much better evaluation of YN(x = 1) is possible by an integration of (10): y(l)

= c~~~) - c~o,o)

(a,f3) _

(28)

i=1

2

2 fori = j = {0 for i ¥- j

1

bikKk

bik are correct for k s 2N + 1 - i. In particular, bik is accurate k s 2N. The ordinate at x = 1 can be obtained from

r

f

I

k=O

Matrix B is tridiagonal since i < j - 1 or j < i - 1 both give The main diagonal of B is

B" =

M

bi =

P~ 11P,_ 1 dx

= pipi-116 = Aij

The only difference between (17) and the corresponding equation in · n 2.5 is that B is nonsymmetric and this is of no consequence for a by word repetition of the proof in section 2.5 that leads to the ,nuJmbc~r of correct terms in a power series expansion of bi in powers of K:

2

N= 3:

(27)

2 3 4 (1) = 840 + 480K + 120K + 16K + K 2 y 840 - 360K + 60K - 4K3 3

7

I

1

.

-KJ J=oj!

Ks

+-39,200

154

Even the first erroneous term is quite well represented (for N = 3, the denominator of the ninth term in the series should be 40,320). An explicit formula for the difference A between the first erroneous term in the series derived by MWR and the corresponding Taylor series term can be found:

d = 2[ (N + 1)! (2N + 2)!

]2(-1)NK2N+2

For small values of K (K < 1), A is approximately equal to the y(1) - YN(1) = exp (K) - YN(1). We shall now return to (13) and (14) and deduce a collocation principle that corresponds to this excellent MWR. It is first noted that each wi(x) is zero at x = 1 [equation (22)] that consequently 1 - x is a factor in wi(x) for j = 1, 2, ... , N. This means that

where Qi_ 1 is a polynomial of degree j - 1 or at most of degree NIf theN constants of the Nth-degree polynomial RN are determined that RN is proportional to P~· 0 >(x ), all orthogonality relations (16) automatically satisfied. It is thus seen that the procedure (13) to (16) 0 identical to a collocation procedure at the N zeros of P~' )(x) when RN is a polynomial of degree N in x. The very accurate value of YN(1) was obtained in (31) by an .1·.1. tion of YN(x ), which showed that only b 1 entered into the final result. the collocation ordinates are used in a quadrature analog of the integration (31) of P~o,o)(x), we must resolve the dilemma that the 0 collocation ordinates are available at the zeros of P~· \x) and not at zeros of pco,o)(x ). ILI.JI!.La-

For (31):

YN(l) = 1 + K

r

YN(x) dx

The Lagrange interpolation polynomial for the Nth-degree YN(x) is N

YN(x)

= I

155

Integration of Initial Value Problems

Solution of Linear Differential Equations by Collocation

li(x)yi

It is, however, easier to use the Gaussian-type weights that are found chapter 3 and for which efficient computational routines are available. One method is to use (35) to find YN(x) at the N zeros of P<;J-· 0 )(x) and te (34) by a Gauss-Legendre quadrature. A more convenient is to evaluate a preliminary value of YN(x = 1) from (35) and suo~;eq11en.t1y use this preliminary YN(x = 1) in a Radau quadrature-all ordinates are now at their correct positions.

is immediately clear that all three integration procedures give identical when the integrand of (34) is a polynomial of degree N in x. The Radau quadrature method based on extrapolation of YN(x) to = 1 and using the preliminary y-value at x = 1 in a corrector formula is, however, not only the simplest to use but it is also the most te whenever the integrand of (34) is a nonpolynomial function or a Jotv·nomt';:tl of degree higher than N. All ordinates Yb y 2 , ••• , YN+ 1 = 1)] are of equal accuracy N, and all terms up to and including in the integrand of (34) are correctly integrated. This important of the recommended method is discussed further in chapter 6. At first sight the weight functions (14) appear to be polynomials of 1, 2, ... , N. The normal method of moments of chapter 2 uses functions that are monomials u'-\ j = 1, 2, ... , Nor polynomials j = 1, 2, ... , N. A closer analysis of (14), however, shows that = (1 - x )Qi-1 with Qi- 1 a polynomial of the usual type and (1 - x) lntP•.rnr·Ptt:•rt as a weight function in (16). A strictly conventional weight function for the method of moments is wi(x)

0

Using the N + 1 available quadrature points it is, of course, oo!;sible. to compute the weights of a quadrature formula that exactly u·1te~~rat~es the Nth-degree polynomial YN(x )-this can be done with an ...... h.;+...,. ....,. choice of N + 1 quadrature points.

P}G__._~\x) - P}

0 0 • \x)

j

= 1, 2, ... , N

- 1

wN(x) = P<;J.:!!i (x)

(37)

this wi(x), the integrand of (13) is at most a polynomial of degree - 1 and collocation at the zeros of P<;J-· 0 ) is identical to (12) and (13) wi(x) given by (37). Note that since RN is orthogonal on each weight function of (37), it is orthogonal on their sum:

i=O

with (xo, Yo) = (0, 1) and PN+1 = xp~' )(x ).

=

N

Wo(x)

=I w/x) = (Po- P1) + (P1

- P2)

+ ... + PN-1 = 1

1

w0 (x) may be conceived as the "missing" zero-degree weight function (37). The approximation error for y(1) can be derived in the same way as collocation at the zeros of P~· 0 )(x) following the development of (17)

156

Solution of Linear Differential Equations by Collocation

Chap. 4

to (28). The elements of matrix A in (17) are the same as in (23). B is again tridiagonal but BNN = cN-1 = 1/(2N- 1). It is easily shown that the expansion of b1 in powers of K is correct up to and including the term K2N-1. The dependent variable YN(x) is now available at the correct values of x for optimal integration of (34) and

L

WkYk

where the {wd are the Gauss-Legendre quadrature weights. When the collocation points are zeros of P~'O)(x ), we can prove that the same value of y(1) is obtained by quadrature and by direct extrapolation from -the N + 1 known ordinates Yo, Yb ... , yN: N

L

N

li(x)yi

YN(1) =

and

L

li(1)yi

(39)

z=O

i=O

The residual RN is orthogonal on w0 (x) = 1 and

( Jo YN(1)

RNdx

=0=

= YN(O) + K

fl d Jo (;;- KyN) dx

f1

Jo

YN(x) dx

= Yo+ K

2+K Y1(1) = 2 - K =

K1

2

1

L l1. + -4K3 + ...

]=0

12 + 6K + K 2 1 Yz( ) = 12 - 6K + K 2 =

1

Kj

4

1 ~o ll + 144 Ks + ...

3

k=1

YN(x) =

A few examples may illustrate the results obtained for YN(1):

120 + 60K + 12K2 + K 3 1 y ( ) = 120- 60K + 12K2 - K 3 =

N

YN(x = 1) = 1 + K

157

Integration of Initial Value Problems

L1YN(x) dx

(40)

K1 1 7 j~o j! + 4800k 6

Collocation at the zeros of P~'O)(x) is a simple, consistent method with a high integration accuracy (41) per step. Collocation at the zeros of 0 P~' )(x) is even more accurate and only slightly more difficult to apply: A preliminary YN(1) must be found by extrapolation from y 0 and the internal ordinates and subsequently used in (36) to find the corrected, very accurate YN(x = 1). If an even higher accuracy is desired, one'may use the known value of y and of (dy I dx) = f at the left-hand interval end point. In this way a collocation method is constructed with YN(1) accurate up to and including K2N+2. y is approximated by the following (N + 1)-degree polynomial: YN+1(x)

= Yo+ f(xo, Yo)(x - Xo) + (x -

xo)y~x)

(42)

The auxiliary polynomial y ~x) is given by On the left-hand side of (40) the interpolation value (39) for YN(1) appears and the integral on the right-hand side of (40) is the same as is evaluated by (38). It is noteworthy that the identity between the interpolated value (39) for YN(1) and the quadrature value (38) for YN(1) also holds when the original differential equation is nonlinear as in (9). The internal consistency of the MWR based on weight functions (37) or collocation at the zeros of P~' 0 )(x) is an attractive feature of this procedure in comparison with the MWR based on weight functions (14) or collocation at the zeros of P~' 0 )(x ). This consistency of a true method of moments is observed in chapter 2, p. 90 where the same value of the effectiveness factor was found from dyN/dxlx= 1 and from J~ yNdx when YN was determined by the method of moments [collocation at the zeros of 0 P~' )(u)] rather than by Galerkin's method [collocation at the zeros of P~' 0 )(u )].

t~

The method based on (37) is, however, less accurate than the method based on (14), as seen by comparison of the dominant error term for small K with the corresponding formula (32) for (14). For (37):

exp (K) - y (1) N

=(

N! 2N!

)z 2N(-1)N K2N+1 +1

N+1

y~x) = L li(x)y~ with li(x) = ( 1

Xz

XPN

X x=x;

(43)

Differentiation of (42) shows that yK,{x 0 ) = 0 since we demand that y~~1(x = Xo) = f(xo, Yo). Consequently collocation might be applied to the differential equation to find N values of y ~x) at the collocation points {xJ. The reason for choosing zeros of p~' 1 ) as interior interpolation points in (43) is that we wish to compute a high accuracy value of YN+ 1 (1) by extrapolation followed by quadrature, just as in the Radau method but now working with Lobatto quadrature to include y 0 and thus obtaining an error term O(K2N+ 3 ) for the test example y< 0 = Ky. For this equation, with y(x 0 ) = y(O) = 1, one obtains YN+1(x)

r

= 1 + Kx + xyKr{x)

YN+t(l) = Yo

+K

[Yo

K2

(41)

X-

xp 0 ' 0 (x) )( N o,o( ))Cl)

+ Kx + xyUx)] dx

= 1+K +2 +K

N+1

1 ~0 [xiy~x~)]wi

(44)

158

Solution of Linear Differential Equations by Collocation

The lower limit of the sum can just as well be taken as i = 1 since x 0 and y*(xo) = 0. yt{xN+I) = yt{1) is the preliminary value of y~(l) obtained by extrapolation and wi are the Lobatto quadrature weights. The first approximation for y (1) by this Lobatto method is

Integration of Initial Value Problems

159

A few Pade approximations [from Villadsen (1970), pp. 152-53] are shown to illustrate this point: Collocation at zeros of P~,o) and at x = 1:

4+K YI(1) = 4 - 3K K2 =

+

K'

2

1

L l1. + -8K3 + ...

]=0

which may be compared with 2+K YI(1) = - - = 2- K

with an error exp(K)- y1(1) ~ -(K5 /480) for small K. dominant error term is

+ 2)! 2N+3 exp (K)- YN(1) ~ (-1) (2N + 2)!(2N + 3)!K N

N!(N

The approximations that we have obtained for exp (K) by (36), (38), (39), or (44) are all of the following form: On(K)

exp (K) ~ YN(x = 1) = OAK)

4.2.3 Collocation of right-hand interval end point

Approximations of the type (45) are called Pade approximations and they are widely used in the synthesis of process control mechanisms. The Pade approximations that we have constructed by three different collocation principles are all characterized by On being of a degree higher than or equal to the degree of Od. Formulas with n < d can be constructed if x = 1 is inclqded as a collocation point besides the N interior collocation points. Ther~ qre certain advantages to this procedure when solving coupled differential equations, and the end point ordinate is obtained without the extrapolation or integration step. But a serious drawback of all collocation procedures with the right-hand interval end point included as a collocation point is that the approximation error is of the same order of magnitude as that obtained with interior collocation points only.

2

j=O

obtained by (38) or (39) on the basis of interior points only. Collocation at zeros of p<J,o) and at x = 1: 6 + 2K YI( 1) = 6- 4K + K 2 =

3

K'

1

]~0 7! + 36K + ... 4

A combination of YI(1) and YI(t) using the quadrature formula (36) to find a "better" value of YN(1) does not succeed in improving the accura/cy: ,y 1

On and Od are both polynomials inK. The numerator polynomial On is of the same degree N as the denominator polynomial Od for the Gauss method (38) or (39). In the Radau and the Lobatto methods, the degree of On is, respectively, one and two higher than the degree of Od.

K' 1 3 L -:+ -K j! 4

(31) = 6 - 6-4K2K+ K

2

and [yl(l)Jcorr = 1 +

~[ 3yl(~)

+ Y1(1) J = 1 + K 6 _ ~_;! K2 = YI(1)

It is again seen that y 1 (1) has the same order of accuracy as that obtained in (32) for N = 1, but two algebraic equations have to be solved rather than one. Collocation methods with right-hand interval end point included as an extra collocation point are thus seen to be less attractive for integration of (10)-and in general for integration of any single differential equationthan collocation methods based on interior points only. For integration of coupled differential equations with a large span of eigenvalues, the end point collocation methods might be preferable to collocation methods without end point collocation. The reason is that end point collocation methods have very desirable stability properties while collocation methods where the end point is determined by an extrapolation process may be unstable unless very small steps are used. This is further discussed in chapter 8. In the present context, however, we do conclude that with regard to single-step error the Lobatto method is best followed by the Radau and

160

Solut1on of Linear Differential Equations by Collocation Integration of Initial Value Problems

the Gauss method-all based on interior points only. The local truncation error for the three methods is O(h ZN+ 3), O(h ZN+ 2 ), and O(h ZN+I) for stepsize h. 4.2.4 Detailed solution of collocation point

From (3.83), Wi =

y 11=y with one

withy (0) = Yo = 1

y

t) = x 2 -

x(x -

lo(x)

Instead of explicit evaluation of K', we may determine that the sum of the weights must be 1, yielding

= ~'

Wz

= i,

Pi

p(x) = x(x -

!)

