2. Elementary Heuristic Reasoning in Singular Perturbations.

2 . ELEMENTARY HEURISTIC REASONING I N SINGULAR PERTURBATIONS.

2.1.

The e l e m e n t a r y method of c o n s t r u c t i o n . The p u r p o s e of t h i s c h a p t e r i s t o d e s c r i b e , a n a l y z e and i l l u s -

t r a t e by examples t h e h e u r i s t i c r e a s o n i n g which i s i n common u s e i n s i n g u l a r p e r t u r b a t i o n problems and which i n g r e a t many s u c c e s s f u l a p p l i c a t i o n s h a s l e d t o t h e c o n s t r u c t i o n o f asympt o t i c a p p r o x i m a t i o n s of t h e s o l u t i o n s .

We s h a l l a t t e m p t t o f o r m a l i z e t h i s r e a s o n i n g and f o r m u l a t e exp l i c i t l y t h e underlying hypotheses. W e study i n t h i s chapter r e l a t i v i t y s i m p l e cases i n which v a r i o u s c o m p l i c a t i o n s ( t o be a n a l y z e d t h e n e x t c h a p t e r ) do n o t y e t a p p e a r . These e l e m e n t a r y

cases a r e i n e s s e n c e c h a r a c t e r i z e d by two p r o p e r t i e s which s i m plify the analysis:

I . The s t r u c t u r e of t h e a s y m p t o t i c a p p r o x i m a t i o n s v a l i d i n v a r i o u s subdomains of t h e domain of d e f i n i t i o n i s e n t i r e l y d i c t a t e d by t h e s t r u c t u r e o f t h e d i f f e r e n t i a l e q u a t i o n s and t h e boundary c o n d i t i o n s ( i n a s e n s e t o b e s p e c i f i e d s h o r t l y ) . 11. The a s y m p t o t i c a p p r o x i m a t i o n s i n t h e s e subdomains d h e r e

t h e r e g u l a r a p p r o x i m a t i o n i s n o t v a l i d , a r e g i v e n by l o c a l approximations of a r e l a t i v e l y simple s t r u c t u r e .

We s h a l l d e v e l o p h e r e t h e l i n e o f r e a s o n i n g w h i l e c o n s i d e r i n g a c l a s s o f l i n e a r p r o b l e m s ; n o n - l i n e a r problems w i l l be d i s c u s s e d

a t . t h e end o f t h i s c h a p t e r . We c o n s i d e r

where @ = 8(x,&), 5 E

5 ; L 1 i s a second o r d e r l i n e a r d i f f e r e n t i a l

o p e r a t o r and Lo i s f i r s t o r d e r l i n e a r d i f f e r e n t i a l o p e r a t o r .

44.

Elementary heuristic reasoning in singular perturbations

h ( X I a r e c o n t i n u o u s f u n c t i o n s for 5 E n -

r

of

5 we

5. 3 n t h e boundary

impose +(X,E)

=4

on

r

I f L1 and L o a r e p a r t i a l d i f f e r e n t i a l o p e r a t o r s , t h e n L1 i s assumed t o b e e l l i p t i c . The s t r u c t u r e of t h e d i f f e r e n t i a l e q u a t i o n s u g g e s t s a r e g u l a r approximation of t h e s t r u c t u r e

L e t u s assume t h a t t h e r e g u l a r a p p r o x i m a t i o n h a s i n d e e d t h e

s t r u c t u r e g i v e n a b o v e , i . e . t h a t no o t h e r terms o c c u r . Then t h e f o r m a l r e g u l a r a p p r o x i m a t i o n must s a t i s f y Lo$o

= ho(&) = -Ll$n-l

+ h n (-X I , n > 0. Next boundary c o n d i t i o n s f o r Qas must be s p e c i f i e d . S i n c e L o i s a f i r s t o r d e r o p e r a t o r , it i s i n g e n e r a l i m p o s s i b l e t o impose on Qas a l l t h e boundary c o n d i t i o n s t h a t must b e s a t i s f i e d by

@.

T h e r e f o r e a c h o i c e o f boundary c o n d i t i o n s t o b e imposed on gas must be made. T h e r e a r e u s u a l l y no h p r i o r i i n f o r m a t i o n

suggesting

how t h i s c n o i c e m u s t be made, a l t h o u g h i n c e r t a i n p r o b l e m s a p r e l i m i n a r y g l o b a l s t u d y o f t h e problem may p e r m i t a m o t i v a t e d choice

( S e e : Eckhaus

and

De J a g e r

(1966) ).

On

t h e o t h e r h a n d , and f o r t u n a t e l y , a wrong c h o i c e of boundary c o n d i t i o n s

f o r @as o f t e n m a n i f e s t s i t s e l f by t h e f a c t t h a t f u l l c o n s t r u c t i o n of a s y m p t o t i c a p p r o x i m a t i o n of @ a l o n g t h e l i n e s o f h e u r i s t i c reasoning does not succeed.

For t h e c l a s s o f p r o b l e m s c o n s i d e r e d here w e c a n impose

45.

Elementary heuristic reasoning in singvlar prturbations

Where I'

rS

=

r must yet be chosen. (It can occur that

r - rr

rr

= d ) . If

# d , then Oas cannot be an approximation of 0 in

is.

As outlinedin chapter 1, we shall assume that @as fails to be an approximation of @ in a small neighbourhood Ds of Ts. In Ds local approximations of

@

will be studied.

It will appear in the sequel that for large classes of problems it is sufficient for the construction of the asymptotic approximation of

in Ds to study local approximations is special local

variables, to be named the boundary layer variables. The determination of the boundary layer variables is one of the fundamental problems in singular perturbations. Large parts of chapter 3 will be devoted to this problem. In the elementary method that we study in this chapter, we proceed with the following simple reasoning: We remark that the construction of the regular approximation is in fact based on the implicit assumption that €L1@ is much smaller than La@. (If ho introducing

@

=

3

0 then the above statement needs a modification:

+

E@'

it is easily seen that the construction

of the regular approximation with La@, = 0 is in fact based on the implicit assumption that

L1@' is much smaller than Lo@').

In the region in which Qas fails to be an approximation the underlying implicit assumption is no more true. There are then two possibilities: either €L1@ and La@ are of the same order of magnitude or L a @ is smaller than €L1@. In this chapter we shall proceed on the basis of the hypothesis that it is sufficient to study local variables such that cL1@ =La@

are of the same order of magnitude.

However, we emphazise here already that the hypothesis may fail to be true, Problems in which this is the case will be studied in chapter 3 where it will be shown by a more fundamental analysis that the hypothesis is a special simple case of a more general criterium.

46. Elementary heuristic reasoning in singolar perturbations

E x p l i c i t l y now, l e t

rs,

represents

x

-+

6' be a t r a n s f o r m a t i o n s u c h t h a t x i = 0

s o t h a t xi c a n b e u s e d t o measure t h e d i s t a n c e

from i n t e r n a l p o i n t s o f D t o i n same neighbourhood o f

rsq

The t r a n s f o r m a t i o n i s d e f i n e d

rs.

Next i n t r o d u c e a t r a n s f o r m a t i o n t o l o c a l v a r i a b l e s 5'

-+

as f o l l o w s :

I n comparing w i t h e a r l i e r d e f i n i t i o n it i s clear t h a t t h e l o c a l v a r i a b l e i n t r o d u c e d h e r e i s o n e o f t h e most Simple p o s s i b l e t y p e , w i t h o n l y one o f t h e components o f 5' b e i n g " s t r e t c h e d " . The t r a n s f o r m a t i o n i n d u c e s t h e t r a n s f o r m a t i o n L -+

where L;

Lo

-+

and L$ a r e d i f f e r e n t i a l o p e r a t o r s i n t h e v a r i a b l e

s i d e r now any s u f f i c i e n t l y d i f f e r e n t i a b l e f u n c t i c n L;f

L;

and Lgf a r e n o n - t r i v i a l ,

The o r d e r f u n c t i o n 6

II ELifIl

--

1 I L$fll

1

in J

E

5; where 5;

L:

4. Con-

f(s),s u c h

contains

5, =

that 0

,

w i l l b e d e t e r m i n e d by t h e r e q u i r e m e n t

- OS(l)

We r e c a l l t h a t F ( E ) = O J l )

-5 E 6; means F ( E ) = O(1)

F ( E ) # o(1).

The l o c a l v a r i a b l e t h u s d e t e r m i n e d w i l l b e c a l l e d t h e boundary

laver v a r i a b l e . The l o c a l a p p r o x i m a t i o n i n a boundary l a y e r v a r i a b l e w i l l be c a l l e d a boundary l a y e r . (Some a u t h o r s u s e t h e t e r m "boundary l a y e r f u n c t i o n " o r "boundary l a y e r c o r r e c t i o n " , and d e n o t e by "a boundary l a y e r " t h e r e g i o n of

5

i n which t h e r e g u l a r

a p p r o x i m a t i o n i s n o t v a l i d . Although making t h i s d i s t i n c t i o n h a s some m e r i t s , w e s h a l l u s e h e r e f o r s i m p l i c i t y t h e term "boundary layer" i n t h e sense

d e f i n e d a b o v e 1.

c e d u r e d e s c r i b e d above d e t e r m i n e s

W e assume t h a t t h e prouniquely

(modulo and e q u i -

47.

Elementary heuristic reasoning in singular perturbations

v a l e n c e c l a s s ) . Problems i n which t h i s i s n o t t h e case w i l l be d e a l t with i n t h e next chapter. We now s t u d y formal l o c a l a p p r o x i m a t i o n s i n t h e b o u n d a r y l a y e r e introduce a formal expansion of t h e d i f f e r e n t i a l variable. W

o p e r a t o r E L * + L$ as f o l l o w s : 1 L e t f(2) b e a n y ( s u f f i c i e n t l y d i f f e r e n t i a b l e ) f u n c t i o n , f o r which Llf

and L o f a r e n o n - t r i v i a l .

E L1f + L o f =

1 -

%*

p Z

-

n-0

8;

bp,f %

%

where 6 * i s a n o r d e r f u n c t i o n c h o s e n s u c h t h a t 6 8 = O s ( l ) ; furthermore

%A+l

= o(%*) p may b e f i n i t e o r i n f i n i t e , yn a r e n

d i f f e r e n t i a l operators not containing Similarly, l e t the transformation 5 mation h ( 5 , ~ )

-+

E.

+. 5

induce t h e t r a n s f o r -

h*(S,E), a n d assume a n e x p a n s i o n

The d i f f e r e n t i a l e q u a t i o n t h e n t r a n s f o r m s t o

The most s i m p l e case a r i s e s when

%*n

= En

and

b*~S(~)=6~ n

The s t r u c t u r e o f t h e t r a n s f o r m e d d i f f e r e n t i a l e q u a t i o n s u g g e s t s i n t h i s case a l o c a l a p p r o x i m a t i o n o f t h e s t r u c t u r e

48. Elementary heuristic reasoning in singular perturbations

L e t u s assume t h a t t h e l o c a l a p p r o x i m a t i o n i n d e e d h a s t h e s t r u c t u r e g i v e n a b o v e , i . e . t h a t no o t h e r terms o c c u r . The f o r m a l l o c a l a p p r o x i m a t i o n c a n be d e f i n e d a l o n g t h e l i n e s d e s c r i b e d i n s e c t i o n 1 . 4 . E x p l i c i t l y we h a v e :

2090

h8

- Xi90

aP0$1 = h i

etc.

