Theory of Singular Perturbations

THE THEORY OF SINGULAR PERTURBATIONS

E.M. DE JAGER, emeritus Department of Mathematics, Computer Science, Physics and Astronomy University ofAmsterdam The Netherlands JIANG

FURU

Shanghai Institute of Applied Mathematics andMechanics Shanghai University People's Republic of China

1996 ELSEVIER AMSTERDAM

9L A U S A N N E

9N E W Y O R K ~ O X F O R D ~ S H A N N O N

~TOKYO

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ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 EO. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0-444-82170-8

9 1996 ELSEVIER SCIENCE B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B. V., Copyright& Permissions Department, P.O. Box 521, lO00 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and~or damage to persons orproperty as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

This book is printed on acid-free paper. PRINTED IN THE NETHERLANDS

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To o u r wives

Carien and Yongzhen

Tai

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PREFACE Much scientific endeavour is aimed at the relation between causes and their effects. This becomes the more intriguing whenever the cause is small and the effect large. The study of this relation in the field of the theory of perturbations in mathematical or physical systems has already a respectable history, which can be retraced to the time of Lindstedt, Poincar6 and Prandtl about a century ago. Despite this long history the subject is still in state of a vigorous development and it is known as the theory of singular perturbations, where the meaning of a "small" perturbation causing a "large" impact is to be made explicitly clear. This book is about singular perturbation problems, depending on a small parameter such that the solutions show a nonuniform behaviour as the parameter tends to zero. Because of a very large variety of succesful applications of perturbation methods in the physical and engineering sciences and the recognition of the subject in pure and applied mathematics there exists a vast amount of literature on singular perturbations among which several treatises and textbooks. However, it is not well possible to present in a single volume a comprehensive survey including the latest developments. Instead of this we give here an introductory selfcontained text that acquaints the reader not only with topics well treated in other books but also with topics which to our knowledge have not been recorded up till now in already existing textbooks; the latter subjects have been chosen according to our experience and interest. We consider in the first half of the book singular perturbation problems of so-called cumulative type, where the influence of the small perturbation is observable only after a long time interval. The methods of averaging and multiple scales are treated in detail together with several applications from the theory of nonlinear oscillations. The second half is devoted to singular perturbations of boundary layer type, where the nonuniformity is observable only in a small neighbourhood of the boundary or parts of the boundary; as well ordinary as partial differential equations of elliptic and hyperbolic type, either linear or quasilinear have been considered. For a much more detailed description of the contents of this textbook the reader is referred to the table of contents. In the treatment of all these subjects the emphasis lies primarily on rigorous mathematical proofs for the justification of the perturbation methods to obtain valid approximations of the solutions of perturbation problems. The theory of singular perturbations is from the mathematical point of view a very interesting subject because it is possible to apply with success the results of the more abstract theory of differential equations, in particular one needs a priori estimates of solutions which may be obtained by Gronwall's lemma, maximum principles, energy

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integrals, fixed point theorems or Gs inequality. We hope that we meet in this way not only students and researchers who are mainly interested in the mathematical aspects of the theory, but also physicists and engineers who want to understand the mathematical justification of their clever and well devised formal approximations of solutions of practical perturbation problems. This book is the result of lectures given by both authors at their home universities and by the first author E.M. de Jager at the University of Shanghai, already in 1986. In fact it was in consequence of the latter occasion and with the support of the President of this University, dr Chien Weizang, that a contract has been signed between the University of Amsterdam and the Shanghai University of Technology concerning the cooperation of the authors. We are both indebted to the Royal Netherlands Academy of Arts and Sciences (KNAW) and the National Natural Science Foundation of China (NSFC) for the financial support needed for the expenses of travelling and lodging. Also both universities are acknowledged for providing their hospitality. Further we are very much indebted to the department of mathematics of the University of Amsterdam for putting at our disposal the help of a secretary in the person of mrs Y. Voorn; she prepared the whole typescript with great dedication and without complaints; therefore she deserves here a special word of thanks. The second author Jiang Furu is very grateful to professor H. van der Tweel and his wife for their very generous hospitality during his two visits to Amsterdam, in particular during his second stay of about half a year in 1989-1990, in which also his wife enjoyed this token of hospitality. Finally, it is our duty to acknowledge the pleasant cooperation with the publication manager D.J.N. van der Hoop and the technical editor E. Oosterwijk of the NorthHolland division of Elsevier Science, which has resulted in the publication of this book in the North-Holland Series in Applied Mathematics and Mechanics. We are aware that this text could not have been written without the use of many sources from the literature and so we are not in the least very much indebted to many colleagues as well in our own countries as abroad.

E.M. de Jager, Amsterdam Jiang Furu, Shanghai

CONTENTS Preface ........................................................................... vii C h a p t e r 1 General I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Asymptotic Expansions ............................................... 9 1 Order Symbols ...................................................... 9 2 G a u g e F u n c t i o n s a n d A s y m p t o t i c Sequences . . . . . . . . . . . . . . . . . . . . . . . . 12 3 A s y m p t o t i c Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 C o n v e r g e n c e versus A s y m p t o t i c C o n v e r g e n c e . . . . . . . . . . . . . . . . . . . . . . . 16 5 E l e m e n t a r y O p e r a t i o n s on A s y m p t o t i c E x p a n s i o n s . . . . . . . . . . . . . . . . . 18 6 O t h e r T y p e s of E s t i m a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 Generalized Asymptotic Expansions ................................ 21 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 3 Regular Perturbations ............................................... 27 1 Regular Perturbations ............................................. 27 2 A N o n l i n e a r I n i t i a l Value P r o b l e m C o n t a i n i n g a S m a l l P a r a m e t e r . . . 28 3 Applications ....................................................... 37 3.1 Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 T h e M o t i o n of a P l a n e t a r o u n d t h e S u n . . . . . . . . . . . . . . . . . . . . . . . . 38 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 C h a p t e r 4 The M e t h o d of the Strained Coordinate ............................. 43 1 Introduction ....................................................... 43 2 A p p l i c a t i o n s of t h e M e t h o d of t h e S t r a i n e d C o o r d i n a t e . . . . . . . . . . . . . 44 2.1 T h e N o n l i n e a r Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 T h e M e t h o d of t h e S t r a i n e d P a r a m e t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Lighthill's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5 Temple's Method .................................................. 55 6 L i m i t a t i o n s of t h e L i n d s t e d t - P o i n c a r ~ M e t h o d . . . . . . . . . . . . . . . . . . . . . . 57 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 5 The Method of Averaging ........................................... 61 1 Introduction ....................................................... 61 2 The Krilov-Bogoliubov-Mitropolski Theorem ....................... 63 2.1 I n t r o d u c t i o n to F i r s t O r d e r A v e r a g i n g . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2 G e n e r a l i z a t i o n of T h e o r e m 2; K . B . M . T h e o r e m - S e c o n d V a r i a n t .. 66 2.3 T h e K r i l o v - B o g o l i u b o v - M i t r o p o l s k i T h e o r e m for N o n p e r i o d i c Fields; K.B.M. Theorem-Third Variant ................................ 70 3 W e a k l y N o n l i n e a r Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1 T h e G e n e r a l Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 T h e Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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3.3 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4 T h e L i n e a r O s c i l l a t o r w i t h S m a l l D a m p i n g . . . . . . . . . . . . . . . . . . . . . 78 3.5 T h e Free van der Pol E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4 W e a k l y Forced N o n l i n e a r Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1 T h e Case w i t h o u t D a m p i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 T h e Case w i t h D a m p i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5 A Linear Oscillator with Increasing Damping . . . . . . . . . . . . . . . . . . . . . . . 87 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 C h a p t e r 6 The M e t h o d o f Multiple Scales ...................................... 91 1 Introduction ....................................................... 91 2 W e a k l y N o n l i n e a r Free Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.1 T h e Duffing E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.2 T h e P e r i h e l i u m P r e c e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3 The Linear Oscillator w i t h Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 T h e E q u a t i o n of M a t h i e u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 F l o q u e t ' s T h e o r y for L i n e a r E q u a t i o n s w i t h P e r i o d i c Coefficients 102 4.3 A p p l i c a t i o n to Hill's E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4 A p p l i c a t i o n to M a t h i e u ' s E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 T h e T r a n s i t i o n C u r v e s for t h e M a t h i e u E q u a t i o n . . . . . . . . . . . . . . 106 4.6 T h e A p p r o x i m a t i o n of t h e S o l u t i o n O u t s i d e t h e T r a n s i t i o n C u r v e s 110 5 T h e G e n e r a l Case a n d t h e E r r o r E s t i m a t e . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 E s t i m a t e of t h e R e m a i n d e r T e r m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6 A v e r a g i n g a n d M u l t i p l e Scales for P e r t u r b e d W a v e E q u a t i o n s . . . . . 123 6.1 T h e A p p r o x i m a t i o n by C h i k w e n d u a n d K e v o r k i a n . . . . . . . . . . . . . 123 6.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2.1 W a v e E q u a t i o n w i t h L i n e a r D a m p i n g . . . . . . . . . . . . . . . . . . . 126 6.2.2 W a v e E q u a t i o n w i t h C u b i c D a m p i n g . . . . . . . . . . . . . . . . . . . . 127 6.3 J u s t i f i c a t i o n of t h e C h i k w e n d u - K e v o r k i a n P r o c e d u r e . . . . . . . . . . . 130 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 C h a p t e r 7 Singular Perturbations o f Linear Ordinary Differential Equations . . . . 137 1 T h e initial Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 1.3 T h e A P r i o r i E s t i m a t e of t h e Solution of a S i n g u l a r l y P e r t u r b e d O r d i n a r y Differential E q u a t i o n w i t h G i v en I n i t i a l D a t a . . . . . . . . 140 1.4 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . 142 2 T h e B o u n d a r y Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2 T h e M a x i m u m P r i n c i p l e for O r d i n a r y Differential O p e r a t o r s . . . 145 2.3 A n A P r i o r i E s t i m a t e of t h e S o l u t i o n of t h e B o u n d a r y Value Problem ....................................................... 146 2.4 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.5 T h e A P r i o r i E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s 151 3 B o u n d a r y Value P r o b l e m s w i t h T u r n i n g P o i n t s . . . . . . . . . . . . . . . . . . . . 158 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

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3.2 T h e T u r n i n g P o i n t P r o b l e m w i t h f ' ( x ) < 0 . . . . . . . . . . . . . . . . . . . 158 3.3 T h e A s y m p t o t i c A p p r o x i m a t i o n a r o u n d t h e T u r n i n g P o i n t a n d t h e Case / 3 r m=0,1,2 ..................................... 161 3.4 T h e A s y m p t o t i c A p p r o x i m a t i o n in t h e Case of R e s o n a n c e . . . . . 164 3.5 T h e T u r n i n g P o i n t P r o b l e m w i t h .f'(x) > 0 . . . . . . . . . . . . . . . . . . . 168 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 C h a p t e r 8 Singular Perturbations of Second O r d e r Elliptic Type. Linear Theory 175 1 Introduction ...................................................... 175 2 T h e M a x i m u m P r i n c i p l e for Elliptic O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . 177 3 The Formal Approximation ....................................... 179 4 E s t i m a t i o n of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . . . 185 5 Domains with Characteristic Boundaries .......................... 191 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.2 T h e S i n g u l a r P e r t u r b a t i o n P r o b l e m in a R e c t a n g l e . . . . . . . . . . . . 194 6 Elliptic B o u n d a r y Value P r o b l e m s w i t h T u r n i n g P o i n t s . . . . . . . . . . . . 200 6.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2 E x a m p l e s of T u r n i n g P o i n t P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.2.1 C u r v e s of T u r n i n g P o i n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.2.2 I s o l a t e d T u r n i n g Points; N o d e s . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.3 A Saddle T u r n i n g P o i n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 C h a p t e r 9 Singular Perturbations of Second Order Hyperbolic Type.

Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4

Introduction ...................................................... Characteristics and Subcharacteristics ............................. The Formal Approximation ....................................... A P r i o r i E s t i m a t e s of Solutions of I n i t i a l Value P r o b l e m s for P a r t i a l Differential E q u a t i o n s w i t h a S i n g u l a r P e r t u r b a t i o n of Hyperbolic Type .................................................. 5 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 210 213

215 223 227

C h a p t e r 10 Singular Perturbations in Nonlinear Initial Value Problems of

Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction ...................................................... 2 A Fixed Point Theorem ........................................... 3 T h e Q u a s i l i n e a r I n i t i a l Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 4 A G e n e r a l N o n l i n e a r I n i t i a l Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 5 Q u a s i l i n e a r I n i t i a l Value P r o b l e m s w i t h a S i n g u l a r P e r t u r b a t i o n of Second O r d e r H y p e r b o l i c T y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . .

229 229 230 232 232 232 235 239 239 240 244 250 250 250 253

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Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a p t e r 11 Singular Perturbations in Nonlinear Boundary Value Problems of

Second Order

.....................................................