=

x2

-

!x,

0

p \x)

X-

t

1-

-3

= 2x -!

X-! = -2x + 1 1-2

tx,

=-

+ 1)yo +

Y1(1) = 1 + lf3yl + YI(l)preiJ = 2.75

or

lo(x) = -

=

-3x

+1 YI(x)

Y1(x) = (-3x

)]2

Notice that the "corrected value" y 1(1) = 2.75 is much closer to = 2.5.

One collocation point (N = 1) is used, and the values obtained = 1) are used to compare the accuracy of the three methods. 1. Collocation at the zero x 1 = t of Pi1'0 ).

=

(

N+l Xt

e = 2.718 than the "predicted" interpolated value y 1(1)prei 2. Collocation at the zero x 1 = ! of 0 •0 )(x ).

y 1 (x

p(x)

Xi 7

W1

=

K' (p(l)

WI= ~ K'

The mechanical application of the three collocation methods of subsection 4.2.2 are illustrated for equation (10) with K = 1: dy dx

161

=

(-2x

+ 1)yo +

(2x)y 1

dy1(x)

(3x)y1

-----;;;-- = -2yo + 2y1

dy1(x)

-----;;;-- = -3yo + 3y1 dy1(x)

R1(x)

=-----;;;--- Y1(x) = -3yo + 3y1 -

Y1(x)

R1(!) = 0 ~ -2yo + 2y1 - Y1 By interpolation:

3

By interpolation, a preliminary value YI(1)prei 3 Y1 = 2! is obtained. Correction of end point value by quadrature: Y1(l) - Yo = j

l

0

YI = 2 = (-3 + l)y

0

+

=

W1Y1

= (x - 1)pl \x) = p)J~1(x) = 2x - t p)J~I(l) = ~ PN+I

(x - 1)(x -

Y1(1) - Yo =

+ WzYI(l)prei

t)

=

X2

Y1 = 2

Evaluation of y1(1) by (Gaussian) quadrature:

The weigJt factors must be determined from the node polynomial: 1 0 '

0,

Y1(1) = (-2 + 1)yo + 2y 1 = 3

1

Y1(x) dx

=

-1x

+t

r

Y1(x) = W1Y1 = Y1 = 2

or y 1(1) = 3

which is the same result as obtained by interpolation. 3. Collocation of modified equation at the zero x 1 = ! of Again

yf(x)

= (-2x + 1)yt + 2xyf

dyt(x) - --Y2o* + 2 Y1* dx

and

Pi1·1)(x).

From (43), Y1(x) = 1 + x + xyf(x) dy1(x) _

~-

R 1 (x) =

1

with y*(x = 0) = 0

+ *( ) + dyf(x) Y1

X

dx

X

dy~x)- y 1 (x) =

+ yf(x) + xdy~x)]- [1 + x + xy

[1

=!, R1(~) = 0 = [1 + YI + ~(-2y~ + 2yi)] - (1 + ~ + ~yf)

and, for x

By interpolation, .

.

+ 2yi

yf(1)prel = -y6'

dominant error term of the approximate y value for each component at the end of an integration interval is O'(lh.Aijq) where q is 2N + 3, 2N + 2, or 2N + 1, depending on the choice of the N collocation points. Consequently the largest eigenvalue ih.Amaxl of Q is a measure of the overall accuracy of the integration of (46) from x 0 to x 0 + h. A system of equations with constant coefficient matrix would, of course, be solved by the methods of section 1. 7, but the approximate methods of this section are equally well applied in less trivial circumstances. The application of the Lobatto method for solution of two coupled equations is shown below in detail, while the corresponding calculations for the Radau and Gauss methods are referred to as an exercise. Integrate dy1 dx

2

=

163

Integration of Initial Value Problems

Solution of Linear Differential Equations by Collocation

162

3

= Y1 + Yz = /1(x, Yb Yz) (48)

and, by (42),

r

Yo = 1,

Y1 = 1

y,(1) - Yo

=

+ 21 + 21

1

Y 1(1) pre! =

5

· 3 = 3'

8

3

= WoYo + w,y, + w2y,(1)P'"'

y,(x)

The Lobatto weights are 4

w1 = 6, = 6,1 1+~ ·1+~ ·~ +~ ·~

Wo

Y1(1) =

=

~

2.7222

4.2.5 Solution of coupled and higher-order initial value problems

In section 1. 7 a set of M coupled first-order differential equations with constant coefficients ,

y

=

y0

atx

=

d(U- y) dx

= A(U- 1 y)

Yz(x)

= Y10 + /1(0, Y1o, Yzo)x + xyf(x) = Yzo + /z(O, Y10, Yzo)x + xyf(x)

(49)

Index 1 or 2 is used to designate component y 1 (x) or y 2 (x) of the solution vector. Y10 and Yzo are the values of Y1(x) and y 2 (x) at x = 0. We have dropped the index N + 1 or N that designates approximation order. With N = 1, Y1(x) and Yz(x) are both second-degree polynomials, while yf(x) and yf(x) are first-degree polynomials. With the initial conditions of (48), we obtain Y1(x) = x

+ xy'[(x)

(50)

Yz(x) = 1 +x +xy!(x)

0

dyi(x) = 1 + *( ) + dyr(x) dx Yz X dx X

was solved ky 'diagonalization of Q: 1

one step L\x = 1 forward from x = 0 with initial element y 1 (0) = 0 and Yz(O) = 1. The auxiliary functions (42) that are used in the Lobatto method are defined by Y1(x)

=

There is a remarkable difference between the accuracy of Y1(1)prei and the corrected value Y1(1).

dy -- Qy dx

dyz dx = Y1 + (1 - x)yz = fz(x, Yb Yz)

(51)

Q = UAU- 1 or dY = AY dx

Each of theM decoupled equations in (47) is of the type (10), which has been treated in this section. If collocation is used to solve (47), the

yf(x) is interpolated at the zero x 1 = ~of YT(x)

= (-2x +

Pl1 ' 1\x):

1)Yi'o

+ 2xy[1

(52)

164

Solution of Linear Differential Equations by Collocation

Chap. 4

Integration of Initial Value Problems

The derivative of y t(x) is

dy'J'(x) _ * * _ * ---;I;--2yi0 + 2yil - 2yil

that the particular system of collocation equations that result from an nth-order equation can be solved with only slightly more work than is neces~ary to solve the N collocation equations from a single first-order equat~on. We h~ve oc~asion to solve several complicated higher-order equat~ons, ~.g., m sections 4.5 and 5.5. Here a simple second-order equat!on with constant coefficients and a numerical example suffice to explam how the reduction in dimensionality is achieved. Consider a second-order differential equation 2 d y dy dxz + kl dx + kzy = 0 (57)

(53)

since y 'to = o by the definition of y t (x ). Equations (53) and (51) are inserted on the left-hand side of (48) and (50) is inserted on the right-hand side of (48) with x = t:

1 + YTl + t . 2yfl = t + tyi\ + 1 + t + ty~\ 1

+ YI1 + t · 2yfl = t + tYf1 + t + ~ + ~YI1

(54)

1 with given values [yo, y& )] of {y, dy/dx) at x Yz(x) = dy(x)/dx.

The solution of (54) is

(yfl, Yf1)

= ({i, }?g)

dyt(X)

Preliminary values of y'J'(x) at x = 1 are found from equation (52):

(yTz, Yiz)

= (2yfb 2yfl) =

(~,

n)

---;;-- = 0

0

1

0

2

1

2 17

19

32

52

19

= y(x)

= Yzo + -k
These values are quite good considering the large stepsize dx = 1. The true solution of (48) is y 1 (x) = xex and y 2 (x) =ex. The Radau method gives (y 12 , y 22 ) = (3.18, 3.04), while the Gauss method is very poor [(y 12 , y 22 ) = (8, 7)]. The step size is obviously too large for this comparatively low-order method. A Runge-Kutta fourth-order method has a truncation error proportional to h 5 ~ !Just as the Lobatto one-point method. It yields (y 12, y 22 ) = (2.61, :2.625), a result of about the same accuracy as that obtained with the Lobatto method. A linear nth-order differential equation can be interpreted as ncoupled, linear first-order equations. Normally collocation at N points for each of the equations would lead to a system of nN algebraic equations for the collocation ordinates. It is of great practical interest

and

+ Yz(x) (58)

---;;-- = -kzyl(x)- ktYz(x) The discrete version of (58) is

Y~l) = Ayt + AoY10 (t)-

Yz -

19

Corrected values of Yiz are finally obtained by an integration of (48) with a Lobatto quadrature: 18 Y12 = Y10 + -k
Yzz

Let y 1 (x)

dyz(x)

49

19

· Y1(x)

= 0.

(56)

y;'k is transformed to corresponding values Yik by (50): i/k

165

A

Yz

+ AoYzo

= Yz = -k2Y1

(59)

- k1Y2

(60)

Equation (59) is substituted into (60):

A(Ay1 (A

2

+ AoYlO) + AoYzo =

-kzYt - kt(Ay 1 + A 0 y 10)

+ k1A + kzl)yl = -(AAo)YlO-

AoYzo- k 1A 0 y 10

(61)

In (59) to (61) A is the (N X N) matrix that results when the first row and colu~n of the discretization matrix for the first derivative are erased. Ao contams the weights of the left-hand end point ordinate in the N collocation p(l,o) or pO.l) · N ' N , N VIa . . equations. Collocation at the zeros of p(O,o) h~ auxtba~y functions gives N linear equations (61) for y • The solution 1 IS mserted.Into (59) to give Yz· Preliminary and corrected values for y 1 (1) and Yz(l) m the Radau and Lobatto methods are obtained just as in the case of one first-order equation. As an example, take

!

dyl(x) = Yz(x) dx dyz(x)

d;- =

1

3

-2yl{x) - 2Yz(x)

with, initial element [y 1 (0), y 2 {0)] = (2,

-!).

(62)

Solution of Linear Differential Equations by Collocation

166

. h We use one collocation pomt at t e zero x =

1 3

o

f pO,O)( ) 1

x ·

= (-3x + 1)yw + 3xyi1 yW(x) = -3yw + 3yi1 Yil(x)

Vectors yO) of (59) and (60) represent values of the first derivative at internal p~ints only. Only the last N rows of the full A .matrix (63) are needed. Also the influence of YiO has been separated out m (59) and (6~) through the term AoYiO· Consequently the ~atr~x A of (59) and (60) IS obtained from the full A matrix (63) by deletmg Its first row and c~lumn. In our example with only one unknown ordinate (y1r, Y21) we obtam (32

+ ~. 3 + k. 1)yll = -[3 . (-3)] . 2- (-3) . (-~)- ~. (-3) . 2

with solution y 11 = ~; from (59), 45

y 21 = 3 . y 11 + (- 3) . y 10 = 3 . 28 -

6

33 = -28

y 12 and y 22 are found by extrapolation to x = 1:

y 12 = -2 y 10

135

= -2y2o + 3y21 = 3 -

Y22

23

+ 3 y 11 = -4 + ~ = 28 99

28

=

15

-28

The preliminary end point values (64) are finally corrected through an integration of (62): y 12

=2+

y 22 = --

r

-~-!

y2 1 (x)

r

dx = 2 +

yu(x) dx -

-2.l - 1(135 8 28

+ 2.1 28

- 297 28 -

~

~(3(-¥.) + (-¥s)J = ¥.

r

= 0.982

Y21(x) dx

12) 28 =

-~ = 56

-0 • 679

The result may be compared with the exact solution y(x) = Y1(x) = e-x + e- 112 x of (62). For x = 1 one obtains (y 12 , y 22 )

In the present section we shall develop methods of attacking this model by MWR in much the same way as a two-point boundary value problem is treated in chapter 2. We find N-term analogs of the truncated Fourier series (1.141) of the solution. The coefficients (bi) of the MWR series are not identical to the true Fourier series coefficients (bt), and the approximation functions are close approximations to the true eigenfunctions, but not identical to them. This is exactly the situation that was encountered in subsection 2.5 .1: The combination of N approximate eigenfunctions will be shown to represent the solution of the model very well not only for large values of the unrestricted variable z but also for quite small values of z where the Fourier series is slowly convergent. By a specific choice of approximation function, certain functionals derived from the solution of the partial differential equation can be particularly well determined. In section 4.4 the solution for the initial flow (or entry flow) problem is treated separately. Neither a true Fourier series nor a polynomial approximation to this can be used to calculate the heat (or mass) transfer for z ~ 0 since the derivative of the profile at the solid wall increases to infinity at this z-value. Finally in section 4.5 the initial value methods of section 4.2 are used to calculate the true eigenvalues and eigenfunctions of the differential equation one by one with an arbitrary accuracy. As indicated above, this imitation of the true Fourier series is not the most efficient method of solving the partial differential equation, but the principle of the method of section 4.5 can be used in many other circumstances. Cylinder geometry with eigenfunctions that are even functions of x has been treated in most of the preceding examples. In order that the application of MWR in circumstances with less obvious symmetries shall also be sufficiently well exposed, a close analog of (1.140) is solved rather than the Graetz problem in cylinder geometry. (1 - x2) ay

az

y = 0

Equatio~~ The Graetz problem of section 1.2 is discussed furth~r in s~ctio~ 1. 7 a~ a typical nontrivial representative of the linear parabolic partial differen.ttal equations that turn up every so often as models of important chemical engineering systems.