We c a n now impose on ( t h a t i s f o r El

=

0).

@is t h e

boundary

However, s i n c e

condition for

along

rs

ZQis a s e c o n d o r d e r d i f f e -

r e n t i a l o p e r a t o r , @as i s n o t u n i q u e l y d e f i n e d by t h i s boundary condition. I n a f i n a l s t e p we assume t h a t f o r some q a n d m t h e r e g i o n s

@is o v e r l a p

and impose m a t c h i n g con-

d i t i o n s as d e s c r i b e d i n s e c t i o n 1 . 6 .

I f t h i s l e a d s t o unique

and

o f v a l i d i t y of Oas

d e t e r m i n a t i o n o f @ : s , t h e n t h e c o n s t r u c t i o n by h e u r i s t i c r e a s o n i n g h a s s u c c e e d e d . A l t e r n a t i v e l y , o n e c a n s t a r t from a h y p o t h e s i s on t h e s t r u c t u r e o f t h e u n i f o r m a p p r o x i m a t i o n and u s e c o r r e s ponding m a t c h i n g r u l e s ( s e e sec. 1 . 6 ) t o d e t e r m i n e

@&.

I n t h e c o n s t r u c t i o n d e s c r i b e d above one of t h e f u n d a m e n t a l h y p o t h e s e s c o n c e r n e d t h e s t r u c t u r e o f t h e r e g u l a r and l o c a l a p p r o x i m a t i o n s , as s u g g e s t e d by t h e s t r u c t u r e o f t h e e x p a n s i o n s o c c u r r i n g i n t h e d i f f e r e n t i a l e q u a t i o n . For s i m p l i c i t y w e h a v e c o n s i d e r e d t h e case i n which a l l e x p a n s i o n s were i n terms of i n t e g e r powers o f

E.

However, t h i s c a n b e e a s i l y g e n e r a l i z e d .

I n f a c t , l e t t h e boundary c o n d i t i o n be € - d e p e n d e n t and p o s s e s s a n a s y m p t o t i c e x p a n s i o n i n t e r m s of some o r d e r f u n c t i o n s , l e t t h e e x p a n s i o n o f h a n d h’

be i n terms of o t h e r o r d e r f u n c t i o n s a n d

49. Elementary heuristic reasoning in singular perturbations

let

b*

and

3;

n o t n e c e s s a r i l y be powers of E. Then t h e s t r u c -

t u r e of t h e problem s u g g e s t s t h a t sions contain a l l

t h e r e g u l a r and l o c a l expan-

powers and p r o d u c t s o f a l l t h e s e o r d e r

f u n c t i o n s . By h y p o t h e s i s no o t h e r t e r m s o c c u r and w i t h i n t h i s h y p o t h e s i s t h e r e i s no d i f f i c u l t y i n c a r r y i n g o u t t h e c o n s t r u c t i o n i n t h e more g e n e r a l c a s e a l o n g t h e l i n e s d e s c r i b e d above. However, i n v a r i o u s problems t h e h y p o t h e s i s i s n o t v a l i d and t h e l o c a l and r e g u l a r e x p a n s i o n s do c o n t a i n t e r m s of a n o r d e r o f magnitude t h a t i s n o t d i c t a t e d by t h e s t r u c t u r e of t h e d i f f e r e n t i a l e q u a t i o n and t h e boundary c o n d i t i o n s . Such problems

w i l l be s t u d i e d i n c h a p t e r 3 . I n a p p l i c a t i o n s it i s o f t e n a d v a n t a g e o u s t o modify somewhat t h e p r o c e d u r e as o u t l i n e d i n t h i s s e c t i o n i n o r d e r t o s i m p l i f y t h e c a l c u l a t i o n s . One s u c h m o d i f i c a t i o n w i l l be mentioned h e r e ( a n d used i n t h e s e q u e l ) b e c a u s e when a p p l i c a b l e it g r e a t l y s i m p l i f i e s t h e problem o f m a t c h i n g . ( I n f a c t t h e s i m p l i f i c a t i o n c a n be s o c o n s i d e r a b l e t h a t i t may a l l o w t h e s o l u t i o n o f l a r g e c l a s s e s o f problems w i t h o u t e v e r m e n t i o n i n g m a t c h i n g e x p l i c i t l y . T h i s h a s been done i n Eckhaus and d e J a g e r ( 1 9 6 6 ) . ) Suppose t h e f u n c t i o n a p p r o x i m a t i o n of @ i n

ads,

which by h y p o t h e s i s i s t h e r e g u l a r

6 0 c 5 , w i t h 5 - 50

i s d e f i n e d i n t h e whole domain

-

@ = @ -

5.

C o n s i d e r now

@as

W e have

w i t h boundary c o n d i t i o n s

d, i s a function that

m

50. Elementary heuristic reasoning in ringlar perturbations

@as on The r e g u l a r a p p r o x i m a t i o n of t o t h e o r d e r of magnitude

7

ern+',

in

rs

.

50 i s (by h y p o t h e s i s ) z e r o up

T h i s may v e r y much s i m p l i f y t h e

matching r e l a t i o n s €or t h e l o c a l a p p r o x i m a t i o n of

5.

Clearly,

t h e c o n d i t i o n f o r a p p l i c a b i l i t y of t h i s m o d i f i c a t i o n i s t h a t

-@asbe

defined i n

5.

Elementary heuristic reasoning in singlar perturbations

2.2.

51.

Applications t o l i n e a r ordinary d i f f e r e n t i a l equations with constant c o e f f i c i e n t s . 1 be a f u n c t i o n s a t i s f y i n g

x

L e t @ ( x , E ) ,0

€L1@ + L O O

where L1

a

0

n

2 dL + a l h -d; ; +

a0

dx

and t h e boundary c o n d i t i o n s @(O,E)

O(1,E)

a

B

where a and B a r e c o n s t a n t s i n d e p e n d e n t of

E.

I n a f i r s t a p p l i c a t i o n o f t h e method d e s c r i b e d i n t h e p r e c e e d i n g s e c t i o n w e s t u d y t h e case i n which t h e c o e f f i c i e n t s o f t h e d i f f e r e n t i a l o p e r a t o r €L1

+ Lo (i.e. a o ,

al,

a 2 , boy bl) are constants.

( N a t u r a l l y w e must have a

# O , since otherwise t h e equation 2 would be o f f i r s t o r d e r and it would be i n g e n e r a l i m p o s s i b l e t o s a t i s f y t h e two boundary c o n d i t i o n s ) . Although s i n g u l a r p e r t u r b a t i o n problems f o r l i n e a r c r d i n a r y d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s may seem v e r y s i m p l e , t h e i r d e t a i l e d s t u d y i s n e v e r t h e l e s s v e r y r e w a r d i n g and h e l p f u l i n g a i n i n g u n d e r s t a n d i n g o f t h e method and t h e phenomena. The r e a d e r who i s e x p e r i e n c e d i n s i n g u l a r p e r t u r b a t i o n s w i l l no d o u b t f i n d t h a t v a r i o u s d e t a i l s which seem t r i v i a l t o him a r e t r e a t e d i n t h i s section with considerable a t t e n t i o n . This i s done d e l i b e r a t e l l y : t h e p u r p o s e of t h e a n a l y s i s i s n o t t h e e f f i c i e n t construction of t h e approximation, but t h e i n v e s t i g a t i o n o f t h e s t e p s o f h e u r i s t i c r e a s o n i n g and t h e ( s o m e t i m e s i m p l i c i t ) underlying hypotheses.

52.

Elementary heuristic reasoning in singular perturbations

We commence w i t h t h e s t u d y of t h e r e g u l a r a p p r o x i m a t i o n . The s t r u c t u r e of t h e e q u a t i o n s u g g e s t s a f o r m a l r e g u l a r a p p r o x i m a t ion

Lo40 = 0

and

-

Lo+,

Ll$n-l,

n

I

depends on t h e c o e f f i c i e n t s o f L n

Explicit solution

. We

first

c o n s i d e r t h e case b o # 0 and bl # 0

We i n t r o d u c e

bo u = -

bl

and f i n d

- ux

4 0 ( x ) = coe

Suppose now t h a t t h e r e e x i s t s a r e g u l a r a p p r o x i m a t i o n of @ ( x , E ) valid i n

6 0 ,with

-

Do = { x ( O d x

p < 1)

and t h a t t h i s r e g u l a r a p p r o x i m a t i o n c a n be o b t a i n e d as a f o r m a l r e g u l a r a p p r o x i m a t i o n of a s t r u c t u r e d e f i n e d above. Imposing t h e bou'ndary c o n d i t i o l ~w e have -ux

4 0 = a e

F u r t h e r e l a b o r a t i o n shows t h a t 2

-

ala +

-ax xn e

j

n

1

53.

Elementary heuristic reasoning in sinylar perturbations

In general Q a S ( l # , ~8 ), s o t h a t ' i n d e e d p < 1. W e therefore s t u d y now l o c a l a p p r o x i m a t i o n s i n t h e neighbourhood of x = 1. Introduce t h e l o c a l variable c = - 1-x

,

&(El

t h e d i f f e r e n t i a l o p e r a t o r €L1 which w e f i n d EL;

= € -62

Lg = -6

a2

dL d .E2

-1 d bl

t

L O t r a n s f o r m s t o EL;

-1 d € 6 al F~t

+

bo

L e t f ( E) be any f u n c t i o n i n

58

={el0 Q

i f and o n l y i f 6

E.

5 < A < -1,

belonging t o t h e

W e find t h a t

p ( T h i s d e t e r m i n a t i o n i s v a l i d under t h e

c o n d i t i o n s al # 0 , bl # 0 , as a l r e a d y i m p o s e d ) .

We t h u s c h o s e as boundary l a y e r v a r i a b l e c = - 1-x E

The f o r m a l e x p a n s i o n o f t h e o p e r a t o r EL; n

EL;f

with

t

L$f

LZ, f o r

Baa

appropriately r e s t r i c t e d class defined i n 2 . 1 .

o

t

l L I;

n=O

Enxof

t

La r e a d s

54.

Elementary heuristic reasoning in singular perturbations

The s t r u c t u r e of t h e problem s u g g e s t s a f o r m a l l o c a l a p p r o x i -

m a t ion

with

E x p l i c i t s o l u t i o n for Jlo y i e l d s J ~ ~ ( s= )B:

e"

t

ct

where p = -bl

a2 F u r t h e r m o r e , it i s n o t d i f f i c u l t t o e s t a b l i s h t h a t f o r n

where P n ( 5) and S n ( 5) a r e

polynomials

Of

>

1

n-th order, with

c o e f f i c i e n t s depending on t h e c o n s t a n t s B* and C * , P P

p

n-1.

L e t u s now suppose t h a t t h e f o r m a l l o c a l a p p r o x i m a t i o n O * as as d e f i n e d above i n d e e d i s t h e l o c a l a p p r o x i m a t i o n of @ ( x , E ) i n t h e boundary l a y e r v a r i a b l e , v a l i d i n

6;.Imposing

t h e boundary con-

dition we find

BQ

t

CQ = 6

B;

t

Cf, t P n ( 0 )

t Sn(0)

= 0; n 2 1

We t h u s o b t a i n q t 1 r e l a t i o n s f o r t h e 2 ( q t l ) unknown i n t e g r a t i o n ,; c o n s t a n t s B * and C n

mined.

and c o n s e q u e n t l y

@is i s

not yet f u l l y deter-

55.