1 Introduction ...................................................... 2 B o u n d a r y Value P r o b l e m s for Q u a s i l i n e a r O r d i n a r y Differential Equations ........................................................ 2.1 T h e F o r m a l A p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 T h e E s t i m a t e of t h e R e m a i n d e r T e r m a n d F i n a l R e s u l t s . . . . . . . 3 T r a n s i t i o n Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Autonomous Conservative Equations .............................. 5 A M o r e G e n e r a l Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 B o u n d a r y Value P r o b l e m s for Q u a s i l i n e a r P a r t i a l Differential E q u a t i o n s of Elliptic T y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 T h e N o n l i n e a r G e n e r a l i z a t i o n of t h e M a x i m u m P r i n c i p l e . . . . . . . 6.3 Elliptic E q u a t i o n s w i t h o u t F i r s t D e r i v a t i v es . . . . . . . . . . . . . . . . . . 6.4 Elliptic E q u a t i o n s w i t h F i r s t D e r i v a ti v es . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a p t e r 12 Perturbations of Higher O r d e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction ...................................................... 2 P e r t u r b a t i o n s of H i g h e r O r d e r in O r d i n a r y Differential E q u a t i o n s .. 2.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 T h e F o r m a l A p p r o x i m t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Elliptic P e r t u r b a t i o n s of Elliptic E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Elliptic P a r t i a l Differential E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 S o b o l e v Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Elliptic O p e r a t o r s , Bilinear F o r m s a n d G s Inequality 3.2.3 G e n e r a l i z e d Dirichlet P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 E x i s t e n c e a n d G e n e r a l i z e d Solutions . . . . . . . . . . . . . . . . . . . . . 4 Elliptic S i n g u l a r P e r t u r b a t i o n s of H i g h e r O r d e r . . . . . . . . . . . . . . . . . . . 4.1 T h e B o u n d a r y Value P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 E x i s t e n c e a n d A P r i o r i E s t i m a t e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 T h e a p p r o x i m a t i o n of t h e S o l u t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 T h e E s t i m a t e of t h e R e m a i n d e r a n d F i n a l R e s u l t s . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index ..................................................................

260

261 261 263 263 267 273 283 288 291 291 291 293 300 304 307 307 308 308 309 315 315 315 315 317 318 319 323 323 324 325 328 330 331 339

Chapter 1 GENERAL

INTRO D U CTIO N

The theory of perturbations, in particular of singular perturbations, has a memorable history. As so many branches of mathematics it has its roots in remarkable phenomena in physics. These phenomena are characterized by transitions in the observables which are due to a small parameter in the m a t h e m a t i c a l model. Let us write this model for the moment symbolically as an equation

Ps[ue] = 0 ,

(1.1)

where ue is the relevant physical quantity and e the small parameter. Physicists developed an approach for calculating ue in the form of an expansion into powers of e and this expansion is continuous for e > O, but it may well be discontinuous for e = 0. This is related to the circumstance t h a t the so-called reduced problem P0[u0] = 0

(1.2)

is in general of another type as the problem (1.1) and so it is not a priori sure whether the solution of (1.2) is a reasonable approximation of ur even for e very small. Problems of this kind are related to the well-known question:

"Is the limit of the solution equal to the solution of the limit?' or phrased in the case of differential operators: "Is the limit of the integral equal to the integral of the limit?' The subject of this textbook is the study of perturbation problems where the solution is not uniform in e whenever e approaches zero t h a t is lim u~ r u0.

e--~0

(1.3)

We distinguish two classes of p e r t u r b a t i o n problems, viz. singular perturbations of cumulative type and singular perturbations of boundary layer type.

Singular Perturbations of C u m u l a t i v e T y p e This class concerns oscillating systems where the influence of the small parameter becomes observable only after a long time, for instance after an interval of order O(~).

Chapter 1. General Introduction Let us take a nonlinear spring as an example; the displacement of its mass is given by the equation

d2ue +u~+Eu 3=0, dt 2

0 e -1 for 0 < e < 1.

2. G a u g e Functions and A s y m p t o t i c

Sequences

The order definitions give us a tool to compare the values of two functions which depend on the variable x and the p a r a m e t e r e. By taking a special privileged set of functions we get a set of comparison functions with the values of which we m a y compare the values of a large class of functions f(x,r This special set is chosen as simple as possible (depending on the class of functions f(x,r and with this set we have obtained, so to say, a yard stick or measuring rod to be used to measure the values of our functions f(x, r as ~ $ 0. The elements of the special set are called gauge functions. DEFINITION 4

A gauge function 5(~) is a function of the p a r a m e t e r e with the properties of being positive, monotoneously decreasing or increasing for ~ $ 0 and continuously differentiable in a right neighbourhood (0, 60) of e - 0. Sets of frequently used gauge functions are the positive and the negative powers of ~ : g,`(c) -- c,`, n - 0, + l , .... Other useful sets are e.g.: e~'gn(6), ]loge[Ogr,(e) or exp(-1/e)g,~(e) with a,/3 9 R and a > O. After this introduction of gauge functions we introduce some ordering in sets of gauge functions. So we are led to the following definition of ordering: DEFINITION

5

The sequence {5,, (~)}~=0 of gauge functions is called an asymptotic sequence whenever 5,~+1(~) = o(5n(r

Vn, as ~ $ 0.

(2.2.1)

Examples {e,`}~--0, {r with p > 0, {[loge[~e"}.~1760 with j3 e R, {8,`e-1/e}n~176 exp(-l/r = o(r for all values of n and hence e x p ( - X / e ) is smaller t h a n all gauge functions of the set {~n}n~176 0. Therefore we call e x p ( - 1 / r asymptotically zero and this is denoted by e x p ( - 1 / e ) ~ 0.

3. Asymptotic Series

13

3. A s y m p t o t i c S e r i e s We start immediately with some definitions. DEFINITION 6

Let f(x,e) be a function defined in D • (0, e0], and {6n(e)}n~__0 an asymptotic sequence as 6 $ 0. If there exists a function gg(x) and functions {a,~(x) },~=1 g finite in x 0 E D , such that N

f(xo, s) -- Zan(Xo)5~(~) + 9N(xO) " O(hN(~)), as e $ 0,

(2.3.1)

n--O

then we call the right-hand side "the asymptotic expansion of f(x, ~) at xo E D up to the order 5N(~ ) aS e .~ 0". In case (2.3.1) is valid for all x0 E D, then we drop the suffix 0 and we call the right-hand side the asymptotic expansion of f(x,r in D up to the order 6N(r as e$0. DEFINITION 7

If (2.3.1) is valid for all x0 E D and for any positive integer N, then we write oo

f(x,e) ,.~ Z a , ( x ) 6 , ( e ) ,

x E D,

e $ O,

(2.3.2)

n----O

and the infinite series in the right-hand side is called "the asymptotic series of f (x, e) in x E D with respect to the asymptotic sequence 5~(e) as e $ 0". The series is called

asymptotically convergent. DEFINITION 8

In case (2.3.1)ia valid for all x E D and 9N(X) is uniformly bounded in D by some number KN the relation (2.3.1) maybe written as N

f(x,e) = Za,~(x)6,(r

+ o((~N(6)) ,

VX E

D as ~ $ 0,

(2.3.3)

n--O

and the asymptotic expansion of f(x,r up to the order 6N(r is uniformly valid in D. In case (2.3.1) is valid for all x E D and all natural numbers N and the set of functions {gu(x)} is uniformly bounded in D by some number K the asymptotic "equality" (2.3.2) is uniformly valid and the infinite series (2.3.2) is called "the uniformly asymptotic series of f (x, e) with respect to the asymptotic sequence 6, (e) as e $ 0". oo

In the special case {6n (e)}oon=0 = { en }oon=0 we call the asymptotic series y] an (x)r n the n--0

Poincar6 expansion of the function f(x, e) at the point x E D. It is easily shown that the absolute uniform convergence of the series (x)

f(x,r

= Zan(x)6n(r n--O

with 0 < e < Co,

14

Chapter 2. Asymptotic Expansions

uniform with respect to x E D, implies t h a t this series is also a uniformly asymptotic series for f(z,e). An example of an asymptotic series not uniformly valid in its domain of definition is given by the function

f (x,6) = "~/x + 6 = v/'x 9

+-,

x>O.

X

This function has for all x with x > 0 the asymptotic expansion 6

. 6. .2.

+ (-1)'-l(2n-

~-,~

f(x, 6),~ V'~(1 + 2--x - 8x 2

3)!! 6 n

(;) +...),

uniformly valid in any left bounded interval x > ~ > 0; however, the expansion is not uniformly valid in x > 0 because the remainder after n terms is not 0(6~), uniformly in x > 0. ( ( 2 n - 3 ) ! ! - 1 - 3 . 5 . . . ( 2 n 3)). From the definition of the o-symbol and from definition 7 it follows t h a t the coefficients an(x) of the asymptotic series (2.3.2) are uniquely determined by the function f ( x , 6); one proves simply t h a t these coefficients are defined by the recurrent relations

ao(x)

=

lim

f (x, 6)

~,o ~o(~)

and

(2.3.4) n--1

f (x, 6) - ~ ai(x)e~i(c) an (x) = lim ~,0

i=0 ~n(e)

,

n=l,2,....

However the inverse is not true as is clear from the following counterexample. Suppose oo

f(x,6)

has the asymptotic expansion,

~ an(x)6 n, with respect to the asymptotic n=0

sequence {6n},~~176, then the function f ( x , 6) + exp ( - 1 / 6 ) has the same asymptotic expansion with respect to {6n}n~176. This consideration leads to the following definition. DEFINTION 9 Two functions which have the same asymptotic expansion with respect to the same asymptotic sequence are called asymptotically equal with respect to this sequence. A function is called asymptotically equal to zero with respect to a certain asymptotic sequence whenever all coefficients an(x) are zero. This definition m a y be applied as follows: i) Two functions are asymptotically equal in a point x E D. ii) Two functions are asymptotically equal in all points x E D. iii) Two functions are uniformly asymptotically equal in D. T h e latter applies only in the case t h a t b o t h functions possess uniform asymptotic expansions.

3. Asymptotic Series

15

We finally remark, that the asymptotic expansion changes, when the asymptotic sequence changes . We illustrate this with the following example, given by the function ( 1 - l +exe )-1 ' defined for Ixl < 1.

f(x,r r

Expanding with respect to 1-~ we obtain oo

-C ) nxn

S(x,e) = E ( 1

+ e

r~--O

and N

oo

Z( n=N+l

If(~' ~) - ~ - ~ ( 1 + ~ n=O

l+e

)N+llxIN+lf(x'c) 0,

00

I = m~x I,'"~-~1--08(1). ~_>0

A series of the type (2.7.2) is not of Poincar6 type and so we call it a generalized

asymptotic expansion. Exercises 1. Show for 6 small and positive and for all n _ 0

x"e -~- = 0 ( 6 " ) uniformly in any x interval in [0, c~). 2. Determine the asymptotic expansion of the function G

f(x; 6) = (x + 6)(1-- x)'

0 < x < 1,

with respect to {6"}n~176 0 < 6 0,

5>0,

n=l

and

f ( t ) < K e bt f o r t _ > a ,

K>0,

b>0,

then o0

oo

F(s) - / e - ~ t f ( t ) d t ~ 0

~-~a"r(n-)s-e'r n=l

uniformly valid for Is[ large and [arg s[ _< 2 - A, ~ > O. Hint" the remainder of the right-hand side of F(s) cx~

reads

N

RN(S) = / e - ~ t { f ( t ) 0

Ea~t(~-l)}dt n=l

and there exists a constant C such that

N If(t)- Eant(~-l)

J ~_ Ct (N+~-l)ebt,

Vt>0.

n=l

6. Give the asymptotic expansion of oo

/ e-wy dy, F(w) = 1 + y o

w --+ +cr

Derive from this result oo

g(~) .= f0

l +~-~ xt

dt ~ E ( - 1 ) ' ~ n ! x n n~0

0 0. In C h a p t e r 5 we extend the theory in such a way t h a t we obtain an approximation for x(t) with error 0(e), uniformly valid in an interval [0, be-i], with b an arbitrary number, independent of e.

3.2

The Motion of a Planet around the Sun

In this subsection we study the motion of a planet under the influence of the m u t u a l force of gravitational attraction. It follows from mechanics t h a t the planet moves in a fixed plane and we introduce in this plane polar coordinates (r, 0) with the center of the sun as origin

P

J O Figure 3.1 We neglect the influence of the other planets on the motion and we use Newton's law, viz:

Mm F = G ~ r2

'

where F is the force of attraction, G the constant of gravity, and M and m the masses of the sun, respectively the planet. An elementary calculation yields for the

39

3.2. The Motion of a Planet around the Sun radial and the transverse components of the acceleration the formulae:

ar

d2r dt 2

(dO) 2 r --~

1 d(dO) a # - - r - ~ r2

and so it follows from Newton's law of motion:

d2r dt 2

(dO) 2 Mm r -~ - -G r 2 , r2

r dt

-~

(3.3.7)

-- O.