=

a2y

ax 2

(65)

with boundary conditions.

= (0.9744, -0.6711)

4.3 Linear Pprabolic Differential

167

Linear Parabolic Differential Equat1ons

atx

= O,z > 0 y

=1

dy = 0 at x = 1, z > 0 , dx at z

=0

(66)

Equations (65) and (66) are the mathematical model for desorption a liquid film that flows in the z -direction. The flow is laminar with a developed parabolic velocity profile also at z = 0. The free surface at x = 0 and there is no gas film transport resistance. Transport in the -direction occurs by liquid diffusion with zero diffusion gradient at the SOlid wall X = 1.

Solution of Linear Differential Equations by Collocation

168

This model has been discussed in Bird (1960), p. 537. Several simplified versions of the model are solved in the reference. Finlayson (1972) has used Galerkin's method to solve the problem for large z an? a penetration depth concept combined with the method of moments to give a first approximation to the solution for small z. . . . We are interested in the solution y(x, z) and also m certam denved properties: 1. The bulk mean concentration defined by ji(z)

L

Vz

dA

=

L

r

YVz

dA

The coefficients ai (z) of (70) are determined through the solution of N coupled ordinary differential equations, and the problem of finding suitable initial values ai(O) arises. It is immediately clear that no set of N constants ai (0) can be chosen to satisfy the initial condition y = 1 at z = 0 exactly: N

1

=1:-

I

ai(O)[(i

=

~

Sh =

(ayjax)x=o _ y

Integration of (65) yields an expression similar to (1.55): 2 (dy/dz)

Sh =

3

4.3.1 Trial functions for MWR solution

The Nth approximation YN(x, z) is chosen in the same way as with ordinary differential equations: N

YN(x, z)

=

N

y(xh 0) = 1 =

(1 - x )y(x, z) dx

X

To(x, z)

+ I

xi+

1

]

For each MWR however, it is possible to make YN approximately equal to the initial condition y = 1 at z = 0. Using a collocation method, one would require the residual at z = 0 to be zero at the collocation points:

2

2 The local Sherwood number defined similarly to (1.53) without the . factor 2 which enters in (1.53) through the use of 2R as diffusion length s~ale and with a plus sign due to reversal of the direction of

+ 1)x -

1

or for a laminar velocity profile and dA = dx: ji(z)

169

Linear Parabolic Differential Equatrons

I

ai(O)[(i

1

+ 1)x1

-

xJ+

1

]

In. the method of moments one would require that the initial residual N

1

-I ai(O)[(i + 1)x

- x 1+ 1]

1

is orthogonal on N weight functions wi(x), which may be xi- 1 or theN fifst trial functions Tj(x), j = 1, 2, ... , N in close analogy with the method of moments and Galerkin's method as defined in section 2.3. We (1 - x 2 ) Tj(x) as weight functions in a discrete analog of (1.151). The arbitrary constants b T were determined by a Fourier expansion of the initial condition function y (x) = 1 in the eigenfunctions of the differential equation. At each z the true eigenfunctions are here represented by linear combination of J:(x), i = 1, 2, ... , N and 1 - x 2 is the weight function W(x) of the Sturm-Liouville problem that is obtained from (65). The choice of weight function is, however, not critical. In any case, a or less accurate approximation of y (x, z = 0) is obtained, but when is increased, the differences between the various approximate initial functions have a progressively diminishing influence on the profile.

ai(z)J:(x)

i=1

The trial functions J:(x), i > 0, are functions of x alone and we intend to determine the z-dependent coefficients ai(z) by MWR applied to the residual RN(x, z ). The function T 0 (x, z) must satisfy the boundary co~d~­ tions on x [thp upper equations of (66) in the present example], and ItlS convenient if !"0 (x, z) also satisfies the differential equation (in which c~se it disappears from the residual). In the case of (65) and (66) one obtams T 0 (x, z) = 0. .. J:(x ), i > O, must satisfy homogeneous boundary conditions [here y(O) = 0 and y 0 )(1) = 0]. A polynomial trial function T;(x) may be J:(x)

=

(i

+ 1)x -

xi+

1

4.3.2 Solution by Galerkin's method

The residual RN of the differential equation (65) is 2

RN(a, x, z)

2

ayN iJ YN )a;ax

= (1 -

x

= (1 -

x 2)

2

f 1

= (1 -

X

2

da;(z) J:(x) dz

Nda·

)

I -d'[(i + 1)x 1•

z

f

2

ai(z)

1

- xi+ 1]

d~

(71)

dx

N

-I a;[-(i + 1)ix 1

1 1 - ]

2. The eigenv~lues A, .eigenvectors U, and eigenrows u- 1 of A - 1 B are det~rmmed. Thts can be done directly from A and B without ca~~ulatmg ~he product matrix A-lB [Wilkinson (1965), p. 54]. 3. U ao = b Is calculated. 4. ai (z) is now given by

RN is made orthogonal on theN trial functions 7j(x):

r

i = 1, 2, ... 'N

= 0,

RN(a, X, z)1j(x) dx

Inserting (71) in (72), one obtains da A-=Ba AH

r

Bi, = -i(i

N

ai (z) =

dz

[(i

=

+ l)x

+ 1)

- xm][(i

r

x'-'[(i

1

- xi+ ](1 - x

2 )

, 1

=

r

T'; 21 (x)1j(x) dx =

o

= -

r

4.3.3 Solution by collocation

1

r

Tj 0 (x)1j(x)i~ -

The Nth appro~ima~ion for the dependent variable YN(x, z) is now represented by N mtenor ordinates y (x 1, z) at the collocation points x. and by the boundary ordinates y(O, z) = y 0 = 0 and y(1, z) = YN+l:

1

1

T';' (x)Tj

0

(x) dx

N+l

n~

= L

YN(x, z)

li(x)y(xu z)

0

0

(81)

Tj (x)Tj (x) dx

Consequently B is also symmetric. We note that these properties of A and B are also found in chapter 2 for Galerkin's method. The coefficients ai(z) are found as the solution of (73):

da = A- 1 Ba = Ca

dz

with solution (cf. 1.130): 1 a(z) = U exp (Az)U- a 0

1-

~ a;o[(i + 1)x -

x•+']

The trial functions of (70) were chosen such that the boundary conditions y(O, z) = 0 an~ .Y 0 )(1, z) .= 0 were automatically satisfied. In (81) these bo~ndary condttlons provtde two extra equations for the N + 2 ordinates. I~ IS known from chapter 2 that (81) and (70) are equivalent representations of YN(x, z ). The initial values y(x 1, 0) are immediately available: y(x" 0) = 1

Wilkinson (1965), p. 35, shows that the eigenvalues of the product C of the inverse of a positive definite matrix A and a symmetric matrix B is diagonable with real eigenvalues (which may or may not be distinct). The eigenvectors ui can therefore be used as a basis for N -space. The solution of (65) by Galerkin's method consequently involves the following steps: 1. a 0 = a(z = 0) is determined as described in the previous subsection:

rt

}(1 - x )[(j + 1)x -xi+ I] dx = 0 2

for i

=

1, 2, ... , N

(82)

In the collocation procedure the residual is equated to zero at ~N and a set of N ordinary differential equations for the >Im~ruJr ordmates yj are obtained:

x:, ···,

2

dyj

(1 - x J dz

N+l

= ~~o Bjiyi

(83)

is an element of the discretization matrix for the second derivative lt found by the methods of chapter 3 (NO = N1 = 1). · The boundary conditions yield y(O, z) = Yo = 0 (84) N+l

L

AN+l ,I·Y·I =

0

0

or Aa = c with A given by (7 4) and

c1 =

(80)

- x +'] dx

o

0

uij exp (AjZ )bj

5. YN(x, z) is finally determined from (70).

dx

A is clearly symmetric and may also be shown to be positive definite. B

L j=l

+ 1)x

+ 1)x

171

Linear Parabolic Differential Equations

Solution of Linear Differential Equations by Collocation

170

1

1 !v· + 1)- ( -- - - -) 4 j+2 j+4

If the .differential equation contains an explicit function f(x, z ), it is convement to subtract a particular solution from y and collocate on the

Solution of Linear Differential Equations by Collocation

172

resulting homogeneous equation. Also if the boun~ary conditions inhomogeneous, it is advantageous to subtract a solutiOn of the ..............,....... tial equation that satisfies this particular inhomogeneous boundary .. tion. . The unknown boundary ordinate YN+ 1 (z) IS ehmmated from (83) means of (84) and Yo(z) = 0: (1 -

x~)1 dyi = ~ B~iYi dz

dy = Cy dz

Bt

=---2

]I

The solution of (86) is y = U exp (Az )U- 1 y0

1 -X j

withy'{; = (1, 1, ... , 1)

Matrix C is not symmetric and the eigenvalues of C in general are no~ ~eal for an arbitrary choice of collocation points x 1 even though the ongtnal problem is known to have real eigenvalues only. . In all examples that we have studied, an orthogonal collocatiOn scheme with any (a, {3) reasonably close to zero (e.g., a and {3 both than 2) does, however, lead to a matrix C with real eigenvalues when the differential operator eigenvalues are real. 4.3.4 Computation of derived quantities It is desired to develop expressions for y and dyI dz from .t~e approximate solutions YN(x, z) in order to compute Sh .Cz) by (6~). This lS easily done from either the Galerkin or the collocatiOn solutiOn of the partial differential equation. . . . The trial functions of the Galerkin approximatiOn are mtegrated one

by one: N y = -3 L: a;(z)

2

=

1

dy = ~I 2 1

dz

[i + 1 _ 4

=

Jda;

2

(i

+ 2)(i + 4) dz

(88)

~vr uA exp (Az )u- 1ao

the components of v are given in the square bracket of equation

1

We finally obtain a set of N .homof~ne~~s first-order e~~ations constant coefficients when row 1 of B IS divided by (1 - xi).

c.

173

Linear Parabolic Differential Equations

Jl [(i + l)x -

x

i+l

2

](1 - x ) dx

0

3N [j + 1 2 ] 2~ai(z') - 4 - - (i + 2)(i + 4)

In the collocation method the integral in (67) may be found by a quadrature based on the internal ordinates and the known ordiat x = 0. (1 - x 2 )yN(x, z) can be used as integrand and in this case interior ordinates should be at the zeros of P
We choose collocation at the zeros of

P
where Yo(z) = 0. The interpolation points are obtained from JCOBI using NO = N1 = 1 to include the two interval end points. The RADAU program is used generate the weight coefficients W; of (89). These are multiplied by !(1- x7) and stored in vector V2 (see the complete computer program in Appendix, A18). Note the argument list of RADAU and compare with the description of the program in subsection 3.4.4: ID = 2 (since we know y(O) for all z) and consequently a' = a = 0 and {3' = {3 + 1 = 1. The value of NO is 1 since ID = 2, and the value of N1 = 1 since we use x = 1 as an interpolation point. Next vector AN+l and the matrix B for the second derivative are found from DFOPR. These are combined to form B* of (85), and finally of (86) is constructed. Note that the argument I in DFOPR refers to the Ith interpolation point, that is to collocation point I - 1. Therefore the Ith row of matrix C is determined from

Collocation Solution Compared to Exact Solution

Solution of Linear Differential Equat1ons by Collocation

174

+ 1, 2, DIF1, DIF2, DIF3, ROOT, Z2) and the coefficient of collocation point J is the (J + 1)th element of Z2. CALL DFOPR (ND, N, 1, 1, I

It is seen from the computer program that the setting up of the collocation problem is an almost trivial matter. The major computer work is done in subroutine EISYS in which the diagonalization of C is performed

The remainder of the computer program for collocation solution of partial differential equations consists of manipulations on U, u- 1 , exp (Az ), and the initial vector y 0 = 1. The differential equation is

d~ (U-1y) = A(U-1y)

u- 1y = u- 1 • 1 for z = 0.

C = UAU- 1

e specifically wish to determine

y of

(89):

Y = L [~wj(1 - x])]yj = V2ry 1

= (V2 TU) exp (Az )Z1 = Z2 T exp (Az )Z1

INTEGER N:

Row and column dimensions of A (see description of GAUSL and JCOBI). Current size of A = number of collocation equations.

INTEGER INDEX:

This parameter can have the values - 1, 0, or 1.

=I

INDEX = - 1:

INDEX = 0: INDEX= 1:

REAL EPS:

Only the eigenvalues of A are found. transformed into a diagonal matrix (or in case of complex eigenvalues to a block diagonal matrix) A. The eigenvector matrix U is also found and stored in rows N + 1 to 2N of A. Eigenrows u- 1 and eigenvectors U are found besides eigenvalues. The eigenrows are stored in row 2N + 1 to 3N of A. Eigenvalues and eigenvectors are stored as for INDEX = 0. Desired accuracy of eigenvalues. Suggested value for EPS (in double precision) is 10-

12

ARRAY A:

N

N

Z2(/) · Z1(I) · exp (Aiz)

=I

1

Z3(/) exp (Aiz)

is easily seen that the coefficient Z3(/) of the approximate eigenfuncexp (Aiz) is the product of the /th element of Z2 T and Z1. These two are matrix-vector products V2TU and u- 1 • 1 where u and u- 1 found in row N + 1 to 2N and 2N + 1 to 3N, respectively, of A, the matrix from EISYS. The final part of the computer program describes the calculation of (z) at selected z-values. The derivative is

'Yl.J'L..UIIIi'

(94)

where A is taken from the diagonal Ajj (j Information regarding the eigenvalues is stored in this output vector. If NC(J) = 0, the Jth eigenvalue of A is real. If NC(I) = 1 and NC(I + 1) = 2, the /th and the (/ + 1)st eigenvalues form a complex pair. The computed eigenvalues and eigenvectors of A and AT are stored in A as described under INDEX. The row dimension ND of A must be 2::: N for INDEX= - 1, 2::: 2N for INDEX= 0 and 2::: 3N for INDEX= 1.