Elementary heuristic reasoning in singular perturbations

I n a f i n a l s t e p we suppose t h a t t h e r e g i o n s of v a l i d i t y o f t h e r e g u l a r approximation @

and t h e l o c a l a p p r o x i m a t i o n @ * o v e r l a p . as

as

I n t r o d u c e , as i n s e c t i o n 1 . 6 , i n t e r m e d i a t e v a r i a b l e 1-x Si = -

6i

with

E

<

Ai

3 1.

Writing t h e r e g u l a r expansion i n t e r m s o f t h e i n t e r m e d i a t e v a r i a b l e one e a s i l y sees t h a t f i n t h e n o t a t i o n of 1 . 6 )

C o n s i d e r n e x t t h e l o c a l a p p r o x i m a t i o n . The t r a n s f o r m a t i o n 5 induces a t r a n s f o r m a t i o n @ * ( 6 , ~ ) as

+

by i n s p e c t i o n t h a t @ ( i ) as

+

ci

@ ~ ~ ) ( 6 , 0 and ) it i s o b v i o u s

i s unbounded a s E

+

0 , unless

!J
Succes of t h e

c o n s t r u c t i o n . S u p p o s i n g t h e c o n d i t i o n s a t i s f i e d one f i n d s

T h e r e f o r e , by t h e m a t c h i n g r e l a t i o n

*

Co = a e

-U

The f i r s t term of t h e l o c a l a p p r o x i m a t i o n i s now f u l l y d e t e r m i n e d .

W e remark t h a t t h e r e l a t i o n which d e t e r m i n e d t h i s t e r m i s i n f a c t t h e s i m p l e matching c o n d i t i o n l i m JlO(C)

r”

l i m $a(x) X Y

P r o c e e d i n g now t o matching c o n d i t i o n s f o r t h e h i g h e r t e r m s o f t h e

56.

Elementary heuristic reasoning in singular perturbations

a p p r o x i m a t i o n s one must i n v e s t i g a t e r e l a t i o n s of t h e f o l l o w i n g type :

Here, on t h e l e f t hand s i d e , t h e i n f i n i t e power s e r i e s a r r i s e from t h e e x p a n s i o n o f t h e e x p o n e n t i a l f u n c t i o n e-ax o c c u r r i n g i n t h e r e g u l a r a p p r o x i m a t i o n , w h i l e on t h e r i g h t hand s i d e t h e terms O(E N )symbolize any power o f

terms t h a t a r e a s y m p t o t i c a l l y smaller t h a n

which a r i s e f r o m t h e e x p o n e n t i a l f u n c t i o n e

E

115

o c c u r r i n g i n t h e l o c a l a p p r o x i m a t i o n . I n t h e e x p r e s s i o n above t h e meaning o f t h e e x p a n s i o n o p e r a t o r E i S ) i s t h a t t h e terms

Ci

between b r a c k e t s s h o u l d b e a r r a n g e d a c c o r d i n g t o d e s c e n d i n g o r d e r s o f magnitude and t r u n c a t e d a t s + l terms.

It i s obvious t h a t e l a b o r a t i o n of t h e matching r u l e s r e q u i r e s t e d i o u s c a l c u l a t i o n s b e c a u s e of t h e p o l y n o m i a l s i n

ci

t h a t occur

on t h e r i g h t - and l e f t hand s i d e . The r e a d e r may c o n v i n c e h i m s e l f t h a t t h e c a l c u l a t i o n s r e m a i n t e d i o u s when i n s t e a d o f m a t c h i n g by o v e r l a p - h y p o t h e s i s m a t c h i n g r u l e s l i k e Van Dyke's a r e u s e d . However, a c o n s i d e r a b l e s i m p l i f i c a t i o n o f t h e m a t c h i n g problem i s a c h i e v e d when u s i n g a m o d i f i c a t i o n o f t h e method of c o n s t r u c t i o n t h a t h a s been mentioned i n 2 . 1 .

I n t h e p r e s e n t problem

t h e f u n c t i o n @ ( x , E ) , which by h y p o t h e s i s i s t h e r e g u l a r a p p r o x i mation i n domain

6

as

60c

D , i s a f u n c t i o n t h a t i s d e f i n e d i n t h e whole

{xlO

-Q(X,E)

x 4 1). I t i s t h e r e f o r e p o s s i b l e t o s t u d y

=

@(X,E)

-

Qas(X,E)

R e c o n s i d e r i n g t h e p r o b l e m i n t h i s f o r m u l a t i o n , a n d making u s e o f t h e r e l a t i o n s t h a t were u s e d t o d e t e r m i n e a a S ( x , ~ ) , w e h a v e

57. Elementary heuristic reawning in singular perturbations

5 i n 5 ={xlO < x < p

F u r t h e r m o r e , t h e r e g u l a r a p p r o x i m a t i o n of

i s smaller t h a n t h e o r d e r of magnitude S t u d y i n g now t h e l o c a l a p p r o x i m a t i o n of

E

~

< 1)

.

i n t h e boundary l a y e r

v a r i a b l e w e immediately f i n d

WherePn( 5) and

Sn(c) a g a i n are

p o l y n o m i a l s of t h e n - t h o r d e r . F u r t h e r -

more, it i s n o t d i f f i c u l t t o show t h a t

-

sn

o

if

Sn-l

o

Applying t h e boundary c o n d i t i o n w e have

-

BB + E8

-

BL +

8

-

4o(l)

Ff, + Pn(0) +

Sn(0) =

-

$1~(1)

T u r n i n g t o t h e problem o f m a t c h i n g w e o b s e r v e t h a t

p X

Theref o r e

where

so

0.

= 0

58.

Elementary heuristic reasoning in singular perturbations

Hence w e have

-

cg

so t h a t

Fg

t O(l)

= 0

= 0 . T h i s however, i m p l i e s t h a t 3,

-c ;

0 so t h a t w e f i n d

0

which i n terms i m p l i e s

3,

0, etc.

C o n t i n u i n g we o b t a i n

-

Ci = 0 ; n

< s

The v a l u e of s r e m a i n s unknown s i n c e w e d i d n o t make any h y p o t h e s i s c o n c e r n i n g t h e v a l u e of s up t o which t h e o v e r l a p h y p o t h e s i s i s v a l i d . I n t h e p r e s e n t problem t h e c o n s t r u c t i o n c a n be c o n t i n u e d f o r m andq a r b i t r a r y l a r g e , which s u g g e s t t h a t

z:

= 0 for a l l n.

I n f a c t t h e r e a d e r may v e r i f y by comparing t h e r e s u l t s t h u s obt a i n e d w i t h t h e a s y m p t o t i c e x p a n s i o n of t h e e x a c t s o l u t i o n of t h e problem, t h a t t h e p r o c e d u r e i n d e e d l e a d s t o t h e c o r r e c t r e s u l t , E x p l i c i t l y , i n t h e f i r s t approximation w e have:

We p r e s e n t l y re-examine t h e c o n d i t i o n s t h a t have been imposed on t h e c o e f f i c i e n t o f t h e d i f f e r e n t i a l e q u a t i o n and s t u d y t h e e f f e c t of v i o l a t i n g t h e s e c o n d i t i o n s .

W e have found t h a t matching was i m p o s s i b l e u n l e s s

? J = bl -
p >O.

Reviewing t h e h y p o t h e s i s it a p p e a r s t h a t t h e

c h o i c e of t h e boundary c o n d i t i o n t o b e imposed on t h e r e g u l a r a p p r o x i m a t i o n was i n f a c t a n a r b i t r a r y c h o i c e . Suppose i n s t e a d t h a t under t h e p r e s e n t c o n d i t i o n t h e r e g u l a r approximation i s v a l i d i n

59.

Elementary heuristic reasoning in sin@lar perturbations

We a r e t h e n l e d t o s t u d y l o c a l a p p r o x i m a t i o n s i n t h e v i c i n i t y o f x = 0 . R e p e a t i n g t h e a n a l y s i s a n a l o g o u s l y one f i n d s t h a t m a t c h i n g now s u c c e e d s . I n t h e f i r s t a p p r o x i m a t i o n one f i n d s :

Hence t h e s i g n of p d e t e r m i n e s a t which e n d p o i n t

of t h e i n t e r -

v a l t h e boundary l a y e r i s s i t u a t e d . Next w e s t u d y t h e c o n d i t i o n bo # 0; bl # 0 Suppose w e have bo = 0 ; bl # 0

.

We must now r e c o n s i d e r t h e r e g u l a r a p p r o x i m a t i o n and w e f i n d

that

However, w i t h t h i s m o d i f i c a t i o n t h e c o n s t r u c t i o n r e m a i n s e s s e n t i a l y t h e same a s i n t h e p r e c e e d i n g c a s e s . More d r a m a t i c e f f e c t s o c c u r when bo # 0 ; bl = 0

-

I t i s o b v i o u s t h a t i n t h i s case t h e whole c o n s t r u c t i o n must be

r e c o n s i d e r e d s i n c e u l o s e s i t s meaning and P becomes z e r o . We s h a l l now s t u d y t h i s case i n some d e t a i l . Suppose, as b e f o r e , t h a t t h e r e e x i s t s a r e g u l a r a p p r o x i m a t i o n of the structure

60.

Elementary heuristic reasoning in sinylar perturbations

Oas

=

m

x

n =0

EnOJx)

which c a n b e c o n s t r u c t e d as a f o r m a l r e g u l a r a p p r o x i m a t i o n . We see i m m e d i a t e l y t h a t

, OQnQm O n z o ' Hence t h e r e g i o n of v a l i d i t y o f t h e r e g u l a r a p p r o x i m a t i o n cannot c o n t a i n t h e endpoints

B

f 0.

of t h e i n t e r v a l , when a # 0 and

W e t h e r e f o r e assume

C o n s i d e r n e x t l o c a l a p p r o x i m a t i o n s i n t h e neighbourhood o f x = l and i n t r o d u c e f o r t h i s p u r p o s e a l o c a l v a r i a b l e

The d e t e r m i n a t i o n o f 6 i s done as i n t h e p r e c e e d i n g c a s e s . One f i n d s now t h a t

&. T h e r e f o r e w e t a k e

i f and o n l y i f 6

a

The o p e r a t o r EL;

+ L t i n t h i s boundary l a y e r v a r i a b l e r e a d s

EL; t La =

where

L

2 clnZn n=O

Elementary heuristic reasoning in singular perturbations

61.

This suggests a local approximation of the structure 1

and with boundary conditions $6l)(0) = B; J ~ ; ” C O )

If bo/ao >

=

0;

n

1

0 then the solutions of the equations determining

$:I all ) have oscillatory

behaviour. It is easily seen that

in this case all attempts to establish matching with the regular approximation fail. However, and fortunately, the case can be dismissed, because for bo/ao > 0 existence and unicity of solutions of the complete problem are no more assured. We

therefore take

Then in general

where PA’) and PA2) are polynomials of of the n-th order in

4.

The boundary conditions are Pi1)

t Pi2)

= 6

We now assume overlap hypothesis between this local approximation

62.