(3.3.8)

Integrating the second equation we obtain the second law of Kepler, i.e.

r 2dO -~- = h = constant,

(3.3.9)

which means t h a t the angular m o m e n t u m of the planet is constant. We put 1

~(~) = ~(~),

where r - r(0) denotes the trajectory of the planet. We get the following relations

du _ du ~ d O _ dO dt / dt and

1 dr / h _ r 2 dt / -r 2

d2u d du ~ d O dO 2 - dt ( - ~ ) -~ =

1dr h dt ,

l d2r/ h, h-dr ~ -~

or with the aid of (3.3.7)"

d2u dO 2

Mm h2

1 r

or

d2u Mm +u=G~ dO 2 h2 9 Finally, we introduce a characteristic distance ~ and the dimensionless variable - ~u; ~ satisfies the differential equation

d2~t dO 2

- -

with a = G Mm~ h 2

-~-fi=

a

(3.3.10)

9

The solution is readily obtained and we get the result = a + ( b - a)cos(O - O0),

(3.3.11)

40

Chapter 3. Regular Perturbations

where the constants b and ~0 are determined by the initial conditions, which specify the position and the velocity of the planet when e.g. v9 = 0. It follows t h a t r(0)

is given by

r(vq) = a + (b - a)cos(~ - 0o)

(3.3.12)

and the trajectory of the planet is for b < a a fixed ellips with the centre of the sun at one of its loci. However, observation has revealed t h a t the major axis of the ellips rotates slowly around the sun and so the result (3.3.12) is not completely correct. It was a verification of Einstein's theory of gravity that this rotation could be explained by using the theory of general relativity and the so-called Schwarzschild metric (see e.g. [7], Ch.XIV). In this theory the equation (3.3.10) should be replaced by d2'5 d,02 t- '5 - a + ~'52

(3.3.13)

with e---- 3 Gc2e M m , c the velocity of light, and so 6 is a small positive number. Therefore we are led again to a p e r t u r b a t i o n problem which can be handled in the same way as the nonlinear spring. The equation (3.3.13) is supplemented with the initial conditions d'5 '5(0) -- c~, ~-~ (0) -- ,6. (3.3.14) Applying the theory of this chapter we make the assumption

~(~) = u0(O) + ~ , ( ~ ) + . . . + ~"u.(O) + . . .

(3.3.15)

Substitution into (3.3.13) and collecting the terms with equal powers of e we get a linear system of recurrent initial value problems for the functions ui(tg), i - 0, 1, 2 , . . .

d2uo d0---5- + u0 = a,

u0(0) - c~,

duo - - ~ ( 0 ) --/3

(3.3.16)

d2ul dO----5- + ul = u20, ul(0) = 0,

dul --d-~(0) = 0

(3.3.17)

Solving these problems consecutively we obtain: (3.3.18)

Uo(O) : a + (o~ - a)cos vq + ~ sin z9 d

d

ul (~) = sin 0 / [ u o (~o)]2 cos ~od~o- cos ~ / [ u o ( ~ ) ] 2 sin ~d~o o

o

= a(c~ - a)O sin 0 - afh9 cos 0 + . . . .

(3.3.19)

Exercises

41

According to Theorem 3.3.1 we have the result

~(~) = ~0(~) + ~

(3.3.20)

(~) + 0(~ ~)

uniformly in any bounded domain, say o < ~ < t~b, with Zgb independent of ~. We see again the appearance of "secular" terms ~)sin zg, t9 cos~9 in the contribution cul (v~), and thereforethe expansion (3.3.20) is only meaningful in bounded ~9-intervals. In the next chapter we explain as a consequence of (3.3.20) the perihelium rotation and in Chapter 5 we extend the range of uniform validity. Exercises

1. Let y(x; ~) be the solution of the initial value problem

dy =l+y2+r dx

x>0,

y(0)=r

0 0. This becomes apparent in the occurrence of secular terms; see the examples of the nonlinear spring and the motion of a planet. Nevertheless, by using a modification of the regular perturbation method involving a stretching of the time coordinate as t-

( 1 + ewl + e2w2 + ''')T

(4.1.1)

it is possible to determine the constants {wi} such that the solution of the perturbed differential equation does no longer contain secular terms; in this way one obtains an asymptotic expansion valid for time intervals of O(~). With the aid of this modification it is also possible to obtain information on certain important quantities such as for instance the frequency of the nonlinear spring and the perihelium precession of the planet. The method of straining the coordinate originates from Lindstedt [98, 99] and Poincar~ [116], who introduced it in studies of perturbation problems in celestial mechanics. Therefore it is also called the Lindstedt-Poincard or shortly the L.P.-method. We illustrate the method by treating again the nonlinear spring and the motion of a planet around the sun. The question of the asymptotic correctness of the Lindstedt-Poincard procedure will be dealt with in the next chapter. We conclude this chapter with a procedure introduced by Lighthill [96, 97], who generalized the stretching transformation (4.1.1) by setting t -- T + e f l ( T ) + e2f2(T ) + . . . , (4.1.2) where the stretching functions {fi} are chosen such that an asymptotic expansion of the solution of the perturbation problem becomes possible. This transformation appears to be useful for a class of perturbation problems for which the reduced differential equation with e - 0 contains a singularity, e.g. (t -1t- "CU)du + q ( t ) ~ = ~(t), t > o.

44 2.

2.1.

Chapter 4. The Method of the Strained Coordinate o f the Strained C o o r d i n a t e

Applications of the Method

T h e N o n l i n e a r Spring We consider the initial value problem d2x

dt 2 t - x + ~ x 3 = 0 ,

t>O

(4.2.1)

x(0) = a and -~- (0) = 0.

(4.2.2)

w i t h the initial conditions dx

We introduce the following transformation of the independent variable: (4.2.3)

t = (1 + ewl + e2w2 + " ' ) %

where the w~, i - 1, 2, 3 , . . . , are as yet undetermined constants to be chosen in such a way that the secular terms in the asymptotic expression for x(t) disappear, when x is represented as a function of r. Substituting (4.2.3) into (4.2.1) we get the initial value problem for x as function of T : d2x dT----~ -~- (1 + ewl + g2w2 -~---')2(x -[- Ex 3) -- 0,

T > 0,

(4.2.4)

dx

x(0) = c~,

~--~T(0) = 0.

(4.2.5)

This problem is again of the form treated in the preceding section, whenever 1 + ewl + e2w2 + " "

(4.2.6)

is asymptotically convergent. Under the assumption that this is the case, we may apply the regular perturbation method, valid for finite T intervals. So we expand x(~-) as

~(~) = ~0(~) + ~ ( ~ )

+ ~(~)

(4.2.7)

+-

Inserting this expression into (4.2.4)-(4.2.5) and taking together equal powers of e, we obtain the following recursive set of linear initial value problems: d2xo a~---3- + ~o = o,

dxo -~j(o)

~o(O) = ~,

d 2x 1

dT_-----Y -f- xl = -(Xo3 + 2wlxo),

(4.2.8)

= o, dx 1

xl(0) = 0,

--~-T(0) = 0,

(4.2.9)

d2x2 dT---Y + x2 = --3x2x, -- 2w1(x1 -[- XO3) -- (2~2 -[-W12)Xo, dx2 -jj~ (o) = o

~ ( o ) = o, .

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(4.2.10)

45

2.1. The Nonlinear Spring The solution of (4.2.8) reads x0(r) = a c o s T

(4.2.11)

and we obtain for Xl(T) the initial value problem:

d2xl d ~ + ~1 = - { ~ ( ~ o s ~) ~ + 2 ~ 1 ~os~} C~3 -

- c ~ , , (_3 4 62

+ 2wl) cos T -- --4 cos 3T.

The first term of the right-hand side gives rise to the appearance of a secular term in x l ( T ) unless we take 3 O~2 . (4.2.12) CO1 DE 8

Taking this value for W1 we obtain: C~3

Xl(T) = ~--~(cos 3T

-

COST).

(4.2.13)

Substitution of (4.2.11), (4.2.12) and (4.2.13)into (4.2.10) yields for X2(T) the initial value problem:

d2X2dT 2+

(~57

x2 = a ~ l ~ g

~ 4 _

2w2)COST

+

N.S.T.,

(4.2.14)

where N.S.T. denotes the terms which will not lead to a secular contribution in x2(T); so N.S.T. does not contain a term such as cos T or sin T. Avoiding again the appearance of secular behaviour of X2(T) we should take 57.64" 256 In principle this process may be repeated indefinitely; up to the second order we have 092 ---

OL3

X(T) = a cos T + C~-~ (COS3T -- COST) + 0(~2),

(4.2.15)

uniformly valid in any finite T interval 0 _< T _< TO with TO independent of ~, where the strained coordinate T is given by T = (1 -

57 o4c 2 -~-'-' ) - i t .

c~2~ + 256

(4.2.16)

However, it should be noted that the result (4.2.15) is only mathematically justified if the series (4.2.6) converges asymptotically. Assuming for the moment that this is the case we obtain for the circular frequency of our nonlinear spring w = 1 + _3 862~-

21 O4e 2 -~-0(e3). (4.2.17) 256 We remark that this frequency depends on the initial amplitude a and of course also on The expression (4.2.15) still makes sense for large values of T and in the next chapter we shall prove ~'.

x(t) = a cos{(1 + g3626)t} + 0(~), uniformly valid for 0 < t < to with to a positive number, independent of ~.

(4.2.18)

46 2.2.

Chapter 4. The Method of the Strained Coordinate The Perihelium

Precession

The differential equation for the trajectory in polar coordinates planet reads according to Example 2 of the preceding chapter:

r - r(0)

of the

d2u dO---~ + u = a + eu 2,

0>0

(4.2.19)

with u = ~ and ~ is a characteristic length; a is a positive constant and e the small parameter. The initial values are chosen as

du

(4.2.20)

u(O) = cr and ~-~(0) = ft. We introduce again the strained coordinate

(4.2.21)

0 = (1 + ewl + e2w2 + ' " ) 9 ~ , and the initial value problem becomes d2u + (1 + COdI + ~2~d 2 ~ - ' " .)2 (U -- a -- Cu 2) ~- 0 d~p2

(4.2.22)

with

du

u(0) = a and - 7 ( 0 ) = fl(1 + ewl + e2w2 + ' " "). aT

(4.2.23)

Under the assumption t h a t the coefficients wi will be chosen such t h a t (1 -Jr CO)1 -~ E2W2 - ~ ' - . ) is asymptotically convergent we are again in the situation t h a t the regular p e r t u r b a t i o n m e t h o d may be applied. So we set

u(~) = ~0(~) + ~Ul(~) + ~ ( ~ )

+...

(4.2.24)

Substitution into (4.2.22) and (4.2.23) yields again a recursive system of linear equations: d2u______oo_ u0 = a; dT 2

u0(0) = a,

d2ul d~ 2 + u~ = u 0~ - 2 ~ 1 ( u 0 - ~ ) ;

duo - ~ - (0) = fl, ~(0)

= 0,

(4.2.25)

dltl

-d~ (0) =Wlfl.

(4.2.26)

The solution of (4.2.25) is simply obtained UO(T) = a{1 + e cos(qo -- (P0)},

(4.2.27)

where e and ~o0 are determined by a and /3. The function r0(T) = ~o(~) 1 is an ellips, if e > 0, with the origin as one of its loci.

47

3. The Method of the Strained Parameter

Substitution of u0(cp) in the right-hand side of the differential equation (4.2.26) yields

d2Ul d~o------5-

+

u:

--

a2(1 + l e 2 ) + 2ae(a

-

w:)cos(~o

-

~Oo)

+ l a 2 e 2 cos 2 ( ~ 0 - qo0).

2 This equation gives again rise to a secular term unless we take wl - a.

(4.2.28)

Therefore we get

u = a + ae cos(~o - ~o) + 0(:), uniformly in any bounded ~ interval 0 ~ ~ < 9~0 with q:, = (1 + a: + w2:2 + . . . ) - 1 0 . This process may, in principle, be repeated indefinitely and the asymptotic approximation is valid if the series 1 + a : + 022:2 -~- "'" is asymptotically correct. Assuming the latter condition we have finally u(O) -- a + ae cos{(1

-

ae)O

-

0o} + 0(:),

(4.2.29)

uniformly valid for 0 < 0 _ t~b with 0 b a n arbitrary positive number independent of In the next chapter we shall prove the asymptotic correctness of (4.2.29), which is even uniformly valid for 0 < 0 < o__~bwith ~qb again some positive number, independent of :. The peri- and aphelium of the planet is obtained for the values 0 -- Op with du

--0

(u is maximal, respectively minimal)

dO

i.e. 0p satisfies approximately the relation sin{(1 - a : ) O p - 00} = 0, and it follows that (1 - a:)vqp -- t90 + 7rn, n = 0, 1, 2. Two consecutive values of the perihelium angle : ~ ~ 21r + 21ra:, and so the perihelium precession is approximately given by differ y 2, the formula G2M2m 2 2~ra: - 6 1 r ~ . (4.2.30) c2h 2

3.