On exit the value of INDEX is the number of iterations in the QR algorithm. Consequently the subroutine must be called with INDEX as a variable, e.g., INDEX= 1 CALL EISYS (ND, NCOL, N, INDEX, 1.D-12, NC,A)

(93)

1

.

The output is INTEGER ARRAY NC:

(92)

N

The input parameters are INTEGER ND, NCOL:

(91)

The solution of (91) is

u- 1 y = exp (Az )(U- 1 y0 ) = exp (Az )Z1

The subroutine call is CALL EISYS (ND, NCOL, N, INDEX, EPS, NC, A).

175

=

1, 2, ... , N).

4.4 Collocation Solution of a Linear POE Compared to Exact Solution It appears from the sample output in the Appendix that even for small N the collocation solution yields a stable value of the low-order eigenvalues and expansion coefficients. Sh (z) for z > 0.1 to 0.2 also seems to have a stable value when N = 6. For small z a very significant difference in Sh (z) for N = 4 and 6 is noticed. The collocation solution (70) is of the same form as the infinite series solution of the PDE and the properties of the collocation solut1lOn can be explained by a comparison with this exact representation

176

of y. Specifically it can be shown that a truncated Fourier series will also fail to represent Sh (z) for small z with a satisfactory accuracy. In this section we make this comparison of the collocation solution of (65) and (66) with the Fourier series and with a penetration solution that is applicable for small z only.

Y(z) =

r

~

For large values of i, one may obtain the eigenvalues A~ and the Fourier coefficients c~ from the following relations: ( -

AT+1)

112

-

A ~) 112 ~ 4

( -

A ~c~ ~ - 3.82.. .

0 is given by the following infinite

f

= (1 -

x

2 114 }

F

and

t

=

2

t(l - x )y(x, z) dx

* _ 3 [J~ F:(x)(1

- x

2

z~oo

dxf 2J~ [F:(x)]2 (1 - x 2 ) dx )

2

A (1 - x )F

dF dx = 0

~A! = 3.414446204

For small values of z in the infinite series, representation of y is inconvenient since a large number of terms is required to give satisfactory accuracy. For z = 0,

=0

00

at x = 1 and F = 0

TABLE

1 = Ic~

at x = 0

Values of c~ and A~ are given for i = 1, 2, ... , 6 in table 4.3 with 12-digit accuracy. These quantities have been found by the method of section 4.5, but variants of the Graetz problem have been treated in many numerical investigations and results of comparable accuracy can no doubt be found in the literature. 4.3

EXACT FOURIER SERIES COEFFICIENTS AND EIGENVALUES

2 3 4 5 6

-

(100)

[F:(x), A~] are eigenfunctions and eigenvalues of -

~ dg

for z ~ oo

limSh(z) = -~d/dz[c!exp(A!z)] 3 c! exp (A !z)

1

d 2F dx 2

r

and the asymptotic Sherwood number

00

= L c rexp (A~ z) -

(99)

while (99) is obtained by numerical integration of a definite integral. For large values of z, only the first term of (95) is of importance: y(z) = c! exp (A !z)

c1

(98)

relations are derived by the method described in Courant and (1968), p. 250-the first relation follows almost immediately after performing a variable transformation

4.4.1 Properties of the Fourier series

The exact value of y(z) for any z series:

177

Collocation Solution Compared to Exact Solution

Solution of Linear Differential Equations by Collocation

c;

A~

0.789702616221 0.097255111374 0.036093616457 0.018686373595 0.011401759677 0.007675970198

- 5.121669307365 - 39.66083891418 - 106.2492321836 - 204.8560604864 - 335.4731969788 - 498.0970836812

(101)

1

The sum of the first six cT values of table 4.3 is only 0.961. The derivative at z = 0 is undetermined since d____! dz

N

= lim ( I A~ c ~) = - oo at N~oo

t=1

z

=0

by the result of (99). The sum of the first six ATcT of table 4.3 is 23.212 and one would obtain the (erroneous) result Sh (0) =

212 ~3 23 = · 0.961

16 10 .

from the Fourier series truncated after six terms. This result is only interesting insofar as the six-term collocation solution yields 16.94 for Sh (z = 0.0001). This means that the collocation solution is at least no worse than the truncated Fourier series for small z. A different approach is obviously required if a solution for small z is desired.

178

Solution of Linear Differential Equations by Collocation

4.4.2 A coordinate transformation and the penetration solution

z1

=

y=O at x = 0

z

should not only make x and z = z 1 disappear from the equation, which is now transformed into an ordinary second-order differential equation in 'YJ, but the three side conditions (66) should collapse into two boundary conditions in 'YJ. We shall try 'YJ = f(x, z) = x/ zq with q > 0 since this combination of the independent variables has a qualitatively reasonable form. The partial derivatives are recalculated in terms of 'YJ and z 1:

ay = aya, +~az1 = _ q,ay +~ a~ a, az azl az z1 a, azl ay a, a'Y/ ax

ay ax

ay az 1 az1 ax

1 ay

-=--+--=-q2

Z1

179

side conditions (66) are transformed as follows:

A physical argument suggests that the two independent variables and z of (65) may be combined into a single variable 'YJ for small z: Close to z = 0, y is different from 1 only in a thin layer ax close to X = 0. increasing z, this layer penetrates farther into the liquid film and one might expect to obtain the same y along a curve 'YJ = x/ zq (q > 0) in the (x, z )-plane. If 'TI = f(x, z) is the correct "characteristic" for the PDE, a variable transformation

'YJ = f(x, z)

Collocation Solution Compared to Exact Solution

a'YJ

ay = ax

o

at x = 1

~

~

y = 1 at z = 0

~

y=O ay = a,

o

at 'YJ = 0

(108)

1 at 'YJ = - -

2Jz

(109)

oo

(110)

y = 1 for 'YJ

~

If the liquid film had been of infinite depth, the boundary condition

would have been equivalent to y = 1 at x ~ oo (or 'YJ ~ oo) for any z. In this case (66) would have coalesced into two boundary COJ11<1ru·~ons: one at 'YJ = 0 and one at 'YJ ~ oo. For a finite liquid film y < 1 at x = 1 for any nonzero z-value and the required coalesof the side conditions also fails to materialize. We solve (107) for small values of z = z 1 using only boundary conditions (108) and (110), and we separate the two variables 'YJ and z by expanding y in the following perturbation series: M

Y[TJ(X, z), z]--- YM('YJ, z) =

2:

ik(TJ)Zk

(111)

k=O

The same procedure was used to separate u and pin (2.97). For each M, check should be made to see whether YM( 'YJ, z) fits the neglected boundary condition (109) to a satisfactory degree of accuracy. Substitution of (111) into (107) (with z = z 1 ) and equating the coefficients of equal powers of z yields the following differential equations for A(TJ):

1 a2 y

ay

ax 2 = ziq a, 2 Substitution into the differential equation yields

2

(1 - TJ ziq) ( - q,ziq-

2

1a

a ) a a~ + ziq a: = a;z 1

It is apparent that we shall not succeed in removing z 1 from (106) for any positive q-value. The combination of variables 'YJ = xz -q (i.e., any product of powers of x and z) does not reduce (65) to an ordinary differential e,quation. Let us c~ose q = !, which at least takes care of one of the Zt 1 . 1 f act or 21 IS. - 1/ 2 wh ere t h e numenca dependent terms of ( 106 ) : 'YJ = 2xz used to give a slightly more convenient form of the resulting differential equation.

d2f2 + 2 df2- Sf = 8 3df1- 16 2/ dTJ 2 'YJ dTJ 2 'YJ dTJ 'YJ 1

d2fk + 2 dA d'Y/2 'YJ dTJ

4k~Jk

= 8

'YJ

3 dA-1 -

dTJ

16 2(k - 1)-f 'YJ Jk-1

(114)

(115)

Equation (112) is solved with boundary conditions (108) and (110), while the following equations are solved with homogeneous boundary conditions: A(O) = !k(TJ ~ oo)

=0

fork

> 0

180

Solution of Linear Differential Equations by Collocation

181

Collocation Solution Compared to Exact Solution

is found by standard methods and since / 1( YJ) satisfies the boundary conditions / 1 (0) = / 1 (YJ ~ oo) = 0, it is also the full solution of (121).

The solution of (112) is fo(YJ) = C1 JY)exp (-

g2 ) dg +

C2

0

with

The gradient at x = 0 is determined from (104):

Sh (z)

ax

0

dy

~- 3z + 1z 2

= Sh 1 (z)

1

3 (1 - x 2 )y(x, z) dx and ( ay) y=2

dz

1 - (z/ 2 )

fi'Yl), f 3 (YJ), etc. are found similarly-for every k, the particular solution of (115) is also the full solution. For YJ = 0, we obtain

In (69), we used the following relation (118) between

f

~

3

x=O

(ay)

= - 2 ax

dfiYJ)I = _ 19 'TT-1;2 dYJ 7]=0 12

and

631 -1/2 - 120 '1T

d/3(71)1 dYJ 7]=0

_1_f zk d!k(YJ)

[ay 3(YJ, z)] = ax x=O 2.f;;

x=O

Equation (118) is integrated from 0 to z with y 0 (71, z) = fo('Yl) instead of y:

0

dYJ

= ('TTZ)-1/ 2( 1 _

~ _ 19 z 2 2

24

_

631 z 3 ) 240 (124)

Finally an approximation to Sh (z) is obtained from (117) and (119): Sh (z) = (ayjax)x=o ~

y

1

J;;;-

3z

= Sh0 (z)

Sh2 (z) is obtained by deleting the last term in the numerator and denominator of (124). The sequence of functions Shk (z) are extremely accurate approximations for Sh (z) for small values of z. At z = 0.05, the relative error of Shk (z) is given by

fo(YJ) is inserted into (113): 2

'~

'

d /1 + 2 d/1 _ _ 3 dfo _ -112 3 ( 2) 411 - 8 YJ dYJ - 16 '1T dYJ2 YJ dYJ YJ exp - YJ

A particular solution to (121)

k

0

Error%

4

0.3

2

3

0.05

0.02

The improvement fork = 4 and 5 is still noticeable but, for Sh2 (0.1) and Sh3 (0.1) both have an error of 1%.

z

= 0.1,

182·

The reason why the high-order approximations fail to give an increased accuracy of ShM (z) beyond a certain M is that the approximations YM( T/, z) of (111) violate the neglected boundary condition ayM/ aT/ = 0 at Yi = 1/2-JZ to an alarming degree when M is increased. When z is 0.05, the level of attainable accuracy is reached for a larger M than when z is 0.1 This behavior of the approximate method is somewhat similar to that which is observed in an asymptotic series approximation of a given function, but the approximations ShM (z) do not appear to diverge for increasing M. A perfectly satisfactory solution of (65) and (66) is obtained, however, if the approximations of this subsection are combined with the Fourier series solution of subsection 4.4.1. A truncated Fourier series with three terms yields an approximation for Sh (z) with a maximum error of only 0.004% for z :::::: 0.05.

183

Construction of Eigenfunctions by Forward Integration

Solution of Linear Differential Equations by Collocatron

ShN (z) determined by sixth-order orthogonal collocation and by the first six terms of the Fourier series are compared with the exact function Sh(z) in table 4.5. TABLE COMPARISON

OF

COLLOCATION

4.5 AND

TRUNCATED

FOURIER

SERIES

z

Collocation N = 6

Fourier series six terms

Exact

0.001 0.002 0.005 0.010 0.020 0.050 0.10

14.9 13.06 9.48 6.83 5.160 3.92795 3.503935

13.7 11.99 8.911 6.7513 5.185078 3.927106 3.503894

18.865 13.634 9.039 6.7545 5.185081 3.927106 3.503894

4.4.3 Properties of the collocation solution

The Nth-order collocation method leads to a representation (93) for c y(z) with exactly N terms. The value of Sh (z) for z ~ oo is -~At where A 1 is the eigenvalue of smallest magnitude. Consequently the accuracy of

the asymptotic Sherwood number is solely determined by the accuracy of At· The average value of y is furthermore influenced by the first approximate Fourier coefficient Ct. Values of the first two (A 1, c;) are collected in table 4.4 for collocation at the zeros of P}$' 0 (x): TABLE

The collocation solution follows the exact solution reasonably well to

z

= 0.002, while the six-term Fourier series representation seems to

break down for a somewhat higher z-value. For z > 0.05, the truncated Fourier series is accurate to seven digits and the asymptotic value of Sh (z) is, of course, found exactly by only one term in the Fourier series. The first collocation eigenvalue is very close to At and there is an insignificant difference between the asymptotic Sherwood number found by one or the other of the two approximate methods.