Elementary heuristic reasoning in singular perturbations

-ghere

JE

= o ( 1 ) . I t i s q u i t e o b v i o u s t h a t t h e matching r e l a -

I

t i o n can o n l y be s a t i s f i e d i f

For t h e f i r s t a p p r o x i m a t i o n t h i s y i e l d s p

p

I

0

The r e a d e r w i l l f i n d no d i f f i c u l t y i n c o n v i n c i n g h i m s e l f t h a t h i g h e r a p p r o x i m a t i o n s of V

1

are a l s o uniquely defined

by t h e s e c o n d i t i o n s . I n t h e f i n a l s t e p t h e l o c a l approximation i n t h e v i c i n i t y of x = 0 must be c o n s t r u c t e d . The a n a l y s i s i s e n t i r e l y a n a l o g o u s t o t h e a n a l y s i s of t h e v i c i n i t y of x

1. I n t h e boundary

layer variable 50

=

X

one f i n d s a l o c a l a p p r o x i m a t i o n

with

+(Oo)

= a e- w ' ~ , etc.

Comparing w i t h t h e e x p a n s i o n o f t h e e x a c t s o l u t i o n o f t h e problem i t c a n be e s t a b l i s h e d t h a t t h e r e s u l t s o b t a i n e d a r e i n d e e d c o r r e c t . I n f a c t one f i n d s

Elementary heuristic reasoning in singular perturbations

63.

We see that the case under consideration is characterized by the appearance of boundary layers at both ends of the interval, in contrast with the preceding

case bl # 0. One more

difference between the two cases should also be mentioned: In physical application it i s customary to speak of the !‘thickness”of a boundary layer, as expressed by the order of magnitude of the order function that determines the transformation to the boundary layer variable. It thus appears that the thickness of boundary layers is bl = 0.

E

when bl

f

0 and JE when

64.

Elementary heuristic reasoning in singlar perturbations

2.3.

Applications t o l i n e a r ordinary d i f f e r e n t i a l equations w i t h non-constant

W e s t u d y now

coefficients.

@ ( x , E ) ,0 4

x G 1, s a t i s f y i n g t h e e q u a t i o n

€L1@ + L o @ = h ( x , E ) n

with boundary c o n d i t i o n s = a(€);

Q(O,E)

Q(1,E)

=

B(E)

A s i n The p r e c e e d i n g s e c t i o n , w e s h a l l i n v e s t i g a t e t h e e f f e c t s o f v a r i o u s c o n d i t i o n s c o n c e r n i n g t h e d a t a of t h e p r o b l e m . We s h a l l f i n d t h a t i n c e r t a i n a s p e c t s t h e a n a l y s i s i s a v e r y g e n e r a l i s a t i o n o f t h e c a s e of c o n s t a n t c o e f f i -

straightforward

c i e n t s . However, e n t i r e l y new phenomena w i l l also a p p e a r . We assume a t t h e o u t s e t t h a t t h e c o e f f i c i e n t s a a , a l , a 2 , b o , b l

areat l e a s t c o n t i n u o u s , which i s a n e c e s s a r y c o n d i t i o n f o r e x i s t e n c e and u n i c i t y of s o l u t i o n s . F o r t h e same r e a s o n w e s h a l l assume a 2 ( x ) #

o

o

in

d x

1 (no s i n g u l a r p o i n t s ) .

F u r t h e r m o r e , f o r s i m p l i c i t y o f t h e a n a l y s i s w e assume e x p a n s i o n s m

h(x,E) =

x

n =0

Enhn(x)

00

a(€) =

c

n =0

Enan

00

B(E)

n =0

E

n

8,

T h i s a s s u m p t i o n i s n o t e s s e n t i a l , as h a s b e e n p o i n t e d o u t i n 2 . 1 . C o n s i d e r f i r s t t h e formal r e g u l a r a p p r o x i m a t i o n

Elementary heuristic reasoning in singular perturbations

65.

Explicit solutions read

where q(x) =

x

bo(t)

I 7 bl t )

dt

I f w e wish t h e f o r m a l r e g u l a r a p p r o x i m a t i o n t o be d e f i n e d i n 0

x

<

1, t h e n w e must have

bl(x) # 0 f o r 0

x G 1

Problems i n which t h e c o e f f i c i e n t b l ( x ) p o s s e s s e s z e r o s i n t h e interval 0


G 1 a r e c a l l e d t u r n i n g - p o i n t problems.

Assuming t h a t we have no t u r n i n g - p o i n t s we must f u r t h e r remark t h a t i n g e n e r a l t h e e x p a n s i o n c a n n o t be c o n t i n u e d i n d e f i n i t l y , i . e . t h e number o f terms m + l may be l i m i t e d by d i f f e r e n t i a b i l i t y

p r o p e r t i e s o f t h e c o e f f i c i e n t s of L O and L1. Only i n t h e case of i n f i n i t l y

d i f f e r e n t i a b l e c o e f f i c i e n t s c a n m be t a k e n a r b i -

t r a r i l y large. Suppose now t h a t t h e r e e x i s t s a r e g u l a r a p p r o x i m a t i o n o f @ ( x , E ) valid in

bo, with -

Do = { x ~ O<

X

d p < 11

and t h a t t h i s r e g u l a r a p p r o x i m a t i o n c a n be o b t a i n e d a s a f o r m a l r e g u l a r a p p r o x i m a t i o n of a s t r u c t u r e d e f i n e d above. Then imposing t h e boundary c o n d i t i o n s w e o b t a i n

66.

Elementary heuristic reasoning in singular perturbations

W e n e x t p r o c e e d as i n t h e second p a r t o f 2 . 2 , duce

-a

that is, we intro-

= @ - @as

and s t u d y t h e problem d e f i n e d by

-

O(O,E)

m

=

z

n=m+1

Ena

-@ ( l , E )

=

m

x

n =0

'ngn

-

m C

n =0

P$Jn(1)

hypothesis t h e r e g u l a r approximation of

By t h e p r e c e d i n g

i s smaller t h a n

*

n'

E

~

5

i n 50

s, o it i s s u f f i c i e n t t o c o n s t r u c t l o c a l

a p p r o x i m a t i o n s i n t h e v i c i n i t y o f x = 1. C o n s i d e r f o r t h i s purpose t h e l o c a l v a r i a b l e

D e t e r m i n i n g 6 as i n t h e p r e c e e d i n g s e c t i o n one a g a i n f i n d s &=

E

Next t h e f o r m a l e x p a n s i o n o f t h e o p e r a t o r EL; + Lz i n t h e bound a r y l a y e r v a r i a b l e must be e s t a b l i s h e d . Suppose f o r s i m p l i c i t y t h a t t h e c o e f f i c i e n t s of t h e d i f f e r e n t i a l e q u a t i o n p o s s e s conv e r g e n t 'Paylor s e r i e s e x p a n s i o n s i n t h e v i c i n i t y of x = 1. We t h e n have

a;')

= ai(l)

b:')

= bi(l)

Consequently, f o r any ( s u f f i c i e n t l y d i f f e r e n t i a b l e ) f u n c t i o n f ( c ) m

with

67.

Elementary heuristic reasoning in singular perturbations

etc. The s t r u c t u r e o f t h e problem s u g g e s t s a f o r m a l l o c a l a p p r o x i mation of t h e s t r u c t u r e

etc. A s i n t h e case o f c o n s t a n t c o e f f i c i e n t s , w e f i n d

with

F u r t h e r m o r e it i s n o t d i f f i c u l t t o e s t a b l i s h t h a t f o r n 2 1 one

Pn

yn

are p o l y n o m i a l s of t h e n - t h o r d e r w i t h c o e f f i c i e n t s d e p e n d i n g on B* c * p < n . I n p a r t i c u l a r P' P'

where

sn-l=

and

sn3 0

0.

Assuming now t h a t

5s:

i s t h e l o c a l a p p r o x i m a t i o n o f 0 v a l i d in

t h e v i c i n i t y o f x = 1 w e impose t h e boundary c o n d i t i o n s

if

68.

Elementary heuristic reasoning in singular perturbations

F i n a l l y , a s s u m i n g t h a t t h e domain o f v a l i d i t y o f

Tas * overlaps

w i t h t h e domain o f v a l i d i t y of t h e r e g u l a r a p p r o x i m a t i o n m a t c h i n g c a n b e a c o m p l i s h e d as i n t h e c a s e of c o n s t a n t c o e f f i c i e n t s . A g a i n we f i n d t h a t m a t c h i n g i s o n l y p o s s i b l e i f P < O

Under t h i s c o n d i t i o n

= 0 and

sn

3

0.

Hence, when no t u r n i n g - p o i n t s o c c u r , t h e r e s u l t s a r e e n t i r e l y a n a l o g o u s t o those o b t a i n e d f o r c o n s t a n t c o e f f i c i e n t s , a n d t h e method o f a n a l y s i s i s a p p l i c a b l e w i t h o u t any e s s e n t i a l m o d i f i c a t i o n ( t h o u g h t h e c a l c u l a t i o n s a r e somwhat more e x t e n s i v e ) . One f i n d s i n d e e d , by r e p e a t i n g t h e a n a l y s i s , t h a t f o r p > 0 t h e boundary l a y e r o c c u r s i n t h e v i c i n i t y of x = 0 , w h i l e i n t h e

case t h a t bl

0 , a s i n t h e case o f c o n s t a n t c o e f f i -

c i e n t s , two b o u n d a r y l a y e r s o f " t h i c k n e s s " JE o c c u r a t x

0

a n d x = 1. Summarizing w e c a n s t a t e t h a t t h e e f f e c t o f n o n - c o n s t a n t c o e f f i c i e n t s may be t w o f o l d :

I.

If t h e c o e f f i c i e n t s a r e n o t i n f i n i t l y d i f f e r e n t i a b l e t h e n

t h e r e g u l a r a n d l o c a l a p p r o x i m a t i o n s c a n n o t be c o n s t r u c t e d up t o a n a r b i t r a r y number o f terms. 11. The e l e m e n t a r y method o f c o n s t r u c t i o n f a i l s i f t u r n i n g

p o i n t s occur'. We r e m a r k t h a t t h e r e s u l t s o f t h i s a n d t h e p r e c e e d i n g s e c t i o n can be extended w i t h o u t any e s s e n t i a l d i f f i c u l t i e s t o h i g h e r o r d e r

Elementary heuristic reasoning in singular perturbations

69.

linear differential equations. A summary of results on the higher order problems, obtained by somewhat different method of construction then the one followed here, can be found in O'Malley (1968).

70. Elementary heuristic reasoning in singular perturbations

2.4.

Remarks on t h e t u r n i n g - p o i n t p r o b l e m .

Research on problems w i t h t u r n i n g p o i n t s h a s a l o n g h i s t o r y i n applied

m a t h e m a t i c s . The o r i g i n seems t o l i e i n quantum mecha-

n i c s w h e r e p h e s o c a l l e d " c l a s s i c a l t u r n i n g p o i n t " i s d e f i n e d as

a p o i n t i n which t h e k i n e t i c e n e r g y o f a p a r t i c l e e q u a l s i t s p o t e n t i a l energy. E x p l i c i t l y , f o r t h e one-dimensional Schroedinger equation d2@

E-

dx

x

2

+ Q(x)@ 0

xg i s a c l a s s i c a l t u r n i n g p o i n t when

Q(xo)

0-

I n t h i s connection turning-point

problems h a v e been s t u d i e d as

i n i t i a l v a l u e p r o b l e m s . P a r t i c u l a r l y i n t e r e s t i n g phenomena o c c u r when

The s o l u t i o n s t h e n h a v e e x p o n e n t i a l b e h a v i o u r a t o n e s i d e o f t h e t u r n i n g p o i n t and a ( r a p i d l y ) o s c i l l a t i n g b e h a v i o u r a t t h e o t h e r s i d e . Study o f t h e s e problems h a s r e s u l t e d i n t h e w e l l known W.K.B.J.-method

and v a r i o u s more s o p h i s t i c a t e d methods

o f c o n s t r u c t i o n , s u c h as t h e o n e d e v e l o p e d by R . E .