T h e M e t h o d of t h e S t r a i n e d Parameter

It may happen that in a differential equation, apart from the small parameter e, another parameter, say p, appears which depends on :, but this dependence is quantitatively unknown. The requirement of a certain behaviour of the solution of the differential equation, e.g. no secular terms should be present, may lead to an asymptotic

48

Chapter 4. The Method of the Strained Coordinate

expansion of the p a r a m e t e r p into powers of e. The m e t h o d for obtaining an asymptotic approximation for as well the solution as for the unknown p a r a m e t e r is sometimes called the m e t h o d of the strained parameter. We give as an illustration the following example. Stokes [147, p.486] considered in 1849 weakly nonlinear wave propagation on the surface of a liquid. His model is one dimensional and the height u ( x , t) of the fluid measured from equilibrium satisfies an equation of Klein-Gordon type utt - o~2uxx -~- ,),2~t = / ~ u 3,

(4.3.1)

where a, /~ and 7 are certain constants. We are interested in obtaining the so-called d i s p e r s i o n r e l a t i o n for p e r i o d i c t r a v e l l i n g waves; the dispersion relation relates the wave number and the frequency. The linear case with fl = 0 is almost trivial. Putting u = u(~), ~ = x - ct, (4.3.2) with c > c~ > 0 the wave velocity, we get for u(~) the ordinary differential equation: (c 2 - a 2 ) u '' + 72u = 0, and it follows t h a t

u(~) is a linear combination of sin{ ~/c 27--0~2 ( x -

ct)}

and

c o s { ~/C . . . 72 -- Ot2 ( x - ct)}, and so the wave number is given by k=

X/C

7

-- C~ 2

and hence w 2 = k2c 2 = a2k 2 + 72,

(4.3.3)

which relates the wave number k and the frequency w. It is to be noted t h a t this relation is independent of the amplitude of the wave. We consider now the full nonlinear equation, which reads (C 2 -- O~2)U t' ~- ')'2U = ~ U 3

(4.3.4)

with u = u(~) = u ( x - ct). We assume t h a t the amplitude a of the wave is small and so we put u = aft

(4.3.5)

(c 2 - a2)~" % 72fi - ~a2fi 3,

(4.3.6)

and there results and this would present again a p e r t u r b a t i o n problem of the kind considered before with a 2 instead of e, whenever the wave velocity were to be independent of the amplitude. But as is well known from nonlinear wave p h e n o m e n a the speed of the wave depends on the amplitude. Because a is small we expand c into powers of a C -- CO + acl A- a2c2 + ' ' ' ,

(4.3.7)

49

3. The Method of the Strained Parameter where the coefficients c~ should be chosen properly with co > c~. We bring (4.3.6) into the form ~" U

-V2~2 + ~a2u3 --

c2 _

(4.3.8)

ol 2

and we expand c2_1~2 into powers of a. If the series (4.3.7) converges asymptotically we are again in the situation of C h a p t e r 3 and we may apply the regular p e r t u r b a t i o n method. We assume now t h a t (4.3.7) is indeed asymptotically correct and we put (4.3.9)

~(~) = ~1 (~) + a~2(~) + a2~3(~) + " " or what amounts to the same

u(~) = aul(~)+ a2u2(~) + a3u3(~) + " "

9

(4.3.10)

Instead of substituting (4.3.9) and (4.3.7) into (4.3.8) and taking together equal powers of a, we may as well substitute (4.3.10) and (4.3.7) directly into equation (4.3.6). Taking together equal powers of a, there results the following recursive set of linear equations for the functions ui(~), i = 1, 2 , . . . .

(~] _ ~2)~, + ~ 2 ~ = 0

(4.3.11)

(~ - ~)~

(4.3.12)

+ ~

(~0~ - - ~ ) u g + ~

= -2~o~ (-2~o~

--

--

C l~' )U

"1

--

2CoClU

"2

-'k

Z~.

(4.3.13)

We start with a particular solution of (4.3.11), viz. (4.3.14)

~1 = cos(k~) with k2 _ -

,)12

c2 _ a2 or c~ = c~2 + ~'2k-2,

(4.3.15)

which corresponds with the dispersion relation (4.3.3) for the linear case with fl = 0. We substitute Ul(~) into (4.3.12) which gives

,

2c0cl k2

u~. + k2u2 = Co ~ _ ~.

cos(k~).

Searching for waves a2u2 of small amplitude we should eliminate secular terms and so we must take Cl - 0 and the equation for u2 becomes the same as t h a t for ul 9 The equation for u3 becomes now

,,

U3

~

k2u3

--

c~

-

1 oz 2 (2coc2 k2 +

p) cos(k~) + 4(c2 ~- c~2) cos(3k~) "

Eliminating the secular t e r m again we should take 3

3p

4

8cok2

2coc2 k2 -~- -:~ = 0, or c2 --

(4.3.16)

50

Chapter 4. The Method of the Strained Coordinate

and the differential equation for u3 reduces to u3" + k~u3 = 4 ( ~ -f~

~)

cos(3k~).

(4.3.17)

Under the condition that (4.3.7) is asymptotically correct, i.e. c - c o -4- a2c2 + 0(a 3) - Jo~ 2 A- ~,2]~2 _

3~a2 8k 2 J ~ 2 ~_ ~,2k-2 -4- O(a 3)

we have obtained the approximation u ( x - ct) = a cos k ( x - ct) + O(a2).

(4.3.19)

with the dispersion relation = kc = V/c~2k2 + 72 -

3f~a2 + O(a3). 8 j ~ 2 k 2 A- ,72

(4.3.20)

The dependence of w on the amplitude appears to be only quadratic. It is not difficult to integrate (4.3.6) directly; multiply with fi', integrate and one obtains u as an elliptic function of ~, (cnoYdal wave). Taking the amplitude small one arrives at a result similar as (4.3.20); see also [147], p.486-489. 4.

Lighthill's method

An interesting class of perturbation problems concerns the case where the reduced equation with ~ = 0 is singular while the perturbed equation is not singular. A typical perturbation problem of this type is (x + r

du + q(x)u = r(x), -~x

0 < x _< 1,

(4.4.1)

with u ( 1 ) = b and q and r smooth functions, q ( O ) # O. Using the Poincar6-Lindstedt method the uniformity of an asymptotic expansion does not hold in a neigbourhood of x-O. Putting x = (1 + eWl + ~20.'2 + ' " ")~

~(~) =

~0(~) + ~u~ (~) +

~ ( ~ ) +...

leads to the recursive set of singular linear differential equations ~--~

+ q(~)u, = f,(~,

uo, u l , . . .,

Un--1)~

with solutions not suitable in the expansion of u(~) in the neighbourhood of ~ = 0. To illustrate this situation consider the very simple example (z + r

-du -~x+U--O,

O<x ~c and it is graphically presented in Figure 4.1 and the segment O_<x_< 1 is m a p p e d onto the segment

~ ~_ 2. Because Lighthill's method has been widely applied to a large variety of perturbation problems we have chosen to give a detailed treatment of a simple problem. Our derivations have been formal and we do not give a proof of the validity of the method; we remark only that Wasow [143] has investigated conditions for the convergence of Lighthill's expansion for the solution of the problem (4.4.1). 5.

Temple's method

A method of approximation closely related to that of Lighthill has been given by Temple [132]. Consider the perturbation problem f (x, u;-c) du - g(x,u;~),

X < Xl,

U(Xl)-C, 0 < c < < l .

(4.5.1)

Suppose that it is possible to replace the differential equation by the equivalent system dx = X ( x , u; ~), d~

du -w~ = U(x, u; c), a~

(4.5.2)

where X and U have uniform asymptotic expansions in powers of e with C ~ coefficients in x and u. Then it is possible to apply the regular approximation procedure as discussed in Chapter 3, and in this way one obtains in a systematic way the functions {ui(~)} and {fi(~)} appearing in Lighthill's expansion. The possibility to split the differential equation in the form (4.5.2) was already present in the example of the preceding section, see (4.4.2) and (4.4.3). Here we give another example, but this one does not lead to a simple integrable Hamiltonian system. We consider

(x + cu) du dxx + ( 2 + x ) u = 0 '

(4.5.3)

u ( 1 ) - e -1.

From this equation it follows that u(x) > 0 for 0 _< x _< 1, and so the differential equation does not contain a singularity for 0 _< x _< 1. (4.5.3) is equivalent with the equations

~ dx -~+~nd~

du

=

- ( 2 + x)u,

Ux-- 1

-

-

e -1,

~ ~: 0.

(4.5.4)

Expanding

= ~ + ~I~ (r + o ( ~ ) , = ~0(~) + ~u, (~) + o ( ~ ) ,

(4.5.5) (4.5.6)

56

Chapter 4. The Method of the Strained Coordinate

we get the following differential equations

dYl

~-d-~- -- f l ( ~ ) + u0(~),

(4.5.7)

duo : ~-d~-

- ( 2 + ~)u0,

(4.5.8)

~dul -~=

- ( 2 -]-~)tt 1 -- fl (~)u0(~).

(4.5.9)

The solution of (4.5.8) with the condition uo(1) = e -1 gives 1

~0(~) :

~2

(4.5.10)

~-~"

Inserting this expression into (4.5.7) we pick for fl(() the solution

fl (~) -- ~

dr/= ~ 1

~dr/.

(4.5.11)

1

The results (4.5.10) and (4.5.11) determine now the equation (4.5.9) with the boundary value ~U 1(~*) : e -1 -- ItO(~*),

(4.5.12)

1 = (* + ell (~*) + O(e2).

(4.5.13)

where ~* satisfies the equation

The result is u:(:) =

- ~1 e- ( f

(77)dr/+

fl

e - I __ UO(~*)

However c -1 --

--

e

)(=1(~

--

1) , ~

-3e-

e/l(~*),

and so with the aid of (4.5.11), (4.5.13) and ~* = 1 + O(e) 1 _~ f fl(r/) Ul(~) = --~-~e r/ dr/+ O(e). 1 The final result we have obtained reads now as follows: x -- ~ A- ~ f l (~) -f- O(~2),

1 _~ (1 -- e J

U = ~--~e

1

f l (r/) r/

dr/) -~- O(g'2),

(4.5.14)

6. Limitations of the Lindstedt-Poincar~ method

57

with

.f~ (r = ~ f --~-dn, e-~

1 or

x - ~(1 + e~(~)) + O(e 2)

(4.5.15)

1 -~(1 - e / u = ~-se

(4.5.16)

O(~)drl) + O(e 2) 1

with ~(~) -

f --~e--rbdr/.

(4.5.17)

1

(4.5.15)-(4.5.16) gives a parameter representation of the solution u(x) of the initial value problem (4.5.3) for all values of ~ _> 5 > 0, with 5 arbitrarily small, but independent of e. The approximation contains the secular terms eO(~) and e f : ~)(r/)dr/. 6.

L i m i t a t i o n s of t h e L i n d s t e d t - P o i n c a r 6

method

The Lindstedt-Poincar6 method treated in this chapter has the advantage that secular terms may be eliminated from the expansion of the solution and the method has been successfully applied to vibration and wave problems described by differential equations of the type

d2x

02u

dt 2 + w2x -- ~ f ( x ) or - ~

_ c 2 02u

~ x 2 -- ~ f ( u ) .

However, there may arise difficulties, whenever a small damping term is introduced into the equations, even in the linear case. Let us consider as an illustration the simple initial value problem

d2x dx + 2e-~. + x = O , dt----~ dx x(O) - O, --~(0) = a

t > 0

(4.6.1) (4.6.2)

Its solution is readily obtained and it reads e

--et

x(t) = a-~/1 _ e 2 sin (V/1 - s2t),

(4.6.3)

valid for arbitrarily longtime intervals. Expanding this expression into powers of r leads to secular terms and we cannot expect that the naive regular perturbation method of Chapter 3 yields a satisfactory solution

Chapter 4. The Method of the Strained Coordinate

58

of the initial value problem (4.6.1)-(4.6.2). Also the modification by Lindstedt-Poincard does not give any help as is shown by the following small calculation. We expand the coordinate t as t = (1 + r

+ e2ca2 + . . . ) T ,

and the initial value problem for x as function of T becomes

d2x dT 2 -[- 2e(1 + r

9(o)

=

o,

dx + g2ca2 -~-" "")dT ~- (1 + gcal -~- g2ca2 -~-'" ") 2x = 0

dx = a(1 + ewl + r

-f-'" ")

Putting

~(~) = ~0(~) + ~ 1 ( ~ ) + ~ ( ~ )

+.-

leads to the system

d2xo dT----~ + x0 = 0, d2xl

dxo x0(0) -- 0,

-~T (0) -- a,

~dxo

d~---~ + ~

= -z-jV

dxl (0) - 2~1~0,

~(0)

= 0,

~

= ~1.

So we obtain

Xo(T) -- a sin T and

d2xl + x 1 -- - 2cala sin r - 2a cos r. dT 2 Both secular terms can only be eleminated by taking a = 0 which leads to the trivial solution x - 0. The cause of the trouble is easily understood and it lies in the appearance ax in the original of the t e r m - 2 a cos T, which originates from the damping t e r m 2e 2? differential equation (4.6.1). In the next two chapters we deal with the m e t h o d s of averaging and multiple scales and it will appear t h a t the addition of a damping term is not an obstacle for the construction of an asymptotic approximation, even valid in an unbounded time interval.

Exercises 1. Let x(t)

be the solution of the initial value problem

d2x +x+ex dt 2

3=eAcoswt,

t>0,

0<e O,

1/

y(O) = xo

(5.2.3)

T

fo(Y) = ~

f(T, y)dT.

(5.2.4)

0

Under these conditions the function y(t) belongs on the time scale e -1 dependent interior subset f) C 79 and we have in f) 9 (t) - y ( t ) - o ( ~ ) ,

to an e in-

(5.2.5)

uniformly for e $ 0 on the time scale e-1. Proof The function fo(Y) inherits the Lipschitz property from the function f ( t , x ) and the existence of y(t) in an e-independent interior subset ~ C 79 on the time scale e -1 is

Chapter 5. The Method of Averaging

7O

proved in the same way as in T h e o r e m 2. W i t h the aid of (5.2.2) and (5.2.3) we have on the time scale e -1 t

t

x(t) - y(t) - r / f(T, X(T))dT -- e / fo(y(T))dT 0

(5.2.17)

0

Applying local averaging with To - T, using L e m m a s 1 and 2, and remembering t h a t due to the periodicity of f ( t , x ) the local and the ordinary average of f ( t , x ) coincide, we get from (5.2.17) on the time scale e -1 t

t

f

-:/fo

0

+

0 t

= e / { fo (x(r)) - fo (y(r)) }dr + O(eT). o The Lipschitz condition for f induces a Lipschitz condition for Lipschitz constant, and so we obtain

f0

with the same

t

Ix(t) - y(t)[ < EA / I x ( T ) - y(T)ldT + O(eT). 0

Using Gronwall's l e m m a (Ch.3, L e m m a 3) once again we have on the time scale e -1

Ix(t)- y ( t ) l - O ( e T e ~ t ) ,

(5.2.18)

which proves the theorem.