4.4

FIRST- AND SECOND-COLLOCATION EIGENVALUE AND FOURIER CONSTANT

N

- A1

- Az

cl

Cz

2 4 6 8

5.1358 5.1215 5.1216691 5.1216693

34.6 40.19 39.674 39.661

0.807 0.78967 0.78970249 0.789702616

0.0257 0.09451 0.09737 0.09726

The values ~or N = 8 are accurate for all digits shown in the table. The high. ~igenvalues will also converge to their true values A~ with increasing N but at a slower rate. A general observation for problems of this type is that the first N/2 collocation eigenvalues are reasonably accurate approximations to the true eigenvalues. The high-order (i > N/2)· collocation eigenvalues become much larger in magnitude than the true eigenvalues, as seen by comparison of the results of table 4.3 and the output in A19 and A20.

4.5 Construction of Eigenfunctions by Forward Integration In section 4.4 we have established a frame of reference for evaluating the accuracy of collocation solutions to a particular linear partial differential equation: For z > 0.05, the Fourier series solution of (65) and (66) may be truncated after the fourth term with a negligible (0.004%) error; for z < 0.05, a penetration solution has been constructed that becomes increasingly accurate as z tends to zero. Considering the superiority for large z of the truncated Fourier series over any other approximate solution, it becomes a matter of great interest to compute the true eigenfunctions and eigenvalues to a high degree of accuracy. We only need the first few eigenfunctions since it is intended to use this representation of y for large enough z-values to make the series rapidly convergent.

184

Solut1on of Linear Differential Equations by Collocation

Chap. 4

The eigenfunctions Fj(x) and eigenvalues A; are given by the solution

Construction of Eigenfunctions by Forward Integration

with initial conditions

of

[F(O, A.) = 1 for all A.]