Langer ( 1 9 4 9 ) .

I t i s i n t e r e s t i n g t o remark t h a t t u r n i n g - p o i n t problems as defined i n t h e preceding

s e c t i o n (and i n t h i s we have followed

t h e e s t a b l i s h e d c u s t o m , f o r example O'Malley ( 1 9 7 2 ) ) ,

do not

e n t i r e l y c o i n c i d e w i t h t h e " c l a s s i c a l " t u r n i n g p o i n t o f quantumm e c h a n i c s . T h i s c a n be s e e n f r o m t h e f o l l o w i n g example:

71. Elementary heuristic reasoning in singular perturbations

E -

d 2 @ + x -d @ 2 dx

dx

+

c@ = 0

where c i s a c o n s t a n t . A c c o r d i n g t o 2 . 3 w e h a v e a t u r n i n g p o i n t a t x = 0 . However, i n t r o d u c i n g Liouville-transformation 2

- -X

@ = e we o b t a i n E2

4E

-@

2q + / t x 2 dx

-

&(C-t)]

m=

0

T h e r e f o r e w e have no " c l a s s i c a l " t u r n i n g p o i n t s i f c <

"classical" t u r n i n g p o i n t a t x turning points f o r c >

i.

For

0 if c

$ and

two

t,

one

"classical"

s m a l l t h e s e points l i e very

6

c l o s e t o g e t h e r , and a r e g i v e n by xg = T 2G dzqI n what f o l l o w s t u r n i n g - p o i n t s

w i l l be understood t o be defined

i n t h e sense of s e c t i o n 2.3.

L e t u s c o n s i d e r a problem as s t u d i e d i n t h a t s e c t i o n w i t h b l ( x g ) = 0 . The r e g u l a r a p p r o x i m a t i o n c a n n o t be d e f i n e d i n a n interval containing

xg.

However, w e may a t t e m p t a g e n e r a l i z a -

t i o n o f t h e method o f c o n s t r u c t i o n s o as t o accomodate t h i s c a s e . Suppose t h e r e g u l a r a p p r o x i m a t i o n as d e f i n e d i n 2 . 3 e x i s t s i n m o d i f i e d domain

60which

does not c o n t a i n x =

XO.

This s u g g e s t s

t h a t w e s t u d y l o c a l a p p r o x i m a t i o n s i n t h e v i c i n i t y of x = x g as w e l l a s t h e boundary l a y e r a t e n d p o i n t s o f t h e i n t e r v a l . We would e x p e c t t h a t m a t c h i n g c o n d i t i o n s between t h e r e g u l a r approximation and t h e various local approximations w i l l f u l l y determine t h e s o l u t i o n s . I n f a c t t h i s expectation is not

72.

Elementary heuristic reasoning in singular perturbations

always f u l f i l l e d , as can be s e e n from t h e f o l l o w i n g example: Consider @ ( x , E ) , -1 Q x =G 1, s a t i s f y i n g d2@ €2 + px-do = dx dx

0

w i t h boundary c o n d i t i o n s

@(-l,~) = a ; @(1,~) = 6 a , 8 and p a r e c o n s t a n t s . We f i n d t h a t t h e r e g u l a r a p p r o x i m a t i o n , v a l i d by h y p o t h e s i s i n some r e g i o n s 0 < p o G x G p1 < 1 i s @:s = A+

(constant)

S i m i l a r l y i n -1 < pml d x Q - P o < 0

@as =

A-

(constant)

Near x = 1 t h e boundary l a y e r v a r i a b l e i s

and t h e f i r s t t e r m of t h e l o c a l a p p r o x i m a t i o n r e a d s

Near x = -1 t h e boundary l a y e r v a r i a b l e is

and t h e f i r s t t e r m o f t h e l o c a l a p p r o x i m a t i o n r e a d s

F i n a l l y , n e a r x = 0 , t h e boundary l a y e r v a r i a b l e i s 50

X

I n t h e s i m p l e problem under c o n s i d e r a t i o n h e r e w e r e t a i n i n 5 0 t h e f u l l d i f f e r e n t i a l e q u a t i o n . However, w i t h i n t h e c o n c e p t s o f t h e method t h a t we s t u d y , we i n t e r p r e t t h e e q u a t i o n i n

60 as

73.

Elementary heuristic reasoning in singular perturbations

one generating formal local approximations near x

0. We

We attempt now to determine all the unknown constants by

1) Imposing boundary conditions at x=-1 and x=l. 2 )

$A1)

Matching

3 ) Matching

and @:s

@zs

4) Matching

as well as

@as

and and

(0)

,

$o

@asand

.

(0)

$0

In doing so it is essential to distinguish two cases: p > 0 and p < 0.

(0) = B

Bo

-

= .f

-m

a

t2

G ' r dt

Cko) = I(at6) Hence all constants are determined. 1I.p < 0 (the bad case)

The procedure yields

'A

A- = Cil)

I

C6-l' =

Cia)

A

where A is an undetermined constant. Furthermore

= a - A; Bi-')

= B

- A;

Bi0)2 0

We see that o u r method fails to produce a full determination of the (presumed) asymptotic approximations.

74.

Elementary heuristic reasoning in singular perturbations

The f a c t t h a t e l e m e n t a r y h e u r i s t i c r e a s o n i n g may f a i l t o d e t e r mine a s y m p t o t i c a p p r o x i m a t i o n i n boundary v a l u e problems w i t h t u r n i n g p o i n t s h a s b e e n p o i n t e d o u t by A c k e r b e r g and O'Malley ( 1 9 7 0 ) . These a u t h o r s have s t u d i e d , by a d i f f e r e n t method of a n a l y s i s ,

a c l a s s o f s u c h problems and h a v e r e v e a l e d v a r i o u s u n e x p e c t e d f e a t u r e s i n t h e b e h a v i o u r of t h e s o l u t i o n s . From t h e p o i n t of view of our a n a l y s i s w e must r e t a i n t h e conc l u s i o n t h a t a s u b c l a s s of t h e class o f boundary v a l u e problems w i t h t u r n i n g p o i n t s ( e x a m p l i f i e d h e r e by t h e case p < 0 ) marks one o f t h e l i m i t s o f a p p l i c a b i l i t y o f t h e method b a s e d on t h e elementary h e u r i s t i c reasoning as developed i n t h i s c h a p t e r . The i n t e r e s t e d r e a d e r may f i n d some f u r t h e r i n f o r m a t i o n on t h e phenomena d e s c r i b e d above i n Watts ( 1 9 7 1 ) and f u r t h e r m o r e i n Cook and Eckhaus.

75.

Elementary heuristic reasoning in singular perturbations

2.5. Linear e l l i p t i c problems without turning-points. Consider 0 = @ ( x , Y , E ) ,

(x,y)

E

5

C

IR 2 ,

€L1@ + L o @ = h

with

€L1

t

L o i s assumed t o b e e l l i p t i c i n 5 , on t h e boundary

r

of D

t h e value of @ i s p r e s c r i b e d . I n analogy t o t h e problems f o r o r d i n a r y d i f f e r e n t i a l equations w e s h a l l say t h a t t h e problem h a s no t u r n i n g p o i n t s i f t h e c h a r a c t e r i s t i c s of t h e o p e r a t o r

L O do n o t i n t e r s e c t i n

5.

That t h i s indeed i s t h e analoge w i l l

become c l e a r i n c o n s t r u c t i n g t h e a p p r o x i m a t i o n s . Problems o f t h i s t y p e have b e e n s t u d i e d by L e v i n s o n ( 1 9 5 0 ) , V i s i k and L y n s t e r n i k ( 1 9 5 7 ) , and Eckhaus and d e J a g e r ( 1 9 6 6 ) . I n t h i s s e c t i o n we s h a l l a p p l y t o t h e s e problems t h e r e a s o n i n g developed i n t h e

preceding p a r t of t h i s chapter.

W e assume t h a t by a p r e l i m i n a r y t r a n s f o r m a t i o n o f c o o r d i n a t e s t h e problems h a s b e e n b r o u g h t i n t o s u c h a form t h a t

- r(x,y) aY where p i s a c o n s t a n t . I t i s n o t d i f f i c u l t t o see t h a t i n c a s e s Lo = t p

a

without t u r n i n g p o i n t s such a t r a n s f o r m a t i o n can always be achieved. Furthermore, w e t a k e a > 0 , and i n o r d e r t o a s s u r e u n i c i t y of solution r - c f . 0 I n what f o l l o w s , a n d u n t i l t h e c o n t r a r y i s s t a t e d , w e t a k e The case 1.1

0 , t h a t i s t h e c a s e i n which t h e " u n p e r t u r b e d "

f

C

76.

Elementary heuristic reasoning in singular perturbations

o p e r a t o r i s o f z e r o t h o r d e r , w i l l be t r e a t e d s e p a r a t e r y . A s i n t h e preceding

s e c t i o n s , t h e a i m o f o u r a n a l y s i s is t o

i n v e s t i g a t e t h e a p p l i c a b i l i t y and t h e l i m i t a t i o n s of t h e e l e m e n t a r y h e u r i s t i c r e a s o n i n g . We s h a l l c o n s i d e r convex domains

-

D and u s e r e s u l t s of Eckhaus and de J a g e r ( 1 9 6 6 ) . For s i m p l i c i t y we

t d k e t h e f u n c t i o n h = h ( x , y ) t o be i n d e p e n d e n t o f

6,

snd s i m i -

l a r l y iri t h e boundary c o n d i t i o n

O = @ o n r t h e f u n c t i o n $ w i l l be i n d e p e n d e n t of E. The s t r u c t u r e o f t h e d i f f e r e n t i a l e q u a t i o n s u g g e s t s a r e g u l a r a p p r o x i m a t i o n of t h e s t r u c t u r e

m

For t h e formal r e g u l a r approximation w e o b t a i n :

en,

Where

n=O, 1 , . . . m y a r e as y e t undetermined f u n c t i o n s .

Assuming t h a t t h e r e i n d e e d e x i s t s a r e g u l a r a p p r o x i m a t i o n o f t h i s s t r u c t u r e w e must d e c i d e what boundary c o n d i t i o n s s h o u l d be imposed on @as. Before making t h e d e c i s i o n we a n a l y z e t h e boundary

r

of D .

C o n s i d e r any l i n e x = c o n s t a n t t h a t c u t s

r.

(Such l i n e s c u t

r

t w i c e ) . L e t y = y+ and y = y- be t h e p o i n t s of i n t e r s e c t i o n o f x = c o n s t a n t and y

y ..with y

+

w i t h y-

< y+.

+

r

and

d e n o t e by

> y- and by

r0

r-

r+

the collection of a l l points

t h e c o l l e c t i o n of a l l p o i n t s y = y,

w i l l i n d i c a t e t h e c o l l e c t i o n of a l l p o i n t s s u c h

77.

Elementary heuristic reasoning in singular perturbations

that y- = y+. Finally, if

r

contains segments on which x =

constant, then the collection of these segments will be indicated by

rC.

In fig. 2.1 the situation is sketched in which

r

= d (no

characteristic boundaries). Then

r where

ro

=

r + u r- u r a

consists of the two extremal points A and B.

fig. 2.1 In fig. 2.2 an example is sketched of a situation in which

# d

fig. 2.2 With these preliminaries we return to the question of boundary

.

conditions for @ as It is immediately obvious that we have the choice between r + and r- and that in both cases QaS is uniquely defined. A s has been shown in Eckhaus and de Jager (1966), a global

78.