2.3.

The K r i l o v - B o g o l i u b o v - M i t r o p o l s k i T h e o r e m for Nonperiodic Fields; K.B.M. Theorem, Third Variant In the preceding subsections we treated the system

dx d---t = e f (t' x)

t>0,

x(0)=x0

with f periodic in t. We consider now the more general case where f is no longer periodic in t. In this case the K.B.M. theorem reads as follows, see also [31], [123]. THEOREM 4 (K.B.M. theorem, third variant) Suppose

x(t)

satisfies the initial value

problem d_ff = e f ( t x) t > 0, x(0) = x0, (5.2.19) dt ' ' where f is bounded and continuous in G = [0, oo) x T), with 1) a ball I x - x 0 l ~_ d in ]R~, and further f is Lipschitz continuous in G with respect to x, while the Lipschitz constant A is independent of t.

2.3. The Krilov-Bogoliubov-Mitropolski Theorem for Nonperiodic Fields

71

Further it is assumed that the average T

(5.2.20)

f o ( x ) = Tlim - ~ T1 f f (t, x)dt 0

exists for all x C 79. Let y(t) be the solution of the initial value problem dy d---[ = efo(y),

t > O,

y(O) = xo,

(5.2.21)

belonging to an e-independent interior subset f) C 79 on the time scale e - 1 . Under these conditions the solution of the initial value problem (5.2.19) satisfies in f) -

y(t)

(5.2.22)

=

uniformly on the time scale e -1 with t

5(e) =

sup

sup

x(t)ET)

tE[O,Lr -1]

~1

f

fo(x))dTI,

(5.2.23)

0

where L is some constant independent of e. Proof According to Theorem 1 the solution x(t) of the initial value problem (5.2.19) exists on a time interval d 0 _< t _< inf [t2 e ' e M ( t l ) ]' with tl arbitrarily large positive and M(tl) :

Taking tl 0 ( t (

w

max If(t,x)l

=ET~ O 0, (5.2.2s) dt where f satisfies in G = [0, oo) x D the conditions of Theorem 4 of this section; /9 = {x; x - Xo[ _< d}. Further it is assumed that the averaged equation

dy dt = efo(y) with y(O) = xo,

t > 0

(5.2.29)

has the property that fo(c) = 0 and that the point y = c is an asymptotically stable critical point in the linear approximation of (5.2.29). If c belongs to the ~ independent interior subset Z) C 7), fo is continuously differentiable in l), and xo belongs to the domain of attraction 7)o of y - c with 7)o C I), then x(t) = y(t) + 0(5 89(e)) (5.2.30)

for O < t < c~. Hence also x(t) approaches for e sufficiently small the critical point x = c. For the proof, we refer the reader to the references cited above. In the next sections we apply the theory of this section to a number of oscillation problems.

75

3. Weakly Nonlinear Free Oscillations 3. 3.1.

Weakly Nonlinear Free Oscillations The General Case We consider weakly nonlinear free oscillations governed by the differential equation (5.3.1)

d2u (du) d t---~ -t- u = ~ f u , - - ~

,

with f ( u , v ) continuously differentiable with respect to u and v and e is a small positive parameter. The initial conditions are given by u(0) = a, Substituting u -

du - ~ ( 0 ) - / 3 , with C~2 -~-/~2 r 0.

(5.3.2)

zl and -~ du = z2 we may write (5.3.1) as the system dzl dt = 22,

dz2 _ dt - - z l + ~ f ( z l , z2).

(5.3.3)

For r = 0 the general solution reads Zl -- a cos(t + 0),

(5.3.4)

z2 = - a s i n ( t + 0)

with the amplitude a and the phase 0 as arbitrary constants of integration. We use now Lagrange's method of variation of constants and consider (5.3.4) as a socalled phase-amplitude transformation from the functions (zl (t), z2(t)) to the functions (a(t), O(t)), which is clearly nonsingular as long as the Jacobi determinant, equal to - a , does not vanish. After a straightforward calculation we get for (a(t), O(t)) the equations da2 d d-t or

-- d-t (z12 -~- z22) -- 2ZlZ2 "~- 2 z 2 (

da dt

= -r

-

-

z l -~-

~/(z~, z2))

+ O ) , - a sin(t + 0 ) ) s i n ( t + 0)

(5.3.5)

and dO _ dt -

d (t + arctan dt

~11)-

-

~zlf(zl, z2) z2 + z2

or

dO e = --f(acos(t + 0), - a sin(t + O)) cos(t + 0). (5.3.6) dt a The initial conditions for the system (5.3.5)-(5.3.6) follow easily from (5.3.2) and (5.3.4); we get

a(0) -- V/a 2 +/32 and 0(0) - - arctan -~. a

(5.3.7)

The function f as function of t is periodic with period 2~, and so we have according to the Krylov, Bogoliubov, Mitropolsky theorem (first variant) tl (a, 0) - (5, ~ ) + O(E), uniformly for 0 _ t < -c

(5.3.8)

76

Chapter 5. The Method of Averaging

with tl some constant independent of e and with equations

(a, 0)

satisfying the averaged

21r

da d--t- =

r f 27r j f ( ~ c o s ( t + tg), - ~ s i n ( t + tg)) sin(t +

~)dt

0

and

dr9 dt

271-

a27r

f (a cos(t + tg), - a sin(t + tg)) cos(t +

~)dt

0 or 271"

da- = d-t

r f f (a cos ~0, - a sin ~o) sin ~od~o 21r

(5.3.9)

0

and 211"

dr9 = d-t

a 2e r f f ( a cos ~o, - a sin ~) cos ~ad~a,

(5.3.10)

0

with a(0) = V/a 2 +/32 and zg(0) - - a r c t a n / 3 .

(5.3.11)

The solution of this autonomous system gives a first order approximation for (a(t), O(t)) and hence also for u(t) = a(t)cos (t + ~(t)) and the approximation is uniformly valid for 0 _~ t < tl/r with tl some number independent of e. For practical applications it is useful to have an estimate for the value of tl. Suppose that the greatest of the upper bounds of ]f{acos(t + ~), - a s i n ( t + zg)} I and I~f{ac~ in the domain (D) 0 < t < c ~ , 0_~ l a - v / c ~ 2+/32 I _ d , 0_~ I z g + a r c t a n ~ l < c ~ is Md (in case fT(=0 for a = 0 , we should take d < V/c~2 +/32 ), then the solution of the initial value problem (5.3.5), (5.3.6), (5.3.7) exists for any positive value of e and for d

0 0. 4. 4.1.

T h e Equation of M a t h i e u Introduction

We consider an inverse pendulum with a prescribed force at its lower end S which is permitted to slide freely along a vertical slit. This vertical slit is represented by the y-axis and the prescribed force in the y-direction by the function Y(t) with t the time, see Figure 6.1.

Y

M

Fig. 6.1 The inverted pendulum We assume that the mass of the rod with length I is negligible in comparison with the mass m at the upper end M of the rod and so we may assume that the total mass of the pendulum is concentrated in the point M. Further we suppose that the motion of the point M takes place in the (x, y) plane under the influence of the weight mg, the applied force Y(t) and the reaction force X(t) in the support S, due to the constraint of the rod. The dynamics of the point M is determined by Newton's law and the angular momentum law resulting into the equations

d2x r n ~ = X(t),

(6.4.1)

Y ( t ) t ~ s i n ~ - X(t)~ cos z9 = 0.

(6.4.2)

and

102

Chapter 6. The Method of Multiple Scales

The latter equation results from the fact that the moment of inertia about the centre of gravity M is zero. Assuming that the deviation ~ from equilibrium is small, we get after linearization

m i ~ - Y(t)t9 = O. Putting the applied force Y(t)

equal to

Y(t) = mg-

rap(t)

we get finally the equation g

~

(6.4.3)

In case p(t) is periodic the linear equation (6.4.3) is known as Hill's equation, usually written in the form d2w + q(t)w = 0 (6.4.4) dt 2 with q(t) periodic. If p(t) =_ 0 the solution of (6.4.3) is unbounded and the equilibrium solution 0 = 0 is unstable. Nevertheless, it is possible to choose the periodic function p(t), such that a bounded stable solution exists. Considering a normal pendulum, not upside down, the term -}v~ in equation (6.4.3) changes its sign and the equilibrium solution 0 = 0 is stable. Mutatis mutandis it is again possible to choose a periodic p(t), such that an unbounded unstable solution exists. In this section we consider in particular the equation

d2u dt---$ + (5 + e cos t)u = O,

(6.4.5)

where 5 and e are parameters with 0 < e ~z we have two purely imaginary roots corresponding to the region of stability. The values 6z - =k 1 correspond with transition curves and this is in agreement with the results of the preceding section.

1) The case of instability [61[ < 51 If [~1[ < ~1 we obtain the fundamental set of solutions

(A(1)(~)/= /-r Bo(1) (~)

/

+ g / 1 + ~1 exp [ V / ~ - ~12 ~]

+lexpE

and

-

-

-

+1/ o

113

4.6. The Approximation of the Solution outside the Transition Curves Hence we have

~0/~/= ~111 ~1explll ~ ~1+~11-~ and

~0~) ~il+~ex~ ElI ~ ~l+~i~+~lexp E 1 4 ~1~~1 Inserting the initial conditions (6.4.38) we get a--Clll-

51 -Jr-C 2 1 ~ - 51

2 b = C l l l - + - 5 1 - t - C 2 ~ / 1 +51 and so A0(~)+

(

~a - b

~1 5+5i

( ~ 1 ~+ba

exp [

51) 2zexp[5+51

- 512. ~] il

-52.~]

(6.4.40)

and

( j1 ( i1

B0(~)= -~a ~~ +-5~1 + a

+

+~

g+51

b>exp[ )

-~.~]

1 51 + b exp [ ~_

_ 512 . ~c],

(6.4.41)

Finally substituting these results into (6.4.37) we obtain the unstable leading term of the expansion of the solution of the initial value problem (6.4.31)-(6.4.32) with 5 = ~1 -Jr-51g and

1 For 51 1"+1~ or 51 $ --51 we get the leading term of 151t < 2"

the expansion of the solution along the transition curves. After a small calculation we obtain for 51 1" +51. 1 1 w0(~, rl) = a cos ~ r / + (2b - a~) sin ~rl 1 = a cos-z + (2b2 and for 515

1 2

acz)sin-z

(6.4.42)

21 1

1

w0(~, ,/) = (a - 2b~) cos ~rl + 2bsin ~,/ 1 1 = (a - 2b~z) cos ~ z + 2b sin ~ z

(6.4.43)

"~

o

o

o ..~

~

.~

~

~176 "~

" "~~

o ~o .,-,

~

,_,~.~ ~

~.~

~'0 ~

,,,,~

~

~

I

II

' "~

+

~

I

~

I

~

Ii I

0

~ 9~

l

II

~ 0

~ 0

II

.,~

~'~

I

o

o c.,)

C + 9

'-'

C o,i,

: C ..., ~

8

r

r.

~

~

u

II

"I=I

,

0 C9

0

'-0

~I~

.,,.

.,d

o o

~

,,d

o o

4-

II

I

-l-

I

q

.-.....:>

II

o

.o,i,

,,~ i r 1 6 2

o

o.../,

5.1. Introduction

115

Substituting finally, these results into (6.4.37) we obtain the stable leading term of the expansion of the solution of the initial value problem (6.4.31)-(6.4.32) with 5-

1 ~ + 51e + . . .

and

I~1 >

1 ~.

We leave it to the reader to show that one obtains for 51 $ 1 the result (6.4.42) and for 51 ~'--~1 the result (6.4.43). Sumarizing the results of this section, we have obtained the following: The solution of the initial value problem (6.4.31)-(6.4.32) with 5

-

1

~

+ e51

is for e $ 0 approximated formally by 1 + Bo(ez) sin (~l z ) , w~(z) ,~ wo(~, 77) - Wo(eZ, z) = Ao(eZ) cos (~z)

(6.4.46)

where the functions Ao and Bo are given by (6.4.40)-(6.4.41)in c a s e stability) and by (6.4.44)-(6.4.45)in case 1511 > 1 (stability).

~11 < 1 (in-

From the general theorem in the next section it will appear that the formal approximaz* with Z* some tion (6.4.46) is uniformly valid in the expanding interval 0 < z < -7positive number independent of e. 5.

5.1.

The General Case and the Error Estimate

Introduction

In this section we deal with the two scale method applied to the differential equation of the general form

d2u dt---~ + u = e f ( e , t , u , - ~ ) , with the initial conditions u(O) -- a,

t>0,

0<e0

with du u(0) = cr > 0 and - ~ - ( 0 ) - 0.

2. Give by using two time scales a first order approximation of the solution of the initial value problem

d2u dt 2

du

+u=0,

t>0

with du u(0) = a and - ~ ( 0 ) = 0.

3. Give by using two time scales a first order approximation of the forced van der Pol equation d2u du dt--$ + w21u = r - u 2) --~ + 3" cos w2t with u(O)=a,

du -~-(O)=Oandw2-wl=O(1),

no resonance). Consider the following two cases: 3' is a constant with 3" = eft and 3" = O(1). See also [108, 6.2.3].