G(u = 0) = 0 d2Fj(x) 2 dx 2 - At(1 - x )Fj(x) = 0 Fi(O) = J1°(1) = 0

~~~u=O = O

(125)

d 2Fj(u) du 2 - A.iu(2 - u)Fj(u) = 0

= F(1) = 0

dA.

1. Insert a trial value A. for A. 1 [all eigenvalues of (126) are negative and -At< - A2 < ...]. 2. Integrate (126) with Fj(O) = 1 and FP\0) = 0 from u = 0 to 1. 3. If Fj(1, A.) = 0 and Fj(u, A.) has i - 1 sign changes in the open interval 0 < u < 1, we have succeeded in computing the ith eigenfunction Fj (u, A.J and the trial value A. is equal to A;. 4. If on the other hand Fj(1, A.) =1: 0 and Fj(u, A.) has i sign changes E (0, 1), then -A.~ < -A. < -A.i+t·

By this application of Sturm's equioscillation theorem, one may isolate each eigenvalue in a certain A -interval. Within these intervals the individual eigenvalues can be determined by a search technique: bisection, inverse interpolation on the values obtained for Fj(1, A.), or (most profitably) by Newton's method, which is excellently suited for this type of problem. Define G(u, A.) by

= aA.a [F(u, A.)]

(127)

The relation between the sensitivity function F and G is the same as that existing between (A.) and d / dA when solving an ordinary algebraic equation (A.) = 0 for A. In the present case, (A.) is the value of F;(1, A) and it is gi~en implicitly by the solution of (126) with F(O, A.) = 1, dF/dulu=O = .. Inserting (127) into (126) yields the following equation for G:

d 2G du 2 - A.(2- u)uG- (2- u)uF = 0

=

(126)

Equation (126) defines the eigenfunctions except for a scalar factor. We choose Fj(O) = 1. The solution of (126) can now be found as follows:

G(u, A.)

[ aF(u, au

A.)

Iu=O = 0

for all A.]

(129) (130)

The two coupled second-order initial value problems (126) and (128) are solved by integration from u = 0 to 1. Values of F(1, A.) and G(1, A.) determine the correction to A:

or introducing u = 1 - x by

F~ 0 (0)

185

F(1, A.) 0(1, A.)

(131)

The u-interval (0, 1] is divided into M subintervals of equal length h = 1/M, and the four first-order equations for It = F, l2 = dF/du, g1 = G, g2 = dG/ du that result from (126) and (128) are solved by one of the collocation methods of section 4.2 using N-collocation points in each subinterval u 0 = (K- 1)h < u = Uo + hv < uo + h. dlt = h/2 dv dl2 dv = hA.(2- u)uft

(132) dgl = hg2 dv dg 2 = hA.(2- u)ug1 + h(2- u)ult dv

The first u 0 -value at K = 1 is zero and It = 1 while l2 = gt = g2 = 0

(133)

Values of the four dependent variables at u = u 0 = (K- 1)h are / 2 (u 0 ) = l2o, etc., and the discretization of (132) yields expressions similar to (59) and (60) for l 1 (u) ... at the collocation abscissas v:

f 1(u 0 ) = / 10 ,

Aft+ I10Ao = hf2 Af2 + hoAo = hA Uft

(134)

Agt + g10Ao = hg2 Ag2 + g2oAo

= hA. Ugt + hUft

U is a diagonal matrix with

(128)

Uu

=

(2 - Uo - Vth)(uo

+ Vth)

(135)

186

Solution of Linear Differential Equations by Collocation

Chap. 4

The first and third equations are solved for f2 and g2 , respectively. These are substituted into the second and fourth equations and one obtains

Equation (136) is first solved for f 1 and (137) is thereafter solved for g1• Altogether two (N x N) systems of linear algebraic equations must be solved to obtain the values of F and G at the collocation points of the Kth subinterval. The coefficient matrix of (136) and (137) is the same and the reduction of this matrix to an upper triangular form needs to be done only once per subinterval. Matrix A 2 and vectors (AA0 , A 0 ) are calculated only once, at the start of the integration. The diagonal matrix U depends on the value of the independent variable within the current subinterval and it has to be evaluated at each of the M steps from u = 0 to u = 1. At each K, the right-hand side of (136) is recalculated using the right-hand interval end point ordinates from the previous step to give the scalars / 10 and / 20 • When (136) has been solved for f1, the right-hand side of (137) can also be updated. A slightly different formulation is required if the Lobatto method [with auxiliary functions fi(v) ... that are all zero at v = 0-i.e., u = u0 ] is used, but the system of equations has the same properties as (136) and (137). Once the eigenfunction ~(u, At) has been determined with sufficient accuracy, the Fourier coefficients c~ of table 4.3 can also be found. In each interval, u 0 < u < u 0 + h, ~(u, At) is given at the quadrature points of an optimal quadrature formula and the integrals that enter into b ~ and c~ are easily evaluated. b* =

' d

Jf ~(x)(1 -

So FT(X )(1 =

x

- x

2 ) 2

)

~b~ rF;(x)(l -

dx dx

Sec. 4.5

187

Construction of Eigenfunctions by Forward Integration

If this method should succeed to determine A with a high accuracy, it is imperative that the method of integration of (126) to (128) produces reliable results for f 1(u = 1) and g 1 (u = 1). Otherwise the correction (131) will not lead to a better A -value when AA is in the range of uncertainty of / 1(1) and g1(1). With N interior points in each subinterval the error per step is of the order of h 2 N+ 3 with the Lobatto method. The total number of steps is proportional to h - 1 and the overall error of integration is expected to be 1

(J(h2N+2).

This is verified in figure 4-1, which shows the error of the first eigenvalue as a function of M for N = 1, 2, 3, 4. The straight lines on the log-log plot all have a slope a = -2N- 2 forM > 2 to 3. Log IA. 1

-

A. 1 (N, Ml I

0.5

1.5

Log M

-5

-10

(138)

2

-15

x ) dx

00

y(x, z) =

L h~~(x) exp (Aiz) 1

co

y(z) =

(139)

L c~ exp (Aiz) 1

The Lobatto method was applied to compute A~ and c~ of table 4.3.

Figure 4-1. Error of first eigenvalue of equation (4.125) by an initial value integration technique using different number of subintervals M and different number of collocation points N in each subinterval.

The same dependence of accuracy on N is found for the higher eigenvalues and for the expansion coefficients c~.

188

Solution of Linear Differential Equations by Collocation

=

The total mass balance (141) is integrated from z

4.6 An Ordinary Differential Equation at the Boundary of a POE In the desorption problem considered in sections 4.3 to 4.5, the concentration of the volatile component in the gaseous phase x ::; 0 is assumed to be 0 and gas-phase mass transfer resistance is not taken into account. These assumptions are likely to be valid for desorption (or absorption) of very insoluble gases. If a more soluble gas is desorbed, mass transfer resistance in the gas phase should be taken into account. Furthermore the concentration of the desorbing component in the gas phase also changes in the z -direction provided the flow rate of gas relative to that of the liquid is not extremely high. The liquid-phase concentration profile would still be described by

189

An Ordinary Differential Equation at the Boundary of a POE

Chap. 4

0:

(143) Y + q · Yg = Yz=O + q · (yg)z=O = 1 + q · YgO For cocurrent flow, yg 0 is known, Ygo = Yg,in· For countercurrent flow,

however, yg 0 is unknown. Equation (143) may be applied to express yg by yg = Ygo andy = 1 at z = 0: 1 yg = Ygo + -(1 - y)

y and

the conditions (144)

q

Discretization of the liquid-phase equation at N interior points X; and at the interval end points x 0 = 0, xN+t = 1, and subsequent collocation at the interior points leads to the collocation equation (83): (145)

(65):

(1 _ x 2 ) ay = ily

az

and the boundary relations

ax 2

N+l

X=

and two of the boundary conditions remain unchanged: y

= 1 at z = 0

ay =

ax

o

L

0:

at x = 1

X=

L

1:

j=O

where BiM (see section 1.3) is a modified gas-phase mass transfer coefficient and y g is the liquid concentration that would correspond to equilibrium with the partial pressure of the desorbing component in the gas phase. A total mass balance for the two streams yields

Yg = Ygo

dz

=0

1

+ -(1 q

- y)

= Ygo

1

- q

1

N

L

,=1

WiYi

+-

q

where wi are the Gaussian weights for integration of ~(1 - x ) • y. Radau quadrature is not used in this case, where the dependent variable is given implicitly at the end points. The boundary relations may be written 2

•

+ qdyg = 0

AN+l,iYi

From (144), (140)

dy

(146)

N+l

The third boundary condition is, however, changed to

dz

Ao,iYi = BiM(Yo - yg)

]=0

(Ao,o- B1M)yo (141)

+ Ao,N+tYN+l = j=l _LN ( -

Ao,i

) + -BiM w i Yiq

. ( B1M Ygo

1)

+q

N

where q is a capacity ratio for the gas and the liquid stream. For countercurrent flow, q is negative; fot cocurrent flow, q is positive. The concentration of solute in the inlet gas stream is normally specified. '!;'his additional boundary condition is different for countercurrent and coci.trrent flow: Countercurrent: Cocurrent:

yg = Yg,tn yg

at z = h

= Yg,in at z = 0

where h is a dimensionless column height.

(142)

AN+t,oYo

+ AN+l.N+tYN+l = L j=l

AN+l.iYi

These relations are used to express y 0 and YN+l by a linear combination of interior ordinates Yi and the term BiM[Ygo + (1/q)]. The resulting expression is substituted into (145) to give a set of equations: d

dz y = C · y + d · [ygO + (1I q)] with initial conditions y = 1 at z = 0.

190

Solution of Linear Differential Equations by Collocation

The solution is

Lz exp (- Cx) · d · [yg

y = exp (Cz) • 1 + exp (Cz)

= exp (Cz) · 1 + [exp (Cz)- I]·

c- 1 · d

· [ygo

0

191

Chap. 4

+ (1/q)] dx

+ (1/q)]

(147)

(provided C is nonsingular). For concurrent flow we simply substitute yg 0 = Yg, m but for countercurrent flow the situation is more complicated. Ygo must be determined from the side condition. At z = h:

2. Double-pipe ordinary heat exchangers or chemical reactors with radial and axial temperature gradients and cocurrent or countercurrent flow in the outer tube. 3. Breakthrough curves for fixed bed adsorption under constant pattern conditions (see exercise 9.7).

EXERCISES

1

and or

1

Yg,in = Ygo + -(1 - Yz=h) q

Yz=h = 1 + q(ygO - Yg,in)

1. In this exercise we wish to demonstrate that for dy/ dx = Ky the coefficients Mf) of equation (4.28) are correct when k ::;; 2N + 1 - i, as stated in the text. a. Calculate exact expressions for the first.three coefficients b~ of the infinite series 00 y = exp (Kx) = 1 + L b~[P~o.o) + P~~~)] 1

Letting z = h in (14 7), this yields 1 + q(ygo- Yg,m) = w T exp (Ch)1 + w T [exp (Ch) - I] c- 1 d ( Ygo +

= wr{ exp (Ch) + [exp (Ch)

1

- I]C- d( 1

q·1')

b. Solve equation (4.17) for N = 1, 2, and 3 to obtain b~n (i = 1, 2, 3; j = i, i + 1, ... , N) where h;n is the (j - i + 1) approximation to b~. c. Expand b~n and b~ in powers of K. Confirm that the following terms of b~n are identical to the corresponding terms of b~:

+ ~)} (148)

b~n

=

K'(n1

+

nzK

+

plus incorrect terms starting with K

+ wr[exp (Ch)

2

2

+ ... +

nzi-ziKZi-Zi+l)

i-•+Z.

1

- I]C- d(ygo - 1)

The first term on the right-hand side of (148) is the solution that would apply if yg 0 was 1. In this case, all driving forces would be zero, and the solution would be y = 1 for all z or y = 1 for all z. We thus obtain q(ygo- Yg,m) = wr[exp (Ch)- I]C- 1 d(ygo- 1) = e(ygo- 1)

or Yo= g

n 3K

q · Yg,m - e q- e

where e = wr[exp (Ch) - I]C- 1d

The problem treated in this section as a relatively trivial extension of the standard problem type of the previous three sections is quite commonly encountered in practice. A part~!' differential equation with one boundary condition given as an ordinary differential equation has been discussed several times in chapter 1. Typical examples are 1. Adsorption into a solid phase or extraction from a solid phase with a limited outer volume.

2. The choice of trial function and the choice of the method of moments with a rather unfamiliar weight function in derivation of the Radau method (4.14) to (4.28) does involve a large portion of hindsight: The collocation method at the zeros of P~· 0 )(x) was found to give excellent results for YN(1) when applied as shown in the text, (4.35) and (4.36), and an identical "classical" MWR was derived afterward. Whereas there can be no doubt that the highest possible accuracy is found with the proposed procedure [the accuracy corresponds exactly to the number of free parameters in the resulting Pade approximation for exp (K)], one might well ask why the method of moments was used rather than the Galerkin method, which was found to give very accurate results in chapter 2 for an apparently similar problem. The reason why a Galerkin MWR is less suitable with our present goal-to approximate y(1) well-is given in this exercise. a. Prove-by manipulations on 'Yi of formula (3.8) or more simply by using the orthogonality properties of the polynomials-that

b. Consider the residual RN(a, x) of dy - - Ky = 0 dx

when

y(O)

=

1

Solution of Linear Differential Equations by Collocation

192

Chap. 4

Show that collocation of the zeros of P~· 1 l(x) will give the same value of ai as Galerkin's method. c. If the collocation ordinates are used in a Radau quadrature similar to (36) but now including the known value y(O) as the extra quadrature ordinate, one might think that y(1) is obtained as accurately as in the Radau method of the text. What is the result for N = 1? d. In the present method YN(1) is obtained similarly to (4.31): K YN(1) = 1 + K + 2a1 The disappointing result of part accuracy of a 1· Find the structure of A and B, method. What is the accuracy of The result for N = 1 and N = 2

a (11)

c is explained by consideration of the

dyl dx

=

K = --

1- tK'

ax

ax.

4. Consider the end point collocation method with internal collocation point x1. Write the two collocation equations in terms of Y1 = y(xt) and Yz = y(1) for a general f(x, y ). Show that -

Yo =

1

2(1 - X1)

[1 - xlhfy(Xo, Yo)](y1 - Yo)

h[f(x 0 + x 1 h, y 1 ) + (1 - 2xt)f(xo + h, Yz)]

where Yo = y(xo). . . . It is desired to eliminate y 1 from the expression. We wish to avmd usmg f taken at y = y 1 and y = y2 • The elimination follows in two steps: (1) Collocation at x but with an interpolation polynomial in Xo and x 1 gives one equation for f(x:, y 1). (2) The first term of a Taylor series from (xo, Yo) gives equatrql}s for f(x 0 + Xth, Yt) and f(xo + h, Yz). . . Derive';he following expression for y2 and show that the approximatiOns have not resulted in a decrease of the order of accuracy for the method.

J

1 - 2x1 [ 1 - 2 (1 _ x ) hfy(Xo, Yo) (yz - Yo 1

)

=

x1h[(Xo· + hxt. Yo)

Wz

Show that x 1 = 1 is a particularly suitable collocation point and obtain the following general eJ(h 3 ) method: [I - (1 - tJ2)hA](y1 - Yo) = (1 - Wz)hf(xo + hxt. Yo)

[I - (1 - tJ2)hA](yz - Yo)

~ ~!1[

Xo

+(

1- ~)h, y

= V2 y -

y(O, x) = 1,

<1>

2

2(1 - X1)

+ (v'2- l)l(x 0 +

h,y

0 ))

y with the following side conditions:

y< 1\t, 0)

= 0 and y
and in the three geometries. The solution should be computed at x = 0(0.1)1 and t = 0.1(0.1)3. Case a. BiM ~ oo, = 0. Case b. BiM a given value < oo (e.g., 2, 5, 10, 50, 200). In each case the average value of y = y should be compared to the curves in Crank (1957). Case c. ~ 0 (e.g., 1 and 2) and otherwise as for cases a and b.

6. Determine the first two eigenvalues and eigenfunctions for the model of Exercise 5 using the forward integration technique of section 4.5. Case a and examples of cases b and c should be studied. Compare the approximate eigenfunctions with their known analytical expressions for plane parallel geometry.

7. The usual procedure for calculating a penetration solution is to assume in advance that the two independent variables can be compounded into a single variable 'TI· This is done in Mickley, Sherwood, and Reed (Applied Mathematics in Chemical Engineering) and in Jenson and Jeffreys (Mathematical Methods in Chemical Engineering), two standard texts for chemical engineering students. In fact this procedure might obscure the more general technique that is discussed in subsection 4.4.2. Consider diffusion from a semi-infinite medium y(x, 0) = 1 = y(oo, z)

h

1]

where f = f(xo, Yo) and A = (iJ{;/oyi)xo,yo· This is Rosenbrock's method, one of the semi-implicit Runge-Kutta methods discussed in section 8.2. What are the computational operations that are involved in each integration step? s~ Solve oy/ot

Y1 + Yz

Yz(O) = 1 dyz ( ) - = Y1 + J - X Yz dx by the Gauss and the Radau methods using = 1. Make a computer program that employs the Lobatto, the Radau, or the Gauss method to solve (48) with an arbitrary Find an approximate relation between the error at x = 1 and ax.

y2

where y 1 is given explicitly by

the matrices of (4.17), for the present a 1 with anN-term approximation? is