Elementary heuristic reasoning in singular perturbations

s t u d y of t h e p r o p e r t i e s of t h e d i f f e r e n t i a l e q u a t i o n p e r m i t s i n t h e p r e s e n t problem a m o t i v a t e d c h o i c e (which depends on t h e s i g n Of

11).

I n t h e p r e s e n t a n a l y s i s w e a r b i t r a r i l y c h o s e t o impose

t h e boundary c o n d i t i o n s on

r-. r+

It i s c l e a r t h a t i n g e n e r a l Qas f $ on

and

rc.

In t h e

v i c i n i t y of t h e s e b o u n d a r i e s l o c a l a p p r o x i m a t i o n s w i l l have t o be c o n s t r u c t e d , F u r t h e r m o r e , t h e v a l u e of m up t o which t h e f o r m a l r e g u l a r a p p r o x i m a t i o n as d e f i n e d above c a n be c o n s t r u c t e d , depends on t h e d i f f e r e n t i a b i l i t y p r o p e r t i e s of t h e c o e f f i c i e n t s of t h e d i f f e r e n t i a l e q u a t i o n , t h e c u r v e t i o n $ on

r-.

r-

and t h e f u n c -

T h i s c o u l d s u g g e s t t h a t w i t h s u f f i c i e n t l y smooth

d a t a m c a n be t a k e n a r b i t r a r i l y l a r g e . However, a t c l o s e r anal y s i s a fundamental l i m i t a t i o n o f t h e c o n s t r u c t i o n p r o c e d u r e

ra # of r o .

o c c u r s when

at points

d,

and

lines x

constant are tangent t o

r

Imposing t h e boundary c o n d i t i o n w e have

Consider i n f a c t

where $ - ( X I are t h e v a l u e s o f $ on

r_

expressed as f u n c t i o n o f x .

It i s not d i f f i 1 c u l t t o see t h a t i n cases i n which t h e c o n f i g u r a t i o n i s as

w e must e v a l u a t e L

In order t o c a l c u l a t e

sketched i n f i g . 2 . 1 ,

Ll$0

w i l l i n general contain singularities

a t t h e p o i n t s A and B . (See Eckhaus and de J a g e r ( 1 9 6 6 ) or Eckhaus (1972)).

For t h i s r e a s o n w e d e f i n e

-

-

D; = D

where

V(ro) a r e

boundary I'D

- v(ro)

s m a l l neighbourhoods o f

t h o s e p o i n t s of t h e

a t which t h e c h a r a c t e r i s t i c s of t h e o p e r a t o r L1

79.

Elementary heuristic reasoning in sinylar perturbations

(that is lines x = constant) are tangent t o r . The method of analysis as developed in this chapter is only applicable to

-

Dk if m > 0. Since Qas is defined in

-

0 = 0

Ek,

we introduce

- cpas

We have &LIT + LOT =

-€

m+ 1 Ll@m

with boundary conditions

-@

= 0 on

cp

=

-

@

-

ron I'+

cpas

U

rc

If we assume that Qas indeed is the regular approximation of @

of

in ED such that

r+

U

rc,

5

-

ED only consists of small neighbourhoods in 5 0 is zero

than the regular approximation of

up to the order

ern+'.

approximations along

We now proceed to construct the local

r+

and

rc.

2.5.1. The ordinary boundary layer. Consider the situation as sketched in fig. 2.1, with

r+

a smooth curve. We must introduce a transformation to new variables x,y

+

x',y' such that say y'

r + while D i to r + .

0 represents

Y' measures the distance from internal points of

Within these stipulations various choices are possible. F o r example one may take x' = x; y'

y+(x)

-

y

This transformation can be used in 54, however the transformation introduces singularities in

6.

Another possibility is a transformation x,y

+

v,p

as indi-

cated in fig. 2.1, such that v measures the distance along

r+

80.

Elementary heuristic reasoning in singular perturbations

w h i l e p m e a s u r e s t h e d i s t a n c e a l o n g a normal t o

r+.

The d i s -

advantage of t h i s transformation i s , t h a t it only i s defined i n a s t r i p along o f l’+

r+’

s i n c e t h e normals from d i f f e r e n t p o i n t s

may i n t e r s e c t a t some d i s t a n c e from

r+.

Now, no matter what t r a n s f o r m a t i o n one u s e s , i f t h e c o n s t r u c t i o n by h e u r i s t i c

r e a s o n i n g i s c o r r e c t t h e n t h e r e s u l t s must

be e q u i v a l e n t . One s h o u l d t h e r e f o r e u s e t h o s e new v a r i a b l e s t h a t a r e o p t i m a l f o r t h e p u r p o s e of t h e a n a l y s i s . The v a r i a bles x’, y’ lead t o simpliest calculations i n

5’. r

However,

f o r t h e f u r t h e r developement o f t h e t h e o r y ( c h a p t e r 3 ) t h e v a r i a b l e s V‘,p a r e b e t t e r a d a p t e d

i n s p i t e of c e r t a i n a r t i -

f i c i a l d i f f i c u l t i e s which t h e y i n t r o d u c e . We t h e r e f o r e u s e h e r e x,y

+

w,p.

Let u s c o n s i d e r f i r s t a s i m p l e p r o b l e m i n which t h e r e a s o n i n g

w i l l n o t be o b s c u r e d by t h e c o m p l e x i t y o f t h e c a l c u l a t i o n s . Such p r o t o t y p e problem a r i s e s when w e t a k e

€L1 + L o

EA + u- a

aY where A i s t h e L a p l a c e o p e r a t o r . A s boundary w e t a k e :

r

=

I X , Y ~ X ~ + Y = ~11.

The v a r i a b l e s @ , p a r e now s i m p l y a s s o c i a t e d t o p o l a r c o o r d i n a t e s and a r e e x p l i c i t l y g i v e n by

x = ( 1 - p ) cos w ; y =

(1-p)

s i n b.

The t r a n s f o r m a t i o n x , y e p , V i s d e f i n e d e v e r y w h e r e , e x c e p t a t p = 1 . Furthermore

r+

= cp,e1

P=

o, o

< zp<

We i n t r o d u c e t h e l o c a l v a r i a b l e

TI

Elementary heuristic reasoning in singular perturbations

81.

6.- P

6(E)

and w e f i n d :

I t i s o b v i o u s t h a t f o r any f u n c t i o n f ( c , W )

L f f are n o n - t r i v i a l , I E L i f II

w e have

= o

-rmir

i f and o n l y i f 6

EJ

f o r which L i f and

p

E.

We t h e r e f o r e t a k e

6 = Q E

Expanding EL;

t

L$ w e obtain

EL;

t

Lf

E

where

00

L:

n.0

'

Enz

etc. T h i s s u g g e s t s a l o c a l a p p r o x i m a t i o n i n t h e boundary l a y e r v a r i a b l e

The formal l o c a l a p p r o x i m a t i o n must s a t i s f y XOdJO

0

= -del$n-l

-

....

-%h-l$~ ; q < m t 2

82.

Elementary heuristic reasoning in singular perturbations

It i s c l e a r t h a t a l t h o u g h J1,

= + n ( c , 2 c ) a r e f u n c t i o n s o f two

v a r i a b l e s , t h e y a r e d e f i n e d as s o l u t i o n s o f o r d i n a r y d i f f e r e n -

t i a l e q u a t i o n s i n which Z* a p p e a r s as a p a r a m e t e r . For t h i s r e a s o n t h e l o c a l a p p r o x i m a t i o n i n t h e boundary l a y e r v a r i a b l e a l o n g I'+ i s c a l l e d a n o r d i n a r y boundary l a y e r . E x p l i c i t l y w e have as s o l u t i o n s J ~ ~ ( s , z "=) B ~ W ) e p

sinst +

CQW)

F u r t h e r m o r e , it i s n o t d i f f i c u l t t o e s t a b l i s h t h a t f o r 1

an( S , F )

''5 + c*(z') n

n 4 q

+ Pn ( 5 , ~ ) e' + S~(~,ZP) where Pn andSn a r e p o l y n o m i a l s of t h e n - t h order i n t h e v a r i a b l e 6,

= BA(ZY) e p ' i n

w i t h c o e f f i c i e n t s d e p e n d i n g on B * C*, p < n . I n p a r t i c u l a r P' P sn 5 0 o f sn-l = 0 .

I n o r d e r t o d e t e r m i n e a l l t h e unknown f u n c t i o n s of tC m a t c h i n g c o n d i t i o n s must b e imposed. We r e c a l l t h a t t h e r e g u l a r a p p r o x i m a t i o n f o r 3 i s z e r o up t o o r d e r

ern+'.

A s i n t h e case o f o r d i -

n a r y d i f f e r e n t i a l e q u a t i o n s , one e a s i l y f i n d s t h a t m a t c h i n g i s only possible i f U < O

Under t h i s c o n d i t i o n one f i n d s

C.

0 a n d Sn 2 0 n Hence, t h e f o r m a l l o c a l a p p r o x i m a t i o n i s f u l l y d e t e r m i n e d . I f p > 0 t h e n t h e c o n s t r u c t i o n f a i l s . Again as i n t h e case o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , t h e h y p o t h e s i s must b e r e c o n -

83.

Elementary heuristic reasoning in singular perturbations

boundary conditions along I'+ and as constructing boundary layer along I'- one succeeds again in sidered: imposing on @

determining the formal approximations. Turning to more general problems, that is problems for the general operator

EL^

i

L o and arbitrary shape of the boundary

I', one finds that the same reasoning leads to analogous results. The reader may find the detailed computations in Eckhaus and de Jager ( 1 9 6 6 1 , The only complication arises

(also summarized in Eckhaus (1972)). because the boundary layer is

r+.

determined in a strip along

However the difficulty can be

easily, although somewhat artificially, overcome. Let x,y

+

p , 'Vbe a one to one transformation in 0 Q p Q po

where p o is some constant independent of

E.

At p

=$pa

the

values of T i s are asymptotically small. We now introduce a

t

"smothing function" (or "cutoff function" or "mollifier") such that

X(P)

=

1 for 0 G p Q $ p o 0 for

x( p ) E C

m

,

p > po

Using as local approximation

we have a function that is defined in

-

Dr

and that is equal to

@is

in the domain in which T* is not asymptotically small. as Eckhaus and de Jager ( 1 9 6 6 ) contains the proof of the asymptotic validity of the results outlined above. 2.5.2.

The parabolic boundary layer.

We now study cases in which the boundary that coincide with a characteristic

r

contains segments

of the operator LO (that

is lines x = constant). A typical example is

84.

Elementary heuristic reasoning in singular perturbations

Then T c i s t h e u n i o n of x = 0 and x = 1, 0

y

<

1.

The d i f f e r e n c e between t h e cases p > 0 and p < 0 h a s a l r e a d y been shown i n t h e p r e c e e d i n g a n a l y s i s . W e now t a k e 1.1 < 0 , and specifically

)J

= -1. (The c o n s t r u c t i o n for

p > 0 is entirely

a n a l o g o u s ) . Again w e e x p l i c i t l y s t u d y t h e s i m p l e c a s e

We f i r s t a n a l y s e l o c a l a p p r o x i m a t i o n s i n t h e v i c i n i t y of x = 0 .