This Page Intentionally Left Blank

Chapter 7 SINGULAR ORDINARY

1. 1.1.

PERTURBATIONS DIFFERENTIAL

OF LINEAR EQUATIONS

T h e Initial V a l u e P r o b l e m Introduction

In this section we construct an a s y m p t o t i c approximatio~ of the solution of the initial value problem

d2ue due Le[ue] = ~ ~ + a(x) ~ + b(x)ue - f(x),

x > 0,

(7.1.1)

with 0 < r ~ 1 and the initial conditions ue(0) - a,

due

--~-x(0) - p,

(7.1.2)

where c~ and ~ are a r b i t r a r y constants independent of ~. T h e coefficients a and b and the right-hand side f are s u b m i t t e d to the condition t h a t they should belong to the space C~[0, c~) (continuously differentiable for 0 _< x < c~) and moreover t h a t

a(x)_>ao>O,

x>0,

(7.1.3)

with a0 a constant independent of c. The latter condition is very essential because o t h e r w i s e - as will a p p e a r l a t e r - the construction breaks down. First we construct in subsection 1.2 a formal approximation of ue, i.e. an a p p r o x i m a t i o n t h a t satisfies the differential equation and the initial conditions up to some order of ~. Consecutively we show in the next subsection 1.3 the validity of this formal approximation; the proof consists in an a priori estimate of the error. 1.2.

The Formal Approximation

We expect t h a t the solution ue will be a p p r o x i m a t e d to some extent by the solution w of the reduced problem with ~ = 0. So we consider the initial value problem

a(x) dw + b(~)~ = f(~),

(7.1.4)

138

Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations

with w(0) = a. The solution reads

eq(X){o~2r-e-q(')~(~)~} f(~)d

W(X)--

,

(7.1.5)

0 where q(x) = - y ~b(~) d~~.

0 However this function does not satisfy in general the second initial condition (7.1.2) and so we need a correction t e r m t h a t accounts for this lost second b o u n d a r y condition. Therefore we investigate the differential operator L~ in the neighbourhood of the point x = 0 more closely by using a "microscope" defined with the aid of the transformation x = r

v > 0.

(7.1.6)

The operator L~ expressed in the new coordinate ~ is d2

L~ = ~ 1 - 2 ~ , .

+~-~a(~V~)

d

. +b(~,~).

The most dominant terms are the first two whenever v > 1 and to get a proper balance between these terms we put v = 1. Using now the supposed regularity of the coefficients a and b we have L~ = E - I ~ _ 1 + ~0, (7.1.7) with

d2 s

and

d

= -7~,- +a(0)-7; ",

d f~o = ~--dx (zgE~)-~-~ . +b(e~) .,

0 < ~ < 1.

To satisfy b o t h initial conditions at x -- 0 and the differential equation up to some positive order of e we make the following a t t e m p t for the formal approximation of uE:

u,(~) = ~(~) + ~(~-) + R,(~) = ~ ( ~ ) + , ( ~ ) + R,(~),

(7.1.s)

where the correction t e r m v(() satisfies the b o u n d a r y walue problem

~_1 IV]----0, with the b o u n d a r y conditions

dv 1 dv dw dx (O) = -~-~(0) - ~ - -~-~x(O),

(7.1.9)

139

1.2. The Formal Approximation and

(7.1.10) lim v(~) = O.

~-+oo

The latter condition stems from the consideration that we want to represent the correction term v(~) as a boundary layer term, that has for r $ 0 only significant values in a right small neighbourhood of x - 0. It follows immediately from (7.1.9) and (7.1.10) that v(~) is given by dlu

v(~) = r

~(o)- Z a(O) exp [ - a(O)~-].

(7.1.11)

Putting w and v together we get

x Le[w(x) + v ( ~ ) ] -

f(x) + r

d2w

+ C0[v( )]

d2w

da

= f(x) + ~-j-fi~2 + r162162 with

w(o) + v(o) and

-

~ + ~

~'(o)

~ a(0)

dv

+ b(~r

(7.1.12)

-

dw dv dx (0)+ -~x (O) = ~.

Because the data of our initial value problem (7.1.1)-(7.1.2) are C1[0, cr it follows from (7.1.5) and (7.1.11) that the right-hand side of (7.1.12) is f ( x ) + O ( ~ ) , uniformly in any bounded segment [0,~] with I positive and independent of ~, (n.b. ~exp[-a(0)~] is uniformly bounded in [0, co) ). So w ( x ) + v(~) is a formal approximation of ue in [0, ~]. We call this composite expression a formal approximation, because it satisfies the differential equation and the initial conditions up to O(~), uniformly in [0, ~]. Finally to show that w ( x ) + v(-~) is indeed a good approximation we consider the remainder term Re(x) defined in (7.1.8). It satisfies the initial value problem

d2w da dv Le[Re] = Le[ue - w - v] = -6-~fix2 - ~-~x (O~)--~ - b(r = O(~), uniformly in [0, ~],

(7.1.13)

with the initial conditions

Re(O) = -v(O) = 0(r

dRe and -~-x (0) - 0.

(7.1.14)

Given this initial value problem it is possible to derive an a priori estimate of Re(x) without solving explicitly the initial value problem (7.1.13)-(7.1.14). This a priori estimate is constructed in the next subsection and it will appear that Re(x) = 0(r uniformly in any bounded segment [0, g].

140 1.3.

Chapter 7. Singular Perturbations of Linear Ordinary Differential Equations The A Priori Estimate of the Solution of a Singularly Perturbed Ordinary Differential Equation with Given Initial Data We consider the initial value problem

d2u du ~ x 2 + a(x; ~)-~x + b(x; ~ ) u - f(x; e)

0 < x < oo,

(7.1.15)

with the initial conditionS u(O) = a(6),

du ~x(O) = 13(6).

(7.1.16)

The coefficients a, b and the right-hand side f depend as well on x as on 6 and we assume that for .E sufficiently small a(x; E) is continuously differentiable with respect to x and that b(x; e) and f ( x , e ) are continuous in x for x _> 0. Moreover, there exist positive constants A , B and i, independent of E, such that [a(x; 6)[ + [dxxda(x; 6)[ < A, ]b(x; E)[ < B, for 0 _< x _< l,

(7.1.17)

uniformly for 6 sufficiently small, say 0 O, f o r 0 < x < g a n d 0 _ 6 < 6 0 ,

(7.1.18)

where a0 is some constant depending on ~ and e0, but independent of e. The estimate of the solution u(x) of (7.1.15)-(7.1.16) in the segment [0, i] is obtained by the method of energy integrals. Instead of estimating u we may as well estimate the function fi defined as ~t(x) = u(x) - 13x- oL. (7.1.19) It follows that fi(x) is a solution of the initial value problem

d2u dfi -C~x2 + a(x; e ) ~ x + b(x; e)~t = g(x; 6),

O<x_ ao ~2 -+- 2efifi' + ~ao ~'2 >_ (ao - V/e)~ 2 + e(ao - v/e)fi '2 > m(fi 2 + eft'2),

(7.1.24)

for e sufficiently small, say 0 < e < eo (so is generic!); m may be taken for instance 1 as m - - ~ao. Combining (7.1.23), (7.1.24) and (7.1.21) we get x

m ( ~ 2 + e~ '2) _< f ~ ( a f i 2 + 2 ~ '

+ eaft'2)d~

0

< ] { (1 + a' - 2b + b2)~ 2 + e(2 + a')fi '2 + 2g2}d~ 0 x

_< M { / ( f i 2 + efi,2)d~ + {Igtt~o,l]},

(7.1.25)

0

with M = max[]l + a ' - 2b + b2l, 12 + a'l, 2] [0,~] < 2 + A + 2B + B 2 independent of v, and with I]" [1[o,~] the L2-norm. The inequality (7.1.25) is uniformly valid in [0, g] for 0 __<E _< eo and we may apply Gronwall's lemma from Chapter 3, (3.2.23)-(3.2.24). With the aid of this lemma we obtain the estimate ~2(x ) + efi,2(x ) _< -M M M _< --]]g]l~o, t]e M m e, m- ilgll 2[o,l] exp [~-x] ?n

uniformly valid in [0, t] for 0 _< c < so. Returning to the original dependent variable u ( x ) f u ( x ) l 0; this is not a restriction because the other case bl(X) < 0 can be treated in a similar way. The m a x i m u m principle provides a simple and useful a priori estimate for the solution u~(x) of (7.2.1)-(7.2.2). For this purpose we consider the function u*(x) : ue(x) - fl, t h a t satisfies the b o u n d a r y value problem

L~[u:] : f ( x ) -

t3bo(x)- et3ao(x)= f*(x; e),

a < x ( b,

with the b o u n d a r y conditions ue (a) = c~ - fl, The linear function inequality

(I)(x) = M ( b -

x)

with

u e (b) = 0. M

some positive constant satisfies the

L~[M(x - b)] = E { a l ( x ) M -4- a o ( x ) M ( x - b) } -f- b l ( x ) M -4- bo(x)M(x - b) > M { b l ( x ) A- ~al(x) -+-eao(x)(x - b)}, a < x < b.

147

2.2. The Maximum Principle for Ordinary Differential Operators Hence with 7 - min bl (x) we have the inequality [a,b]

L ~ [ M ( x - b)] > ~1 M ~,, valid for a < x < b and for c sufficiently small, say 0 < c < e0. Choosing

M = max

[~, b],o_ O" Be(Z) exhibits a boundary layer in the neighbourhood of x - - 1 . 1

[

y(z') dx'

y~(x) ~ b exp + x

- 1 + 5 1 _ < x _ < - 5 2 , +52_<x_ 0. If a twice continuously differentiable function (I)(x, y) attains a local maximum in an interior point P of D then M[(I)] ~ 0 in P. If on the contrary (I) satisfies in 7) the differential inequality M[(I)] > 0 then it follows that (I) cannot attain a local maximum in 7); if'moreover (I) is also continuous in 7) then the maximum of (I) can only be attained in some point of the boundary 07) of 7). This is the maximum principle in its most simple form. If the interior local maximum is nonnegative and if f(P) 0, c(&, ~) > 0 and b~- V -< 0, b~ - ~ - 0andv(0,

y)-qo(y),

y>0.

(8.5.7)

The solution is easily obtained and it reads (x)

(8.5.s) ~/v~

192

Chapter 8. Perturbations of Elliptic Type

Hence tvl < M ~

2

erf c[

],

where M is the upper bound of the function I~(Y)]" So v is uniformly bounded and analyzing the expression (8.5.8) for large values of f and finite values of y one gets

,(~, X

yo) = o ( ~ ) ,

(8.5.9)

uniformly for x > 5 > 0 and any fixed value y0 >_ 0. 5 may be any positive number, indpendent of e and the symbol O(e N) denotes again asymptotically zero (see also Exercise 8 of Chapter 2). Therefore the function v ( ~ , y) is a boundary layer function and we call it a parabolic boundary layer function since it is a solution of a parabolic differential equation. We put now

~(~, y)= .(~, ~) + R~(~, y), X

where Re satisfies the boundary value problem

02 Re e

OX 2

02 R~ ~ + Oy2 )

ORe 02 v Oy = - ~ -~y2,

x > O,

y >0

with

R~(~, 0 ) = R~(0, y ) = 0,

~ > 0,

y >__0.

(8.5.10)

O2v To show that v ( ~x, y) is at least a formal approximation of ue(x, y) we need sb--~y2 = o(c), uniformly in x > 0, y >_ 0, or eventually in a bounded part of this region, say x 2 + y2 < r 2. However we have from (8.5.8) oo

02v V~{ ~ exp[ Oy2 -(2y)3/2

1~2 ~yy]~'(O) +

/

e-

892

~2

~o"(y- ~-~)at

}

(8.5.11)

~/,/'N and it follows that ~ has a corner sigularity at the origin x = 0, y = 0 whenever ~'(0) ~= 0. The nature of this singularity is clearly revealed if in equation (8.5.11) the origin is approached along any curve ~ = my ~ with m and c~ arbitrary positive constants. The presence of this corner singularity indicates that in attempting a proof for an appropriate estimate of the remainder Re we may expect a difficulty. This difficulty disappears whenever dd-~y(0) = 0 and therefore we change the boundary. condition ue(0, y) = ~(y) a little bit such that the new boundary value ~5(y) has the properties ~(0) = 0 and -~y(0) = 0. Therefore we define ~(y) -- ~ ( y ) - y~o'(O)exp[-- Y l,

(8.5.12)

5.1. Introduction

193

where a is a positive constant to be determined later on. We have indeed ~(0) = 0 and ~'(0) = 0; moreover the change in the boundary condition is small because

~(y)

-

[-~] ~(y) = ~'(0) --exp ~

= O(~),

uniformly for 0 _< y < oo. Instead of the approximation v that satisfies (8.5.6)-(8.5.7) we use now the function that fulfils (8.5.6) and the boundary conditions ~((,0)=0,

~>_Oand~(O,y)=~(y),

y>O.

Similarly as in (8.5.8) we get oo

~((, y) =

e-~

~(y - - ~ ) d t .