3. Calculate the solution of equation (48) for x = 1: -

193

[f(x 0 + hxt. y 1) + (1 - 2x1)f(xo + h, Yo)]

which has the exact solution y

= erf (xj2.J;).

and

y(O, z)

=0

194

Solution of Linear Differential Equations by Collocation

195

Proceed as in subsection 4.4.2 (17 = x/zq, z 1 = z) and show that the first perturbation function f 0 is erf (x/2h), while all higher perturbation functions !1> [ 2 , ••• are identically zero.

8. The differential equations (115) for the perturbation functions fk were solved analytically in the text. Expansions similar to (111) may be obtained for other entry length problems where an analytical solution of the perturbation equation is not feasible and a numerical solution of (115) is therefore of interest. Compute a numerical solution for fk of (115), k = 0, 1, 2, 3, 4 by either of the following techniques. a. The boundary condition at 17 ~ oo is replaced by a boundary condition at 17max and the differential equation is solved by collocation on the resulting finite interval [0, 17maxJ. Discuss the effect of the choice of 17max· b. The semi-infinite interval [0, oo] for 17 is transformed to a finite interval [0, 1] by a suitable coordinate transformation, e.g., x = exp ( - a17 ), and the transformed equation is solved by collocation. Discuss the effect of the parameter a.

O(l,z)

ao)

ao = 0 and -(O,z) = 0 for ax

=

I

0

00

z

Their final model for the blood phase (capillary) is the same as used in Exercise 9, but a flux balance at the boundary capillary-tissue should be used to tie the two phases together. Only radial diffusion needs to be considered. Compare your computed results with figure 2-4 of the reference. Hint: The problem is solved in two parts. First the ordinary differential equation for the tissue phase is solved with an arbitrary value of the interphase concentration. The result is the flux at the interphase except for a scalar factor. The collocation equations for the capillary phase can now be set up, and the unknown interphase concentration eliminated using the flux condition at the interphase.

(1)

Integrate this equation by the method of section 4.3 to obtain the mean value 1

JI

00

0

df.· 1 v'~ + -17(2- 17v) I iv'[; d11 3 0 .

00

ax

ao ax

11. Davis, et al. (1974) have solved a model for capillary-tissue mass transfer.

> 0

ao a2 0 (1- xn+l)- = -2 0(0, z) = 0 = -(1, z)

dx

ay)

10. MasheJkar, et al. (1973) have made a calculation similar to that of section 4.3 but fo'f a power law fluid (see section 1.1). In the nomenclature of the reference one obtains

az

(}(l - xn+l)

o

and compare with the values obtained by Mashelkar for n = 0, 0.5, 1 (a Newton fluid), 2.5, 10 at z = [0, (0.02), 0.3].

1

d2f. [ 1 v'-;2 = - -17 2(2- 17v) +vI (17v)' d17 3 0

1

D,.L- -1 -a-(x - -Day 2 ay 2(1x)-= a( R 2 Vav X ax ax

can be solved by the method of subsection 4.4.2 combined with the collocation technique of Exercise 8 for the higher perturbation functions. Compute the first four terms of the Leveque series for . J ~ - 4 s~ (aojax)ix=l dz and compare with Newman's result (1969), whtch IS given in chapter 9 [(equation (9.19))]. Hint: A reasonable 17 is (1- x)/zq. It appears that the transformed equa-_ tion in variables (17, z 1 ) is simplified best when q = !. Next v = z~ 13 is introduced as a new variable and finally the perturbation functions appear from the following equation: 00

J

chemical reaction in an empty tube is presented in equation (1.34) of chapter 1:

ao = .!_ _i_ (x az x ax ax

O(x, 0)

2 n +(}- = n +1

12. The following model for laminar Newtonian flow with first-order isothermal

9. The Graetz problem in cylinder geometry (1 - x2)

Compute G = 1 - if where

and

O(x, 0) = 1

y(()

= 4 J x(1 -

2

x )y(x, () d( =

I

A; exp (A;()

0

(2)

Verify the results in table 1.4 and also verify that Al = - Da (1- Da R2Vav) +Dad Da R2vav)2 48D,.L v\ 48D,.L

A1 = 1 +

A;=

~

~

2

R v Da--a_v 48D,L

)2

2

R v Da--a_v 48D,.L

13. Modify the boundary condition at 0

y )(t, 1)

qyb +

)2

(3)

fori> 1 x = 1 of Exercise 5 to

+ BiM [y(t, 1)]

y=

1 and

=

BiM yb(t)

Yb (t = 0) = 0

The model can now be taken to represent leaching of a solute from a solid into a solvent of finite volume q per .unit volume solid. Yb is the solute concentration in the solvent outside the solid and y is the volume averaged concentration on the solid.

196

Solution of Linear Differential Equations by Collocation

197 a. Compute y(t), yb(t), and the extent of the leaching process: E =

1 - y(t) = q + 1 [1 - y(t)] 1 - y(t ~ oo) q

To check the program, insert q = 1, plane parallel geometry, BiM ~ oo, and <1> = 0. Now E can be compared with figure 4.6 in Crank (1957). b. Modify the program to design an isothermal cocurrent desorption from a liquid film in laminar downward flow into a gas phase of volume q per unit volume liquid. As a result the distance to achieve 99% desorption may be plotted as a function of G. Gas film transfer resistance may be neglected and for convenience the solute may be assumed to be equally soluble in the gas and liquid phases.

14.

(i) Use the material of subsection 1.2.3 to derive the following transient model for a tubular reactor. Radial gradients are neglected but axial dispersion is taken into account. The reaction is first order and isothermal ay ay 1 o2 y -+----+D ay= 0 OT oz PeM oz 2 Z

= 0:

1 ay Y - -PeMaz z

=

Yin (T)

= 1: ay = o az

T

= 0:

y

=

Yo(z)

For simplicity let Yin(T) = 1 and Yo(z) = 0. (ii) Find y(1, T) for Da = 1 and PeM = 2, 5, 20, and 50. Hint: Proceed as in section 4.3 with collocation at the zeros of P~·o)(z ). For certain combinations of N and PeM complex eigenvalues may be obtained and you should consult exercise 1.11. (iii) Derive an analytical expression for the jth eigenvalue and compare with the numerical results. Find limPeM-+O .A 1 and limPeM-+oo .A1 and use these results to explain the behavior of the approximate solution for large and small PeM. (iv) Derive an analytical solution for the transient in the two cases PeM ~ 0 and PeM ~ oo.

of Wilkinson (1965). Program packages of this type are available by now almost everywhere. The methods of section 4.2 for solving initial value problems are to well-known implicit or semi-implicit methods. Some references given in chapter 8. The end point collocation methods of subsection 3 were probably first proposed by Villadsen (1968). They are further dts:cussed in Villadsen (1970). Bird (1960), section 17.5, and Finlayson (1972), section 3.4 have both

u~ed the ex~mple of ~ection 4.3. The asymptotic behavior of the large eiiJren,vahJes In subsectiOn 4.4.1 can be derived following a method discussed in Courant and Hilbert (1968). The presentation of the penetration soJ:uti'c.m .in sub~ection 4.4.2 closely follows Newman's derivation (1969) of a sumlar senes for the Graetz problem in cylinder geometry that we have included as Exercise 9. The model appears to have a considerable interest in the design of trickling filters and related units for biological water treatment. An application is given in Exercise 8.13.

1. WILKINSON, J. The Algebraic Eigenvalue Problem. Oxford: Clarendon Press (1965). 2. VILLADSEN, J. Transactions Norddata 68, pp. 138-75, Helsinki (June, 1968). 3. VILLADSEN, J. Selected Approximation Methods for Chemical Engineering Problems. Lyngby, Denmark: Instituttet for Kemiteknik (1970). 4. BIRD, R. B., STEWART, W. E., and LIGHTFOOT, E. N. Transport Phenomena. Wiley (1960). 5. FINLAYSON, B. A. The Method of Weighted Residuals and Variational Principles. New York: Academic Press (1972). 6. COURANT, R., and HILBERT, D. Methods of Mathematical Physics. New York: Interscience Publishers (1953). 7. NEWMAN, J. J. Heat Transfer 91 (1969): 177. 8. CRANK, J. The Mathematics of Diffusion. Oxford: Clarendon Press (1957). 9. MASHELKAR, R. A., CHAVAN, V. V., and KARANTH, N. G. Chemical Engr. Journal 6 (1973): 75.

REFERENCES

eli~iriation

The Gauss program of section 4.1 is similar to many other programs that use row and column pivotation. It may be slightly better to use a LU decomposition program, which is also standard at most computing centers. The program EISYS of section 4.3 for diagonalization of a nonsymmetric matrix by the QR method is made following the recommendations

DAVIS, E. 1., COONEY, D. 0., and CHANG, R. Chemical Engr. Journal 7 (1974): 213.

199

Nonlinear Ordinary Differential Equations

Introduction The examples of previous chapters have generally bdee? line at~~~h~ ~n~~ . . h . 1 onlinear problem use m sec I . exceptiOn bemg t e 1stmpll e nt' by comparison with an approximate introduce orthogona co oca ton Galerkin method. . h · tants or A linear differential equation is charactenzed by .avmg cons

explicit

fu~~tionsdo!1~h~:~e~~:~~e~! vaa~~~\~n~':':i~~~:~~~o et~~a~:'!e~~

~::s~ ~~~aof ~h:~e coefficients is a function of the dependent vana~~~er­

Nonlinearities do not in principle introduce new faspect~s of 5th4e and 5 5 . h' t The examples o sec tons . · ical methods discussed ~~ t Is. tex ·h. ch the derivatives occur linearly is ·n t ate that an equatiOn m w I . 41 I us r .. 1 d'fication of the algorithms of sectiOn . ' ~~:t~~~~ :o~~~~;~~~~~e:~o~linea~ differential equations have been . 1 d b MWR especially by collocatiOn. so v~he kyout the iterative calculations to solve :he no~linear ~~ebraic equations that~ ¥sult from discretization of a nonlmear dtffe~~nti r:q~~~ tion does however, vary significantly from problem t~ pro t~m. 5 5 it . ' f the Weisz-Hicks problem m sec ton . ' roach to the numerical numencal treatment 0 becomes apparent that different ~e~?o~; ~fc:~!es of widely different

of

c:c~lationsTh~a~a:~~dpa~~ oft:~~~;:tational ~ark is, however, always ~a~=~~y~ubroutines that have been described in chapters 3 and 4. 198

The outstanding difference between nonlinear and linear differential equations is not of numerical nature but is of a much more fundamental character. The qualitatively different response of nonlinear and linear differential equations to a change in side conditions is well known from elementary textbooks, but the frequent occurrence of linearly independent multiple solutions to nonlinear differential equations in certain parameter intervals is an intriguing feature of these equations that is not at all well studied in theory or by computer experiments. The occurrence of strange phenomena in physical systems is a major incentive to use nonlinear models in an attempt to obtain bounds on the regions where these phenomena, e.g., unstable flow in a heat exchanger or flickering of a catalyst between several activity levels, occur. Only nonlinear models can give any hope of a mathematical explanation of most of these phenomena. It would be highly desirable if without actually solving the equation one could decide whether any given nonlinear boundary value problem may have multiple solutions in some domain of its parameter space. This could help the investigator to avoid wasting time by computer solution of a model that either cannot explain a certain phenomena at all or can only do so within a certain parameter range that may go undetected in. the numerical experiments. The present state of development of functional analysis and related topics of applied mathematics does not give much encouragement, however, for this rational approach to the model analysis. A large number of investigations have recently appeared in chemical engineering literature where topics from functional analysis such as fixed-point theory have been applied to estimate the size of parameter regions where multiple solutions may exist. We have chosen to illustrate this type of preliminary model analysis by means of comparison differential equations since these are extremely simple to apply and they may lead to approximate solutions that give much insight into the actual solution of the given nonlinear model. Two theorems from the theory of differential inequalities are stated without proof in section 5.2 and the few examples that illustrate the application of these theorems may whet the appetite of those who want to study this promising field of functional analysis more closely. The paucity of the results of general nature concerning nonlinear boundary value problems does, however, emphasize the usefulness of efficient numerical methods for solving the actual problem. In the forseeable future we must expect nonlinear differential equations to be discussed on the basis of individual examples solved on a computer and with only a preliminary guidance from theoretical mathematics. For this reason the various numerical tricks that are mentioned in connection with the Weisz-Hicks problem in section 5.5 may be of value also beyond this example since they represent the semianalytical

Nonlinear Ordinary Differential Equations

200

approach to numerical work that is necessary in order to reac~ the of~en almost inaccessible regions of parameter space where the most mterestmg features of a mathematical model are displayed.

Multiple Solutions of a Trivial Boundary Value Problem

Since YsiYo > 1, a unique (and positive) solution for xs is determined from (5). This value is inserted into (6) and (7), and the (unique) solution B) of the two linear equations is ihserted into (8). The solution of (8)

5.1 Multiple Solutions of a Trivial Boundary Value Problem Consider the following problem: d2 dx;_ + f(y) = 0, f(y)

=

dy = 1 and -

2 { -<1> Kiy -<1>2K~y

K) _ 1t-l1 - 4AB) exp ( 1 A

= y

for Ys :S y :S 1 forO:Sy:Sys

y = A exp (K1x)

where A > 0 and 4AB is either negative or less than 1. The plus sign always be chosen and a unique (positive) value of is determined (9). The numerical results (y 0 ) are shown in figure 5-1 for K 1 = 4, 2 = 20, and Ys = 0.6.

+ B exp (-<1>K1x) for Ys :S

0.25

0.2

0.8

0.6

0.1

0.4

0.05 Vo

0.2

0.4

Ys = 0.6

0.8

Figure 5-1. Solutions (
for Yo :S Y :S Ys Y :S 1

Kb K 2, and Ys are given besides y 0 , and the four constants , Xs, A, and B of (3) and (~ can be determined from

= Yo cosh (K2xs) A exp (K1xs) + B exp (-Klxs) Ys

Ys =

(9)

(1)

It will be shown that (1) has either one or three nonnegative solutions for x E [0, 1] while the linear equation with f given by (2) has one and only one solution y(x). . The differential equation is solved in the same way whether f(y) IS given by (1) or by (2). . .· A value y 0 is chosen for y(x = 0) and the corresponding value IS calculated. For y 0 < Ys the solution of (1) is:

y = y 0 cosh (K2x)

112

2

= 0 for x = 0 dx Kb K 2, <1>, and Ys are given constants, and we are interested only in a solution that is contained in the closed interval 0 :S y (x) :S 1. The differential equation (1) is nonlinear since f(y) = /1(Y )y where the coefficient ft (y) to y is a function of the solution y and not an explicit 1 function of x. The second derivative is discontinuous at Ys, but Y and l ) are both continuous functions of x. A similar but linear differential equation appears when f(y) is given by -<1> 2Kiy for Xs :S X :S 1 f(y) = { -<1> 2K~y for 0 :S X :S Xs

y(1)

201

y 0 K 2 sinh (K2xs) = A exp (K1xs) - B exp (-Klxs) K1 1 = A exp (K1) + B exp (-K1)

2 For Yo > Ys, f(y) of (1) is -<1> Kiy for all x simply y = Yo cosh (K1x)

E

=

[0, 1] and the solution is

with = __!_arc cosh(_!__) K1 Yo

(10)

For any Yo < Ys, the method of solution described in (5) to (9) must be used. When Yo ~ 0, increases to infinity while Xs ~ 1. The interesting feature of figure 5-1 is that y 0 is a multivalued function of <1>, while Xs as well as are single-valued functions of y 0 • Xs increases from 0 to 1 when y0 decreases from Ys to 0.

202

Nonlinear Ordinary Differential Equations

The following conclusions are drawn from the calculations: 1. The initial value problem (1) given by y 0 , y(l)(O), y(1), Kh and K 2 has a unique solution [<1>, y(x)] that passes through (x, y) = (1, 1). 2. The boundary value problem (1) given by yC 1)(0), y(1) = 1, Kh K 2, and has three solutions [y 0 , y(x)] in a certain range:

Multiple Solutions of a Tnv1al Boundary Value Problem

203

The Thiele m?dulus <1>, dimensionless activation energy y, and adiabatic · temperat~re nse {3 have all been defined in chapter 1. Figure 5-2 shows the functiOn F(y) for y = 20 and {3 = 0.8. F(y)

<1>1 < < <1>2

The upper limit of multiple solutions for (Kh Ys) = (4, 0.6) is easily found: <1>2 = ; 1

arc cosh ( _!_) = 0.27465 Ys

The lower limit of multiple solutions can only be found iteratively: <1>1

=

0.0884

at Yo

=

0.45

3. The uniqueness of the linear boundary value problem (2) follows

20