Introduce t h e l o c a l v a r i a b l e

g = k T By t h i s t r a n s f o r m a t i o n €L1 t L o

+

EL*

1

+ L t and it i s n o t d i f f i -

c u l t t o see t h a t f o r any f u n c t i o n f ( g , Y )

f o r which L;f

and Lbf

ai?e n o n - t r i v i a l we h a v e

I1 EL;fll = i f and o n l y i f 6

zz

OS(l)

JE. We t h e r e f o r e t a k e as boundary l a y e r v a r i a b l e

O b v i o u s l y we have

The s t r u c t u r e o f t h e e q u a t i o n s s u g g e s t s a l o c a l a p p r o x i m a t i o n o f the structure

The f o r m a l l o c a l a p p r o x i m a t i o n o f t h i s s t r u c t u r e must s a t i s f y

a

- - -

a c2

a c0 aY

= 0

85.

Elementary heuristic reasoning in singular perturbations

C l e a r l y w e have p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s and f o r t h i s r e a s o n t h e boundary l a y e r i s c a l l e d a p a r a b o l i c boundary l a y e r . A p a r a b o l i c boundary l a y e r h a s a “ t h i c k n e s s ” JE i n c o n t r a s t t o

t h e o r d i n a r y boundary l a y e r which had a “ t h i c k n e s s ”

E.

Next boundary c o n d i t i o n s f o r cln must be s p e c i f i e d . R e c a l l i n g t h e f o r m u l a t i o n o f t h e problem a t t h e b e g i n n i n g of t h i s s e c t i o n

w e have “O(0,Y)

= 4J - [ 4 0 l , = ,

en(OYy)=

-

[ 4 ~ ~ 1 n~ 2=1~ ;

F u r t h e r m o r e , t h e supposed domain of v a l i d i t y of t h i s l o c a l approxi m a t i o n c o n t a i n s ( f o r y = 0 ) a p a r t of

-@

r-

o n which w e must have

= 0. We t h e r e f o r e impose Lon( C , O )

= 0, n 2

L e t us consider f i r s t

‘“0.

o

Writing

bo(O,y) = g ( y )

w e have e x p l i c i t l y

n > 0 one f i n d s t h a t i n g e n e r a l t h e s o l u t i o n P a s s i n g now t o w n’ becomes s i n g u l a r a t 5 = 0, y

0, t h e s i n g u l a r i t y a r i s e s from

-

s i n g u l a r i t y o f ’2vo ( s e e f o r d e t a i l s Eckhaus and d e J a g e r ( 1 9 6 6 ) ) . aY2 T h e r e f o r e t h e method o f a n a l y s i s as d e v e l o p e d i n t h i s c h a p t e r f a i l s i n t h e immediate v i c i n i t y o f t h e c o r n e r p o i n t 5 = 0 , y = O ; t h e f a i l u r e shows e x p l i c i t l y i n t h e h i g h e r a p p r o x i m a t i o n s LPnyn 2 2 .

86. Elementary heuristic reasoning in singular perturbations

The r e a d e r w i l l have no d i f f i c u l t y i n r e p e a t i n g t h e a n a l y s i s f o r t h e v i c i n i t y of x = 1, where t h e p r o p e r boundary l a y e r variable reads 1-x

6 ’ J E

F u r t h e r m o r e , t h e case o f more g e n e r a l d i f f e r e n t i a l o p e r a t o r

EL^

+ L o i n t r o d u c e s no e s s e n t i a l y new phenomena. The c a l c u l a -

t i o n s c a n be found i n Eckhaus and d e J a g e r (19661, where t h e complete a p p r o x i m a t i o n ,

i n c l u d i n g t h e o r d i n a r y boundary l a y e r

a l o n g y = 1, h a s a l s o been g i v e n and t h e a s y m p t o t i c v a l i d i t y o f t h e f i r s t a p p r o x i m a t i o n h a s been e s t a b l i s h e d .

2.5.3.

The case of z e r o t h o r d e r u n p e r t u r b e d o p e r a t o r .

I n t h e s p e c i a l case p = 0 t h e d i f f e r e n t i a l e q u a t i o n as d e f i n e d i n t h e b e g i n n i n g of t h i s s e c t i o n t a k e s t h e form €LIQ

The o p e r a t o r L

-

rQ= h(x,y)

h a s n o c h a r a c t e r i s t i c s i n t h i s case. Again w e

s t u d y e x p l i c i t l y a s i m p l e p r o t o t y p e problem

EA@ - @

h(x,y)

where A i s t h e L a p l a c e o p e r a t o r . A s boundary we t a k e t h e c i r c l e

r

= { X , y l x 2 t y 2 = 11

A f o r m a l r e g u l a r a p p r o x i m a t i o n of a s t r u c t u r e s u g g e s t e d by t h e

d i f f e r e n t i a l equatlon is

with

87. Elementary heuristic reasoning in singular perturbations

The v a l u e o f m up t o which t h e f o r m a l r e g u l a r a p p r o x i m a t i o n c a n be c o n s t r u c t e d , depends on t h e d i f f e r e n t i a b i l i t y p r o p e r t i e s of h(x,y).

If h E Coo t h e n m c a n be t a k e n a r b i t r a r i l y l a r g e and CP a s

is defined i n

fi

( i n c o n t r a s t w i t h t h e case p #

a).

However, i n g e n e r a l CPas

f $I

r

on

r.

where $I a r e t h e p r e s c r i b e d v a l u e s o f @ on b e a r e g u l a r a p p r o x i m a t i o n of CP i n

only

r

E

T h e r e f o r e CP as c a n

c

such t h a t

E,,.

Introduce again

-

@ = @ - aa s

w e have

with

-

CAT

-

-CP

4 -

0

-E

m+ 1

*+m

m

c cnqn on r

n- 0

By h y p o t h e s i s t h e r e g u l a r a p p r o x i m a t i o n of 5 i n 50 i s z e r o up t h e o r d e r o f magnitude

E

~

.

Next w e s t u d y local a p p r o x i m a t i o n s a l o n g

r.

Introduce x

(1-6E)cos 4 ; y

(1-65) s i n ( -

so t h a t

Lo

-*

L8

-1

For any f u n c t i o n f ( 5 , W )

i f and o n l y i f 6

for which L i f and L i f a r e n o n - t r i v i a l w e have

/'?. Hence w e t a k e

88.

Elementary heuristic reasoning in singular perturbations

6.J-Z Expanding now EL; + L$ w e f i n d m

etc

I

This s u g g e s t s l o c a l approximation of t h e s t r u c t u r e

-zl$n-l

Q0+,

-

.. . - x n ~;o1

n =Z q

E x p l i c i t l y w e have

=

$0(5,&)

BO(C)e-‘+

cO(@)e 5

Furthermore it i s n o t d i f f i c u l t t o e s t a b l i s h t h a t for n 2 1

where Pn and Sn are p o l y n o m i a l s i n depend on Bp, p = 0,

1,

Cp, p

... n-1.

Suppose now t h a t

-Do

5 o f which t h e c o e f f i c i e n t s

n. I n p a r t i c u l a r S

@is h a s

n

=

0 if C

P

= 0 for

a domain of v a l i d i t y t h a t o v e r l a p s w i t h

( i n which t h e r e g u l a r a p p r o x i m a t i o n i s z e r o t o t h e o r d e r Em + l )

Then imposing matching r e l a t i o n s one immediately f i n d s t h a t C n E O

O G n < q

89.

Elementary heuristic reasoning in singular perturbations

Consequently a l s o S n = 0 0 ; 1 G n G q Next we impose t h e boundary c o n d i t i o n s : Bo(ty)

= 4I -

B

+ P to,(?)

n

((4)

[4101r

n

BnW) + P ( 0 , P ) n

= 0 i f n i s odd = - [ $n]

i f n i s even.

C l e a r l y a l l t h e unknown f u n c t i o n s Bn((<) a r e d e t e r m i n e d by t i i e s e relations. Hence w e h a v e f o u n d a n o r d i n a r y b o u d a r y l a y e r a l o n g t h e whole boundary I'. U n l i k e t h e case p # 0 , t h e o r d i n a r y boundary l a y e r now h a s a " t h i c k n e s s "

e.

P a s s i n g t o more g e n e r a l o p e r a t o r s L1 and more g e n e r a l domains o n e e a s i l y f i n d s t h a t t h e c o n s t r u c t i o n proceeds analogously, without a n y e s s e n t i a l c o m p l i c a t i o n s ( e x c e p t for c o m p u t a t i o n a l l a b o u r ) i f a l l t h e d a t a a r e s u f f i c i e n t l y smooth. Hence it a p p e a r s t h a t t h e

case p = 0 d i f f e r s from t h e p r e c e e d i n g cases 11 f0

by t h e b e h a v i o u r

o f t h e s o l u t i o n ( b o u n d a r v l a y e r a l o n g t h e whole b o u n d a r y ) , b u t d o e s n o t i n t r o d u c e any new l i m i t a t i o n s on t h e a p p l i c a b i l i t y of t h e elementary h e u r i s t i c reasoning studied i n t h i s chapter.

90.

Elementary heuristic reasoning in singular perturbations

2 . 6 . On n o n - l i n e a r problems.

I n t h e p r e c e d i n g s e c t i o n s w e have developed and s t u d i e d t h e e l e m e n t a r y method o f a n a l y s i s w h i l e c o n s i d e r i n g l i n e a r problems.

We s h a l l now show t h a t t h e r e a r e no e s s e n t i a l d i f f i c u l t i e s i n a p p l y i n g t h e method t o n o n - l i n e a r problems, a s l o n g as t h e s e problems s a t i s f y t h e h y p o t h e s e s on t h e s t r u c t u r e s o f t h e a p p r o x i m a t i o n s on which t h e method is based. L e t u s , once a g a i n , s k e t c h t h e r e a s o n i n g , t h i s t i m e w i t h o u t s u p p o s i n g t h e problems t o be linear. We c o n s i d e r O ( E , E ) ,

~fE

5

satisfying

E L ~ O+ L O O = 0

where L @ and L o @ c a n be n o n - l i n e a r e x p r e s s i o n s i n @ and i t s

1

d e r i v a t i v e s w i t h r e s p e c t t o t h e components o f

~f.

The s t r u c t u r e of t h e problem s u g g e s t s a r e g u l a r e x p a n s i o n o f the structure

W e assume t h a t t h e r e g u l a r e x p a n s i o n h a s i n d e e d t h i s s t r u c t u r e ,

i . e . t h a t no o t h e r t e r m s o c c u r . Formal r e g u l a r e x p a n s i o n c a n now be d e f i n e d as d e s c r i b e d i n c h a p t e r 1. One f i n d s Lo40 = 0

= Rn

n > O

The d i f f e r e n t i a l e q u a t i o n s f o r @

n

a r e o b t a i n e d by s u b s t i t u t i n g

@asf o r @ i n t h e f u l l d i f f e r e n t i a l e q u a t i o n o f t h e problem, expanding and p u t t i n g t h e c o e f f i c i e n t s o f c p , p = 0,1,.

. ., e q u a l

t o z e r o . I t s h o u l d be n o t e d , t h a t LA, n > 0 , always are l i n e a r

o p e r a t o r s (while R n depends on 4 p’ P < n ) .

91.