(8.5.13)

Assuming ~o"(y) continuous we obtain from substitution of (8.5.12) into (8.15.11)

o~ oy~ =

~

~-

~2 2 ,(o)___~. ~ ) + {~-~

r

dt

~/ v ~ = 0(1) 4- 0 ( ~ - ~ ) , uniformly in any quarter disk x 2 4- y2 < r 2. Putting now

X

ue(x, y) = ~ ( ~ ,

y) +/~e(x, y),

(8.5.14)

(8.5.15)

we have instead of (8.5.10) for the remainder term /re the boundary value problem

r

Ox 2 4. (~y2

(~y = --g'~y2 = O(~) + o ( ~ l - - a ) ,

(8 . 5.16)

uniformly in x 2 + y 2 _ < r 2, with /~e(x, 0 ) - 0 , x _ > 0 and /~(0, y ) = p ( y ) - ~ ( y ) O ( ~ ) , uniformly in y _> 0. The exponent a is still free; we take a = 1 and it follows that ~ ( ~ ) is a formal approximation of ue(x, y). To prove that ~ ( ~ , y ) is indeed a good approximation of u e ( x , y ) we make an estimate _ of / ~ invoking again the maximum principle. A simple barrier function for Re is given by -= v/~C(1 4- y), with C sufficientily large and we get / ~ ( x , y) - O(x/~), uniformly in any bounded domain x 2 + y 2 _ _ r 2, X

u~(x,y)=~(~,y)+O(x/~),

x_>0,

(8.5.17) y_>0. Hence

uniformly i n x 2 + y 2 _ _ r 2,

x_>0,

y_>0.

194

Chapter 8. Perturbations of Elliptic Type

However

(x) v(~, y) -- ~(~, y) --

99 (0)

Y-

e- 89t2 (y -- ~-~) exp

~/v'~-~ = O(v/e), uniformly in x >_ O,

2t2

dt

v~

y _> O,

and so we may finally conclude

X

u,(~, y) = v(-~, y) + o(v~),

(8.5.18)

uniformly in x 2 + y2 O, y >_ O, where v ( ~ , y) is given by (8.5.8). The boundary layer term 0(~,y) is called a regularized boundary layer term. In a first approximation of ue the nonregularized boundary layer term is sufficient; the regularization is only needed to prove that the error is O(V/r uniformly in any bounded region of the domain x _> 0, y _> 0. Summarizing we have the following theorem THEOREM 2 Let uE satisfy the boundary value paroblem (8.5.2)-(8.5.3) with qo, du ' ~dy2 , continuous for y > O. Then the solution ue(x, y) is approximated by

oo X V~~ / 1t 2 x 2 u~(~, y) = v ( ~ , y) ~-~ ~(y - 2 - ~ ) e t + o(v~) z uniformly in any bounded region in the domain x > 0, y > 0. 5.2.

The Singular Perturbation

P r o b l e m in a R e c t a n g l e

By now we have carried out sufficient preparations for the treatment of a singular perturbation problem associated with the differential operator e A - ~ and defined in a rectangle. To be specific we investigate the following boundary value problem 9u~

r

(0 ~

02u~ ~ + Oy 2 ]

Ou, Oy

= O,

0 < X

< el,

0

< y < s

(8.5.19)

with the boundary conditions

Ue(X, 0) : fl(X),

0 < X < el,

Ue(X, e 2 ) : f2(x),

0 < X < el,

U,(0, y) -- gl(Y),

0 < V < 12,

U,(ll, y) = g2(v),

0 < y < e2,

where f l ( o ) -- g~(O),

f~(e~) - g2(0),

f2(0) = g~(e2),

f2(e~) - g2(e2).

(8.5.20)

5.2. The Singular Perturbation Problem in a Rectangle

195

Y D

$ .,,.

Ii

Fig. 8.6 Rectangular domain Moreover we take f~(x), g~(y), i = 1, 2, infinitely differentiable; this condition may of course be weakened, but for the m o m e n t we will not bother about too much bookkeeping. It is clear t h a t the solution of the reduced equation with the b o u n d a r y condition ue(x,O) = f1(x), 0 _< x < s is

W(X, y)

= fl(X),

(8.5.21)

and therefore we need correction terms along the three boundaries AC(x = 0), BD(x = ~1) and CD(y = ~2) and we get b o u n d a r y layers along these three boundaries. The b o u n d a r y layer terms along AC and BD are regularized with the aid of (8.5.12)(8.5.13) and the b o u n d a r y layer t e r m along CD is constructed similarly as in (8.3.16). In this way we a t t e m p t the following approximation of u~(x, y)

ue(x, y)

= fl(x) Jr- Vl ( ~

, y) + 32( e l ~ - x, y) -~- v3 (x, ~2 .. c-- y)

Jr_Re(x, y)

(8.5.22)

with x

x2

Vl(-~,y) --

~I--X y) =

e-~t ~I(Y-- ~t2) dr,

/oo e-

(~2(Y--

.1_.2 )dt (8.5.23)

and

!

(~1 (Y) -- gl (Y) -- f l (0) - gl (O)y e

~2(Y) - g2(Y) - f1(~1) -

r

_

-- --Y--

,

g2(0)Y e

,/;,

,

x

- s~(~)-/~(~)-

~1 ( ~ , e ~ )

~1 -

- ~(

v~

x

~2).

'

(8.5.24)

196

Chapter 8. Perturbations of Elliptic Type

From the estimate (8.2.5) or (8.2.6) it is celar that ue(x, y) is uniformly bounded in the rectangle A B e D and so Re(x, y) is also uniformly bounded in this rectangle. Moreover Re(x, y) satisfies the boundary value problem 2Re

(0

OX 2

-t-

02Re ) Oy2

ORe Oy = Se(X, y),

0 < X < el,

0 < y < g2,

(8.5.25)

with

d2 f l ( x )

~- 02el ( X y) -~- 02V2 ( gl --X , y)

d2r ~2 -- Y] } + ~x2 (x)exp [ -

(8.5.26)

and

R~(~, 0) = - ~ ( ~ ) ~ - ~ ,

R~(~,e2) = 0 gl

Re(O,y) = - - v 2 ( ~ , Y )

-

-

v3(0,

gl

g2

_._It_

,

e y) + gl(O)ye ~ -

g2

--

,

__.E_

After careful inspection of the boundary data along the boundary of a.o. (8.5.20), it appears that we have the uniform estimate

R~(~, y)

(8.5.27)

ABCD, using

(8.5.28)

= O(v~). O(ABCD)

To estimate Re(x, y) in the interior of ABCD we have to investigate the right-hand side Se(x, y) of (8.5.25). We have ef~'(x) = O(e) and by means of (8.5.14) with a = ~1 we have also the estimates ~ ~ou2 ( ~x, y ) = O ( v ~ ) and

e ~ 2 ~ v~ ' Y ) = O(v~), uniformly in ABED, and so it remains to inspect d--~(x)exp[-~2-u] in A B e D " dx 2 e J d2(p _ f ~ , ( x ) _ y I , ( x ) ax 2

where

-

-

1{ 0 2

02

}

~ ~12'01(~1, e2) + ~--~'02(~2, e2) ,

(8.5.29)

X ~1 -- X ~1= ~ andS2= V/~ 9

It follows t h a t d2 ~ exp [ - ~ ]e J = O(~) 1 and Se(x,y) = O(1) in smallneighbourhoods of the cornerpoints C and D. An estimate of Re(x, y), uniformly small in the whole rectangle ABCD is not yet possible and therefore we improve the boundary correction along the upper side CD. We remark that it is not surprising that we are at this stage not yet ready to give a uniformly small estimate for the remainder term; the

197

5.2. The Singular Perturbation Problem in a Rectangle

reason is the overlap of two boundary layers in the cornerpoints C and D. Using the local coordinate 7/ - e2-v the boundary layer term V3 has been obtained as a first approximation of the boundary value problem

02v *

Ov*

C~?~2

OT]

- - +

_~2 02v * OX 2 '

=

O < x < ~,

~2 ,

O < rl < - -

(8.5.30)

with the boundary conditions x il-x v*(x,O) = (p(x) - f : ( x ) - f~(x) - ~ , l ( - - ~ , t 2 ) - ~,~.( V'Z ,t~), =o.

v*(~, ~)-

(8.5.31)

(8.~.32)

-

l~ -y

The approximation ~3(x, ,) - r = (p(x)e- ~ satisfies (8.5.30) with e = 0 and the first boundary condition (8.5.31), while the second boundary condition is replaced by lira v*(x, 77) - 0. Instead of ~3(x, r/) we take now as an approximation of v*(x, rl) ~-~c~

~(~,,) + C~(~,,), where V4(X, 77) is a solution of 02V4 OqV4 -cOrl2 ~ (OrI with the conditions immediately that

02~3 cox 2 -

d2r (x)e -~/ dx 2 '

V4(X, O) -- 0 and

0 < x < ~1,

0 < 71 < -~2----c'

limn__~ V4(X,?7) -- O, 0 _~ X ~ ~1. It follows

d25

Vn(X, ?7) "-- ~X277 e-~"

(8.5.33)

Finally we decompose the function u~(x, y) as

u~(~, y) = fl(~) + ~

(~

' Y) + ~ ( ~1v~-- X ' y)

-[-~3(X '

+ k~(~, y). Now we have obtained for /~e(x, y) the following boudary value problem 02

02 )

0/~e = 5 ( x , y),

(8.5.34)

with

{ '/

02Vl

X

02V2 ~,l--X,y)_.~_E2d4~)

= ~ f~ (~1 + -~y~(-~' Y) + -~y~( v~

-~,Te-"

}

(8.5.35/

198

Chapter 8. Perturbations of Elliptic Type

and boundary conditions similar to (8.5.27). With the aid of (8.5.24) and the definitions (8.5.23) we have

~4~

= O(e-~) and ~ x 4 - O(e-2), uniformly in A B e D

and by means of (8.5.14) with a -

1 we get (8.5.36)

Se(x, y) = O(v~), uniformly in A B E D .

The addition of the boundary values of e2~4(x, y) have changed the boundary values of Re(x, y) only up to O(r 2) and so we have accordingly to (8.5.28)

(8.5.37)

/~e(x, y ) = O(v~), uniformly along O(ABCD). Introducing once again the barrier function @(x) = Cv/e(y + 1),

with C sufficiently large, but independent of e, we obtain by (8.5.34), (8.5.36), (8.5.37) and Lemma 1 the result /~(x, y ) - O(V~), uniformly in A B E D . Hence

.~_e2~4(X, g2 - Y) + O(v~), uniformly in A B E D . C

Finally, because

V4(X,7/)=

d-~-e-ndx, 'i =

(8.5.18), uniformly in A B E D ,

ue(x,y)

: fl(x)-q-

~1

0(})

and

v , - ~, = O(v~),

i=

1,2, see

we have also

(~

' Y ) -]- v 2 (

~'~-

~ ' y) +

,~(~, ~

c-

Y) + O ( v ~ ) '

(8.5.38)

uniformly in the closed rectangle A B E D , with

9 Y) VI(--~,

"{ gl (Y -- ~et2 ' ) -- fl(O)

oo -

-

e-2

-

)

dt,

(8.5.39)

z

"'

,Y)=

and ~3( x,

"g2 -- Y) _--r

g.(Y-

2.t-------~-

exp_ [s -- Y], C

(8.5.40)

(8.5.41)

5.2. The Singular Perturbation Problem in a Rectangle

199

where x

-

12) - v2(

' 12).

(8.5.42)

Summarizing our results and keeping track of the required differentiability of the data we have established the following result THEOREM 3

Let the function u~(x, y) be the solution of the boundary value problem {02u~

in the domain conditions

0 < x < s

02u~} + -O-yy

Ou~ =0 oy

0 < y < ~2, with 0 < e 1, x 9 f~, u(x) = 9~(x), x 9 0~2,

(8.5.43)

where L2 is an elliptic differential operator of the second order, L: a differential operator of the first order and ft a bounded domain. Applying the Lapace transformation y

u~ (x) -- / exp[--Ay]v(x, y)dy 0

(8.5.44)

200

Chapter 8. Perturbations of Elliptic Type one obtains a hyperbolic differential equation 0 L2[v] + -~--Ll[v] = 0,

(8.5.45)

ay

valid in the cylinder f l x ]R1 and with the boundary conditions

v(x, y) = 0, y < 0 and v(x, y) = ~(x)~(y) for y > 0. Investigating v(x, y), in particular its support, and approaching u~ (x) for -4 c~ with the aid of (8.5.44) one obtains an approximation of the solution of the singular perturbation problem. Y. Kanai treats a.o. the elliptic problem

" 0 9+c(~). L2 = Ax and L1 = ~-~bj(x)-~-~. j=l

and the parabolic problem

L2 = / k x and L1 = j=l

bj ~xL] 3

Ot '

(Ax is the Laplace operator). Further he presents an outline for a simple turning point problem in one dimension. For related work we refer the reader also to L. Frank, Singular Perturbations I, North-Holland Publ.Cy, Amsterdam, 1990.

6. 6.1.

Elliptic B o u n d a r y Value P r o b l e m s with Turning Points Introduction

In the preceding chapter we have studied perturbed boundary value problems for ordinary differential equations of the form

d2ye dye c~ + f (x)--~x + g(x)y6 = O,

- 1 < x < +1,

(7.3.1)

where the reduced equation (~ = 0) may become singular due to a turning point, say at x = 0, with f(0) = 0. It appears t h a t the approximation of the solution is highly sensitive for the sign of the derivative f~(x) at x = 0 and for the value of the parameter

~= 2

g(O) f'(O)"

(7.3.3)

P h e n o m e n a similar as those demonstrated in section 7.3 may of course also occur in the case of partial differential equations of elliptic type. In the next subsection we give some examples, however without entering into the details of rather complicated calculations.