from the uniqueness of the initial value problem (1) and the monotonicity of X 8 (y 0 ): For a given parameter set Xs, , Kh K 2, only one y 0 value is found.

y

0.8

The very simplicity of the present example allows some important aspects of a nonlinear problem and its numerical solution to be studied without an undue burden of numerical work. We note the uniqueness of the solution of the corresponding initial value problem: Numerical integration of a series of initial value problems with implicit determination of one of the problem parameters (here cl>) permits a tracing of all possible solutions to· the problem. Also the occurrence of multiple solutions in a finite interval of a parameter (here the Thiele modulus), which is quite typical for a nonlinear boundary value problem, is found even in this simple model. In the present example as well as in more complicated problems (b <1> 2) can be determined with any desired degree of accuracy by numerical solution of the differential equation. Here (b <1> 2) was by a relatively uncomplicated method that to a certain degree (e.g., iteration on y 0 to determine <1> 1 or <1> 2) can be applied also when no analytical solution of the differential equation is available. The solution of the trivial model (1) gives some insight into the much more complicai~,model (1.69)-(1.70) that is discussed in detail in section 5.5. For a first-order irreversible reaction, plane parallel geometry, and both BiM and BiM ~ 1: d2y 2 [ dx2- y exp y{31

1- y

] -

and

y(l) = 1

+ {3(1- y) - 0

y 0 )(0) = 0

Figu_re 5-2. ~e function F(y) of equation (5.13) with (y, {3) = (20, 0.8) and 'Its approximate representation [equation (5.14)].

F(y) = {exp

[yf3

1- y ]} 1/2 1 + {3(1 - y) An approximate representation of F(y) by

(13)

F(y) = { 4 0.6 < y ::5 1 (14) 20 0 ::5 y < 0.6 also shown on the figure. Equation (12) ~ith the approximation (14) for F(y) is exactly (1). J:.q:uat:ton (12) admits to three solutions for

0.0691 < < 0.250

(15)

it is seen that the region of multiple solutions obtained from (1), 0.0884 < < 0.275

(16) quite close to the region obtained from the exact solution. Changing from 20 to .30 and Ys from 0.6 to 0.56 yields almost the same region of muJtupte solutiOns for the two equations, but no manipulation on the ppr~oxi.mat~ model can, .h?wever, give more than a qualitative impression the solutiOn to the ongmal nonlinear problem or its region of unicity.

204

Nonlinear Ordinary Differential Equations

True upper and lower bounds <1> 1 and <1> 2 for may be obtained the methods of the next section, but the reader should note that actual values of <1> 1 and <1> 2 are found with many digits accuracy by methods of section 5.5 without excessive numerical work.

Existence and Uniqueness Theorems

205

assumed that the solution u 1 and u 2 of (20), (21) with the conditions of (17) is unique in the x-interval [a, b] or in any part of this interval. nder these conditions on be proved:

t

and on (ub u 2) the following theorem

5.2 Existence and Uniqueness Theorems We consider a second-order differential equation of the following form:

Theorem 1: 0 y, y

)]

There exists at least one solution of (17) in the given region, and any solution of (17) has the property that u1(x)

2

d y

dx2

+ t[x, y, y

(1)]

=0

It is assumed that t is continuous and that it has bounded derivatives with respect to y and y (1) for all x in the interior of x-interval [a, b] and for such y and y 0 ) that are of interest in the · problem (e.g., positive y when y is a dimensionless concentration is said to catalyst pellet). With these assumptions the function Lipschizian. The following four constants K1, K 2, L1, L2 limit the of at/ ay and at! ay (1) within the open x-interval (a, b) and the (y, interval considered:

t

The boundary conditions of (17) at the two interval end points x and x =bare y 0 )(a) y

0

+ Ay(a) = )(b) + By(b) =

2

dx 2

2 d u2 dx 2

(

)

+ h1[x, Ut, u 11 ]

throughout the [x, y, y(

1

1

)] ::5

(24) • • • bl G;[ y, y (1)] Is given m ta e 5.1 by means of the constants in (18): 5.1

0

y

y(l)

~o

~o

~o

$0

$0 s;;O

s;;O

Gl[y, y
~o

Gz[y, /ll]

+ L1Y< 1) + L 2 y< 1) + L1y 0 ) + Lzy 0 l

Kzy Kzy K1y K1y

(l)] _

+ h2[x, u2, u2

t[x, y, y 0 )]

::::;

(23)

u2(x)

t

TABLE

+ Lzy(l) + L1Y< 1) + L 2 / 1l + Lly
- 0

The sid.,cpnditions of (20) and (21) are to be the same (19) as applying to the given differential equation (17) and the two functions and h2 will satisfy the following inequalities: h1[x, y, /

<

FUNCTIONS G1 AND G 2 OF (24) IN DIFFERENT REGIONS [y, y 0 )]

C2

_ -

y(x)

choice of comparison differential equations is completely unreas long as (22) is satisfied. Practical considerations do, however, the choice: In order that the uniqueness of the solutions u 1 and u of 2 comparison equations can be easily determined, a rather simple form and h2 is preferable. It is shown in section 5.3 that (23) can be used an approximation to y (x ), often in an iterative procedure. also calls for a simple, linear form of h 1 and h 2 • As a final ........................·'"" ....... h1 and h2 should be as close approximations to as without sacrificing the simplicity of (20) and (21). In this case and u2(x) may give close bounds on y(x). A specific choice of hJx, y, y 0 )] (i = 1, 2) of (20) and (21) is

C1

We compare the solution y of (17) and (19) with the solutions u 1(x) u 2 (x) of the following comparison differential equations: d u1

<

h2[x, y, y( 1 )]

A theorem on uniqueness of the solution to (17) is based on the G2 of table 5.1:

Theorem 2: Provided the conditions of Theorem 1 are satisfied, the solution of (17) is unique if the linear differential equation 2

)]

interval to be investigated in (17).

d u

dx 2

+ G2[x, u, u

(1)

_

] - 0

(25)

with boundary conditions

+ Au(ai) = 0 u < \b 1 ) + Bu(b1) = 0

u 0 )(a1)

207

Application of Comparison Differential Equations

Nonlinear Ordinary Differential Equations

206

ue of (26) is A l1 ) and A l2 ) with -A }1 ) < -A }2 ), respectively, then parg~t pro~lem (17) has a unique solution for a larger K in the case = A1 than m the case A 1 = A }1 ).

1

has no solution except u = 0 for any (ab b1) E (a, b). Theorems 1 and 2 are the major results of the theory of differentia inequalities. Detailed proofs of the theorems are given by Bailey, et (1968) in chapters 5 to 7 for the case of (A, B) ~ oo in (18) (y known both interval end points) and for either (A, B) = (oo, 0) or (A, B) 0 (0, oo) [y known at one interval end point and y< at the other end An extension to the more general boundary value problem with (A. B) is, however, obvious from the proof of the theorems that is in Bailey. The second theorem concerning uniqueness of the solution to (17) hinged on the solution of a linear eigenvalue problem (25). In a of important problems this equation is of the form

+ Ku = 0 u< ) + Au = 0 at x = a u (1) + Bu = 0 at x = b

V2 u 1

with V2 = (d 2yjdx 2) + (sjx) (dujdx) and s = 0, 1, or 2 for plane par cylindrical, and spherical geometry of the system. The largest value of K in (26) that can be permitted in order that shall have only the trivial solution u = 0 in [a, b] or in any subinterval 2 [a, b] is determined by the first eigenvalue of the operator V :

The value of -A 1 depends on the geometry factors and on the type of boundary conditions (19). The interval [a, b] can be translated to [0, 1] without changing eigenvalues by more than a scalar factor. We briefly discuss the ...· ....u ....... of s and B on the eigenvalues using a = 0, b = 1, and A = 0. For B ~ oo (y known at x = 1 and y(I) = 0 at x = 0) -Al ~ 2 2 2 respective!~ (rr/2) , 2.4048 , and 1T for s = 0, 1, and 2. These are the largest values that can be obtained from (26) in three geometries. If the constant B has a finite value, a smaller value -A is obtained. One interpretation of the effect of a decreasing 1 is that uniqueness of the solution to (17) is guaranteed only in x-interval smaller than 1. More relevant to our problems in which [a, is usually fixed is an equivalent formulation: If in two cases the

Application of Comparison Differential Equations model for diffusion and chemical reaction with heat evolution in pellets is well suited for a demonstration of the theorems of 5.2. Several_results_ concerning existence and uniqueness of the . . can be obtamed with practically no computation. On the other It Is apparent. that only very conservative bounds on the uniqueness can ~e obtai_ned even for this relatively simple example of a single """"'"''' .. differential equation with linear derivatives. It will also be that Theorem 1 can be used for an approximate construction of a sol~tion-but again it must be concluded that the power of the ~ml-::m:~Jyt~cal met~ods of this section is extremely limited compared to collocati~n solutions of section 5.4, at least for quantitative studies. Fo~ sphenca_I catalyst pellets and a first-order reaction, one obtains the . ~quation for the pellet temperature (J when the surface temperIS

8(x

=

1) 2

d 8

dx2

=

1:

+ ~ dx + 2 dO

2

(1 -

(J

+ {3) exp

(

'Y -

~) = 0

(27)

section 5.4_ the ~arne differential equation is analyzed numerically in of the_ ~Ime~s10nless concentration y = 1 - (1/ {3)(8 - 1). This has a tradition m nume:ical studies of the Weisz-Hicks problem ever the first paper on this model appeared [Weisz and Hicks (1962)]between 1 and 0, while (J varies between 1 and the less welfupper limit 1 ~ {3. On the other hand, all theoretical papers on ll1Q1Ien.ess of the solutiOn to the Weisz-Hicks problem have been cast in of 8. A comparison of results obtained by the theorems of the last and equivalent results from the literature is easier when exactly same equation is discussed here. Figure 5-3 shows R(O) = (1 - 8

+ {3) exp ( y

-

~)

for y = 20 and {3 = 0.8

(28)

were the original choice of parameters in the Weisz-Hicks paper. aR

ao

=

-8

2 -

y(O - 1) (}2

+

yf3

( ~') exp 'Y - (j

(29)

208

Nonlinear Ordinary Differential Equations

209

Application of Comparison Differential Equations

of these cases is illustrated by the small figures in figure 5-3. The tangent at (} = 1 + {3 and the chord from (} = 1 to (} = 1 + {3 given by h 2 (8) and h 1(8), respectively: 300

200

h2(8)

= -exp ( :

h1(8)

=

)
1

13

-(8 - 1)

+ {3

(36) (37)

for 1 :::;; 8 :::;; 1 + {3

h1(8) -::::; R (O) :::;; h 2 (8) 100

1 - {3)

The two comparison differential equations

e q {3"( > 2({3 + 1)

1 < 'Yf3 < 2({3 + 1)

(39)

= (20,0.8)

_

-y + [ 'Y2 + 4y({3 + 1)]1/2 2

R(O) has an inflexion point at

1+{3 (}infl

ao

ao

max

= [1 +

4(1 + {3)] exp [y/3 - 2({3 + 1)] 'Y 1 + {3

~

Three {fistinct cases appear according to the position of (} = 1 to Omax and Oinn·

Case 1: {3y > 2({3 + 1)

~

(40)

sinh (x) u 1(x) = 1 + {3 - {3 x sin h()

(41)

u1(x)

1 < Oinn

Case 2:

1 < y/3 < 2({3 + 1) ~ Oinfl < 1 < Omax

Case 3:

y/3 < 1 ~ Omax < 1

1

+ fJa

-

fJ

< O(x) < u 2 (x) for 1 < (} < 1 + {3

The function Gz(u) of Theorem 2 is different in the three cases of 5-3.

= 1 + [2({3 + 1)/ 'Y]

At 8 = Ointh

aR = (aR)

a sinh. (qx) x sirih (q)

U2 (X ) -

and

R(O) has a maximum at

(} =

(38)

1+{3

= (1-

(} = Omax =

y{3/2 + {3 1

unique solutions for any value of . Consequently, by Theorem 1, solution y(x) of (27) lies between the solutions of (38) and (39) with (x) = 1 at x = 1 and duJ dx = 0 at x = 0:

1 + {3

() + {3)exp[y- (y/0)] for (y,{3) for three more general cases. Figure 5-3. R

exp

'Yf3 < 1

e 1+{3

=

2

41 1+ ( ;tl)J u exp [ 'Ytl ~~: + l)J

Case 1:

G2(u) = <1>

Case 2:

G 2 (u) = (y/3 - 1)2u G (u) = ( y/3 - 1)2u

Case 3:

2

[

( y/3

> 1)

( y/3

< 1)

Theorem 2 guarantees that the steady state is unique for all values of in case 3 since G 2 (u) is negative for all positive u. In case 2 or case 1, solutions may be found for some -values-as in case 2, when > '1T/(y{3 - 1) 112 . This result is very poor since it is shown in chapter 6 uniqueness is guaranteed when y/3/(1 + {3) < 4, which covers all ituati011s corresponding to case 2 and even part of the case 1 region.

210

Nonlinear Ordinary Differential Equations

It may furthermore be proved [Luss (1969)] that there exists an upper bound * of the region of multiple steady states 2

(<1>*)2 =

7T

7T

2

(aRja8)o=l

When > *, uniqueness of the steady state is guaranteed. For y = 20 and {3 = 0.8, multiple steady states may occur since y{3/(1 + {3) > 4. The value of G 2(u) is <1> 2 · 1.36 · u · exp(6.8889) = 1334<1>2u while (aRja8) 0 = 1 = 15<1>2. Thus uniqueness is guaranteed for ' 1/2 < * = 7T/(1334) 112 = 0.086 and by (42) for > * = 7T/(15) = 0.811. Even though the comparison differential equation concept in its simplest formulation fails to predict the correct interval of y and {3 for which multiple steady states can be found and cannot directly give upper bound (42) for the region of multiple steady states, it is neverthe· less quite helpful when used with proper ingenuity. Two examples follow. First an upper bound * similar to (42) can be derived. We observe that R 0 is negative when 8max < 8 < 1 + {3 where 8max is defined in (30), For very large the major part of the temperature profile 8(x) may have 8 values greater than 8max· If the x-value for which 8 = 8max is called Xm, · we need only be concerned with the x-interval Xm < x < 1 where Re is positive. If this x -interval is small enough to give uniqueness by Theorem 2, the whole solution 8(x) is unique. Xm is, of course, unknown but an estimate for Xm may be found by means of the comparison functions Ut and u2. The construction of functions u 2(x) and u1(x) that by Theorem 1 are, respectively, above and below the true solution is characteristic of a proper use of the comparison differential equation concept. Our second example shows how this can be done. Define 8t as the abscissa of the R versus 8 curve at which the passes through (8, R) = (1, {3) and let 8* be a value of 8 in the interval (1, 8t). For any such 8 value the chord from (1, {3) to [8*, R(8*)] is always above R(8) and a suitable upper bounding function u2(x) · defined by

Application of Comparison Differential Equations

211

The maximum value of u 2 is found at x = 0. This value u 2(0) must be than 8*; otherwise we fail to meet the condition that the solution from the chord is an upper solution. The smaller we choose 8*, the more likely is u 2(0) to be larger than 8*; but we are obviously cintere~ste~d in choosing 8* as close to 1 as possible in order that the chord be close to R(8) for the 8 values that correspond to the given <1>. Thus the smallest admissible 8* is that for which u 2(0) = 8* or . .,u..au•v.l

a u2(0) = 8* = 1 + 2{3 ( ~ a sm '¥a

-

1)

(44)

which is an algebraic equation in 8*. Its solution is inserted into (43) to the best possible upper solution u 2 (x ). A tangent to R(8) taken at any point [8*, R(8*)] where 8* is between 1 and 8inn of (31) is certainly below R ( 8) for all 8 E (1, 8infl). Hence a lower bound for 8 is found by solution of 2

d u1

2 du1

2

-d 2 +- -d + Ro*U1 X

X

X

Uln(O) = 0

2

=

[Ro*8*- R(8*)]

u1(1)

(45)

=1

Here Ro* is aRja8 taken at 8 = 8*. We wish by an appropriate choice of 8* to obtain u 1 (x) as close to 8(x) as possible. From the solution of (44) we calculate a 8* that is certainly above 8 for any x and the given <1>. Thus 8* of (45) should be smaller than 8*, and a value of 8* = ~(1 + 8*) give a suitable "even" distribution of errors in our approximation of (8) by R 8*U1 - R 8*8* + R(8*). Table 5.2 shows that quite satisfactory results are obtained when the upper solution is constructed by (43) and (44) and the lower solution by (45) either with 8* = 1 or with 8* = ~(1 + 8*). TABLE

5.2

UPPER AND LOWER LIMIT ON CENTER TEMPERATURE SPHERICAL GEOMETRY AND (y, {3)

= (20, 0.8)

u 1 (0)

where 2 a

=

R( 8 *) - {3 8*- 1

and

u 2(1)

= 1,

or u2

=

1+ f!._( 2 si~ ax - 1) a x sm a



u2(0) = O* [solution of (44)]

()* = 1

0.1 0.3 0.4 0.4525

1.001358 1.01451 1.0331 1.0654

1.001356 1.01420 1.0293 1.0417

o* =

!co* + 1)

1.001357 1.01444 1.03153 1.0480

Exact 1.001357 1.01445 1.03178 1.0512

212

Nonlinear Ordinary Differential Equations

For (y, {3) = (20, 0.8), equation (44) has no real solution in 1 < fJ* < 1 + {3 when is slightly above 0.453. Consequently for > 0.453, we cannot construct an upper solution u 2 (x) by the chord process with the property that u 2 (x) ::::; fJ* for all x. The limiting value obtained by this process has no direct connection with the upper and lower bounds for the region of multiple solutions. Rather an approximation is found for the