Elementary heuristic reasoning in singular perturbations

I n r e g i o n s where @

as i s n o t a n a p p r o x i m a t i o n of @ l o c a l a p p r o x i m a t i o n s i n t h e boundary l a y e r v a r i a b l e s a r e s t u d i e d . I n t h e e l e mentary method t h e boundary l a y e r v a r i a b l e s a r e d e t e r m i n e d from t h e h y p o t h e s e s t h a t i n t h e r e g i o n under c o n s i d e r a t i o n it i s s u f f i c i e n t t o s t u d y l o c a l v a r i a b l e s f o r which E L @ and L o $ a r e 1 o f t h e same o r d e r o f m a g n i t u d e . A t t h i s s t a g e , i n a n o n - l i n e a r problem, a d i f f i c u l t y may a r i s e : l e t 5 t o l o c a l v a r i a b l e s , which i n d u c e s

ZO@ +g$@*. In

+

@(x,E)

4

be a t r a n s f o r m a t i o n

+

@*(&,E),

+x:@*,

a n o n - l i n e a r problem t h e r e l a t i v e o r d e r of magni-

t u d e o f E X ; @ * a n d g e t @ *may depend on t h e o r d e r of magnitude of @ * . The d i f f i c u l t y c a n be overcome i f , by some r e a s o n i n g , i n f o r m a t i o n on t h e o r d e r of magnitude of @ * i n t h e r e g i o n under c o n s i d e r a t i o n c a n be deduced. F o r example: i f t h e r e g i o n of v a l i d i t y of t h e l o c a l a p p r o x i m a t i o n c o n t a i n s p a r t o f t h e boundary on which t h e v a l u e s o f @ a r e p r e s c r i b e d , t h e n t h e o r d e r o f magnitude of @ * f o l l o w s i m m e d i a t e l y . The most s i m p l e case a r i s e s i f @ * = Os(l) and t h e d e t e r m i n a t i o n o f t h e boundary l a y e r v a r i a b l e i s t h e n e x a c t l y

as i n t h e l i n e a r case. I n t h e e l e m e n t a r y problems t h i s d e t e r m i n a t i o n

i s u n i q u e ( s i m p l e boundary l a y e r s t r u c t u r e ) . Given t h e boundary l a y e r v a r i a b l e s , f o r m a l l o c a l a p p r o x i m a t i o n s c a n b e d e f i n e d , as d e s c r i b e d i n c h a p t e r 1. One o f t h e h y p o t h e s e s o f t h e e l e m e n t a r y method i s a g a i n , t h a t t h e l o c a l a p p r o x i m a t i o n c o n t a i n s no o t h e r

terms t h e n t h o s e t h a t a r e d i c t a t e d by t h e s t r u c t u r e of

EL;$*

t

L o @ * and t h e boundary c o n d i t i o n s .

I t may be a d v a n t a g e o u s , as i n t h e l i n e a r case, t o u s e a m o d i f i c a t i o n o f t h e p r o c e d u r e o u t l i n e d above i n o r d e r t o s i m p l i f y t h e problem o f matching. If t h e r e g i o n o f v a l i d i t y of @as i s s u c h t h a t boundary

92.

Elementary heuristic reasoning in sinwlar perturbations

c o n d i t i o n s can be imposed which d e t e r m i n e Qas u n i q u e l y , and i f (Pas

t h u s determined i s d e f i n e d i n

study

5 =

-

Q

(Pas.

5 , t h e n i n s t e a d of

The r e g u l a r a p p r o x i m a t i o n of

t h e o r d e r of magnitude

m+ 1

E

.

5 is

@ w e may

z e r o up t o

We i l l u s t r a t e now t h e p r e c e d i n g d i s c u s s i o n by c o n s i d e r i n g a c l a s s of problems s t u d i e d i n O'Malley (1968). ( O ' M a l l e y ' s o r i g i n a l a n a l y s i s d o e s n o t s t r i c t l y f o l l o w t h e l i n e of r e a s o n i n g o u t l i n e d above, b u t c a n be i n t e r p r e t e d i n terms o f t h i s r e a s o n i n g ) .

n

E -

-

d L @+ f ( x , Q ) d @ t g ( x , @ ) = 0 2 dx dx

w i t h boundary c o n d i t i o n s

We assume a t t h e o u t s e t t h a t t h e problem i s one w i t h o u t t u r n i n g p o i n t s . T h i s means t h a t t h e r e e x i s t s a c o n s t a n t K > 0 , independend o f

E

such t h a t If(x,@)l2 K; 0 Q x G 1

I n t h e above c o n d i t i o n t h e as y e t unknown f u n c t i o n @ o c c u r s . However, p u r s u i n g t h e a n a l y s i s on t h e b a s i s of t h e h y p o t h e s i s t h a t t h e r e a r e no t u r n i n g p o i n t s i t w i l l b e p o s s i b l e t o d e r i v e e x p l i c i t c o n d i t i o n s which a s s u r e i n d e e d t h e n o n - e x i s t e n c e of such p o i n t s .

For s i m p l i c i t y w e assume f u r t h e r m o r e t h a t f and g a r e i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s . F u r t h e r c o n d i t i o n s f o r f and g will b e f o r m u l a t e d as t h e need f o r them arises.

93.

Elementary heuristic reasoning in singular perturbations

The problem s u g g e s t s a r e g u l a r e x p a n s i o n o f t h e s t r u c t u r e @as

=

m

z

n =0

En$n(X)

W e f i n d f o r t h e formal r e g u l a r approximation

etc. Suppose t h a t t h e r e e x i s t s s u c h r e g u l a r a p p r o x i m a t i o n i n 0

p

Q

x

1, and t h a t t h i s a p p r o x i m a t i o n i s g i v e n by t h e

f o r m a l a p p r o x i m a t i o n . One t h e n must assume f and g t o be such t h a t t h e ( n o n - l i n e a r ) problem f o r $ o h a s a s o l u t i o n w i t h $ o ( l , ~ =) 8 . We s h a l l assume t h a t $ o t h u s d e t e r m i n e d i s def i n e d i n 0 Q x Q 1. Because of n o n - e x i s t e n c e o f t u r n i n g p o i n t s

, we

have If ( x ,

$ 0 )

I>K>O

i n t h e domain o f v a l i d i t y o f t h e r e g u l a r a p p r o x i m a t i o n , and t h i s i n t u r n a s s u r e s t h e e x i s t e n c e of $ l . We s h a l l s u p p o s e

t h e n tJ1

pursuing t h e a n a l y s i s , t h a t

<

x

w i l l be d e f i n e d i n 0

on,

n

?

1 and i t c a n be shown by 1, a l s o a r e d e f i n e d i n 0 Q x

<

The r e g u l a r a p p r o x i m a t i o n w i l l i n g e n e r a l f a i l i n t h e neighbourhood o f x = 0 . We t h e r e f o r e p r o c e e d now t o d e t e r m i n e t h e boundary l a y e r variable. Introduce

c = 6(E) -

We have @(X,E)

+

@*(5,E)

1.

94.

Elementary heuristic reasoning in singrlar perturbations

I n t h e e l e m e n t a r y method t h e boundary l a y e r v a r i a b l e i s d e t e r mined by t h e r e q u i r e m e n t t h a t EL*@* and Lz@* be of t h e t h e same 1 o r d e r of magnitude. I n g e n e r a l t h e r e l a t i v e o r d e r of magnitude o f EL;@*

and La$* depends on @ * and t h e b e h a v i o u r o f f . However,

by t h e h y p o t h e s i s o f n o n - e x i s t e n c e of t u r n i n g p o i n t s w e have If

6

I

2 K > 0 , and it f o l l o w s t h a t EL;@* E

.

L$@* i f and o n l y i f

Thus

I n o r d e r t o s i m p l i f y t h e problem of matching of t h e l o c a l a p p r o x i m a t i o n s and t h e r e g u l a r a p p r o x i m a t i o n w e now w r i t e

T h i s m o d i f i c a t i o n o f t h e method c a n be i n t r o d u c e d b e c a u s e Oas d e f i n e d i n 0 G x G 1. L e t now x

+

5 induce

T(x,E)

+

~ * ( E , E ) . The r e a d e r may c o n v i n c e

h i m s e l f ( o r v e r i f y by c o n s u l t i n g O'Malley (196811, t h a t t h e s t r u c t u r e of t h e d i f f e r e n t i a l e q u a t i o n f o r

z*

suggests a l o c a l

approximation of t h e s t r u c t u r e

F u r t h e r m o r e , t h e terms o f t h e form 1 1 d e f i n e d by

proximat i

are

is

95. Elementary heuristic reasoning in singular perturbations

where p n depends on $

We now a n a l y z e

d$o

50

I

- = dS

Hence

0

$0.

f(O,@o(O)

P

,p

n . T h e boundary c o n d i t i o n s a r e :

I n t e g r a t i o n of t h e e q u a t i o n y i e l d s t

t ) d t + Co

i s a monotonic f u n c t i o n . We r e c a l l t h a t t h e r e g u l a r

$0

approximation of

7 i s z e r o up t o t h e o r d e r o f magnitude

E

.

m t1

T h e r e f o r e matching i s o n l y p o s s i b l e i f

co =

0

and i f t h e a l r e a d y imposed c o n d i t i o n of n o n - e x i s t e n c e o f t u r n i n g points reads f ( x , @ ) 2- K > 0

.

Suppose now t h a t t h e f o r m a l a p p r o x i m a t i o n s i n d e e d a r e v a l i d a p p r o x i m a t i o n s . Then it i s n o t d i f f i c u l t t o deduce t h a t t h e above c o n d i t i o n c a n be i n t e r p r e t e d as f ( x , @ o ) 2- K > 0 ,

0


1

and f(0,x)

f o r all

x

a

K > 0

between a and $ o ( 0 ) .

P r o c e e d i n g now t o

an,

n > 0, the

terms o f t h e l o c a l a p p r o x i m a t i o n c a n u n i q u e l y be d e f i n e d by u s i n g t h e matching c o n d i t i o n

l i m $,

5%

= 0.

The d e t a i l s o f t h e a n a l y s i s c a n b e found i n O'Malley (1968), where a l s o t h e proof of t h e a s y m p t o t i c v a l i d i t y of t h e r e s u l t i s g i v e n .

96.

Elementary heuristic reasoning in singular perturbations

I t c a n a l s o b e shown t h a t i f - f ( x , O ) 2 K

0 , t h e n i n s t e a d of

a boundary l a y e r n e a r x = 0 o n e h a s a boundary l a y e r n e a r x = 1. Various o t h e r i n t e r e s t i n g n o n - l i n e a r problems belong t o t h e class i n which t h e h y p o t h e s e s o f t h e e l e m e n t a r y method are s a t i s f i e d .

For f u r t h e r i n f o r m a t i o n t h e r e a d e r s h o u l d c o n s u l t O'Malley ( 1 9 7 2 ) and t h e b i b l i o g r a p h y t h e r e i n c o n t a i n e d . I t thus a p p e a r s t h a t n o n - l i n e a r i t y of a p r o b l e m d o e s n o t n e c e s s a r i l y p r o d u c e any e s s e n t i a l d i f f i c u l t i e s i n t h e a n a l y s i s . On t h e o t h e r hand, (and t h i s w i l l be s e e n i n t h e next c h a p t e r ) i f a n o n - l i n e a r problem d o e s n o t b e l o n g t o t h e c l a s s t h a t c a n b e t r e a t e d by t h e e l e m e n t a r y method o f t h i s c h a p t e r , t h e n t h e r e may a r i s e s e r i o u s d i f f i c u l t i e s of a t y p e t h a t i s not encountered i n l i n e a r problems