201

6.2. Examples of Turning Point Problems 6.2.

E x a m p l e s of T u r n i n g P o i n t P r o b l e m s

6.2.1.

C u r v e s of T u r n i n g P o i n t s

An obvious generalization is provided by the following example

0 2u~ 0 2u~~ Ou~ ~ + 0y~ / + :(~' Y)-0~-y + 9(~, y ) ~ = 0,

(~, y) 9 ~,

(8.6.1)

where ~t is again a bounded domain such as considered in section 8.3. The boundary values are given by

ue(x, y) = ~+(x) along the upper boundary y = ~/+(x) with xl _< x < x2, and

ue(x, y) = T_(x) along the lower boundary y = V_(x) with xl 0, we may conclude t h a t

LoIRe]- O(e),

(9.3.13)

uniformly in any fixed bounded domain G in the region t > 0. Moreover R~(x, t) satisfies the initial conditions

R~(x, O) = -v(x, O) = O(c)

(9.3.14)

uniformly in any bounded interval of the x-axis, and

OR~ Ot (x, 0 ) = 0 ,

-cr162

(9.3.15)

It follows that the expression w ( x , t ) + v(x, t) satisfies the initial value problem for u~(x, t) up to O(c), uniformly in any bounded domain with t > 0, and so it is a t formal approximation of u~(x, t). To prove that w(x, t ) + v(x, -i) is indeed a good approximation of u~(x, t) we need an estimate of R~(x, t) and this estimate should be small for small values of r This estimate is obtained in the next sections with the aid of the method of energy integrals introduced by K.O. Friedrichs and H. Lewy, [44], [8]. 4.

A P r i o r i E s t i m a t e s o f S o l u t i o n s o f I n i t i a l V a l u e P r o b l e m s for P a r t i a l Differential Equations with a Singular Perturbation of Hyperbolic Type We consider again our initial value problem (9.1.1)-(9..1.4)

e ~-

(z,t) 02u

valid for - c r < x < +cr

Ou

Ou

+ a(x, t) Ox + b(x, t ) - ~ + d(x, t)u - f(x, t)

(9.4.1)

t > O, and with the initial conditions

Ou (x, O) = h(x), u(x, O) = g(x) and -~-

- c r < x < +cr

(9.4.2)

As to the coefficients, the right-hand side and the initial values, we make the same regularity assumption 1-5 as in the preceeding section. The method to obtain a priori estimates of the solution u(x, t) is due to Friedrichs and Lewy [44], [8], but because of the occurrence of the small p a r a m e t e r c we need here a modification of this method as given in [77, 49, 47]. Multiplication of (9.4.1) with 2u gives

O (bu2 -4- 2r -4- -~-~x(au2 - 2ec2uux) = Ot ( - 2 d + bt + ax)u 2 + 2ca 2 - 2ec2u 2 - 2e(c2)~uu~ + 2fu, where we have omitted the arguments of the functions involved. Multiplication of (9.4.1) with 2but and 2au~ gives respectively

0 (~b~ + ~ b c ~ ) + 0 (-2~bc~u~) =

0-~

216

Chapter 9. Perturbations of Hyperbolic Type

and 0 0 (~au 2 -4- ~ac2u 2) Ot ( 2 e a u t u~ ) - -~x -ea~u 2 + 2eatutux -e(ac2)~u~ 2 _ 2aduu~ - 2abutuz

2a2u 2 + 2 a l u m .

Adding these three equations we get o

o

(9.4.3)

oq-~Q1 + ~xQ2 = Q3, with Q1 - bu 2 + 2 e u u t + ebu2t + 2 e a u t u z + ebc2u2z, Q2 = au2 - 2 e c 2 u u x - e a u 2 - 2 e b c 2 u t u z - e a c 2 u x2,

(9.4.3 a) (9.4.3 b)

and after some calculation Q3 = ( - 2 d + a~ + bt)u 2 - 2(but + a u x ) 2 - 2 d u ( b u t + au~) + 2 f ( b u t + a u z ) + 2 f u + E[(2 + bt - a ~ ) u 2 + (2at - 2 ( b c 2 ) ~ ) u t u ~ + ((bc2)t-

< (-2d+a~+bt)u

(ac2)x - 2c2)u~~ - 2 ( c ~ ) ~ ]

2+(du-f)2+u

2+f2

+ c[(2 + bt - a ~ ) u 2 + (2at - 2 ( b c 2 ) ~ ) u t u x + ((bc~)~- ( a ~ ) ~ - 2 ~ ) = ~

- 2(~)~=u~]

< (1 - 2d + 2d 2 + a~ + bt)u 2 + 3f 2 + c[(2 + bt - a ~ ) u 2 + 2 ( a t - ( b c 2 ) ~ ) u t u ~ + ( ( b c 2 ) t - (ac2)x - 2c 2 + (c2)2)ux2 + u2].

(9.4.3 ~)

Hence we have for E < 1"

OQ1

OQ2

0--~ + - ~ x < Q4 + 3f 2

(9.4.4)

where Q1 and Q2 are given by (9.4.3 a) and (9.4.3 b) and Q4 by Q4 -- ~0 ~2 -~- E(")'ll u2 -~- 2~[12~tUx -Jr-~22'tt2),

(9.4.5)

with 3'0 = 2 - 2d + 2d 2 A- ax + bt, 711 = 2 - az + bt , ")'12 =

at -- (bc2)z,

7 ~ = (b~ ~)~ - ( a ~ ) ~ - 2 ~ + (c ~) x "

(9.4.6)

4. A Priori Estimates of Solutions of Initial Value Problems

217

C

A

B Fig. 9.3

X

The domains G and

To estimate u in any compact subset (~ of the half plane t > 0 we enclose G in a trapezelike domain Ft, bounded by the x-axis, a segment parallel to the x-axis with t = T and two characteristics with slopes c-l(x,t) and -c-l(x,t) see Figure 9.3. We integrate the inequality (9.4.4) over ~t and we obtain by Green's t h e o r e m B

-

C

Qlds+ A

D

A

ds

(Qlc+Q2) x/l+c 2 B

VII ~-c 2 C

D

( b - v ~ ) ~ : + ~ ( b - q~ - v ~ ) ~ Because we made the crucial assumptions _ Vt > 0 , there exists a positive function

instance

q2 _ --

b 2c 2 + a 2 2bc 2 9

b > 0 and

q2(x, t)

2

+ ~(b~~ - a~q-~)~.

with

a 2

1- ~

a ~ b-fi
p02, Vx and q2

< b, V(x, t) C ~;

for

218

Chapter 9. Perturbations of Hyperbolic Type

Since all functions involved are continuous for t > 0 we have for 6 sufficiently small, say 0 < 6 < 6 0 , Q1 >_ m(u 2 + 6u 2 + 6u2), Vx, t 9 ~t, (9.4.8) where m and 60 are generic positive constants, dependent on the coefficients a,b and c and on the domain l~, but they are independent of 6. 60 and m are generic positive constants which means that they may be lowered, whenever desired, without violating preceding results, e.g. (9.4.8). Further it follows from (9.4.3 a) and (9.4.35)

cQ1 + Q2 = (bc + a)u 2 + 26cu(ut - cux) + 6 ( b c - a)(ut - cux) 2 and

cQ1 - Q2 = ( b c - a)u 2 + 26cu(ut + cu,) + 6(bc + a)(ut + cu,) 2. Using again the relation 1 - g~-/-~2> p2 > 0 we obtain for 6 sufficiently small, 0 < 6 < 60

cQ1 + Q2 >_ m { u 2 + 6(ut T cu,)2}, where m depends again on a,b,c On the other hand it follows from coefficients a, b, c and d that there and on ~t, but independent of 6,

IQ41 5

IQ~I +

V(x, t) 9 ~t,

(9.4.9)

and Ft. (9.4.3~), (9.4.5), (9.4.6) and the regularity of the exists a constant M, depending on these coefficients such that

M ( u 2 + 6u 2 + 6u2),

V(x, t) 9 l~ and 0 < 6 < 60.

(9.4.10)

Substitution of the inequalities (9.4.8)-(9.4.10)into (9.4.7)yields m

/C (u 2 + eu2t + eu~)dx + m /C {u 2 + e(ut - cux) 2} -x/1- - -ds+- ~c D

B A

+ m/{u

2 + e(ut + cux) 2}

D B

A

- M/(U

ds ~/1 + c 2

~2

2 + eU 2 + r

+ K(ft),

(9.4.11)

12

where K(ft) is determined by the initial conditions (9.4.2) and the right-hand side S of the differential equation. We have

K(Q) = M{[lg[I~AB] + r + 3I[fll~ABCD ] with

Jig]lAB

+ ellhll~AB]} (9.4.12)

the L2-norm of the function g with respect to the integration interval

A B and similarly for the other terms. Since the estimates of all integrands are valid in the whole domain ~ the inequality (9.4.11) is also valid for any domain ABC*D* < A B C D with the same values of m, M and K(ft), see Figure 9.4.

4. A Priori Estimates of Solutions of Initial Value Problems

219

• T

A

Fig. 9.4

/

..

C

,,

A'

B"

The domains

B

~'

ABCD, AIBICID! and ABC*D*

Therefore it follows from (9.4.11) -~ (t)

t "y~(,-)

f (u2 +eu2t +eu2)dx-M-M-mf f (u2 +eu2t nt-eu2x)dxdT_ O, u(x,t) < e- 88v/C(f~)K(f~)

(9.4.17)

ux(x, t) < ~-1 v/C(~)K(f~)

(9.4.18)

ut(x,t) < e-1 v/C(~)K(~t).

(9.4.19)

The constants C(f~).and K(~) depend on the coefficients a,b,c,d and the domain f~, while K(~2) depends also on the right-hand side f and on the initial data g and h; the latter is given by the relation (9.4.12). 5.

T h e E s t i m a t e of t h e R e m a i n d e r

Term and Final Results

We return to our formal approximation of the initial value problem (9.1.1)-(9.1.4). The result reads

ue(x,t) = w(x,t) + v(x, t) + Re(x,t),

(9.5.1)

c

where w(x, t) is the solution of the reduced problem

a(~, t) O~ Ow -~x + b(x, t) - ~ + d(x, t)w = f (x, t)

--cx3 < x < cx3,

t > 0,

~ ( ~ , o ) = g(~), t and v(x, -i) is the boundary layer correction given by (9.3.10). The remainder term Re(x, t) satisfies according to the equations (9.3.12)-(9.3.15) the initial value problem

( 0 2 R e _ c2(x, t) O2Re ) ORe Ot 2 Ox 2 + a(x, t) -~x + b(x, t)

+ d(x, t)R~ = O(e),

uniformly in any compact domain in t > 0, with the initial conditions

R~(~, 0) = - ~ ( ~ , 0) = - ~

Ot

b(~,o)

= o(~)

(9.5.2)

224

Chapter 9. Perturbations of Hyperbolic Type

and

0--R~(~,0)=0, Ot

also uniformly valid in any bounded segment of the x-axis. We can now estimate the remainder Re(x,t) and its derivatives by applying the result of T h e o r e m 1 of the preceding section. By (9.4.12) we have g ( g t ) = O(c 2) and hence by (9.4.17)-(9.4.19)

Re(x, t) = O(e 3/4) 0

(9.5.2) 0

-~xxRe(x,t) = O(1), - ~ R e ( x , t ) = O(1),

(9.5.3)

uniformly in any compact domain with t > 0. Using finally (9.5.1),

Ov ,(x, ;t) = 0(~), ~(~,;a,t) = 0(~), -b-;(~, ;)t =0(1) we arrive at the following t h e o r e m THEOREM 2

Let ue(x, t) be the solution of the initial value problem \ c9t2 --

c9x2 ] + a(x, t)

+ b(x, t) -g7 + d(x, t)ue - f ( x , t)

- c o < x < +oc, t > O, with the initial conditions

~ ( ~ , O) = g(~),

OUe (x,O) = h(x), -g-f

-cr162

Suppose that all data are of class C ~ with b(x, t) > O, c(x, t) > O, Vx, Vt > 0 and that the subcharacteristics are timelike in the region t > O. Then we have the following approximations of ue(x, t), uniformly valid in any compact domain G in the half plane t > 0 for r sufficiently small and positive. u~(x, t) = w(x, t) +

0(~3/4),

Ou~ (x, t) = Ow cox -~x (x' t) + O(1)' OUe (x t) = cgw Ot ' --~-~(x,t) + 0 ( 1 ) , where w(x, t) is the solution of the reduced problem

a(x, t) Ow O~ -~x + b(x, t) ~ + d(x, t)w = f ( x , t), w(x,O)=g(x),

--cx3 < x < +cx3.

- o o < x < A-co,

t>O,

5. The Estimate of the Remainder Term and Final Results Remarks 1. The approximation of imation of the form

225

ue(x, t) in G may be improved by constructing an approxm

m

4-

(,~(x, t) - E eiw'(x' t) + E eiv,(x, ~), i--0

(9.5.4)

i--0

with Ll[w0] = f and Ll[W.i] = -L2[wi-1],

i = 1, 2 , . . . m ,

while the boundary layer terms vi are obtained from the expansion of r t--~T. Using the notation

Le - gL2 + L1 - e-1 {M0 + gM1 + . . . + gmMm} + gmMm+l,

(9.5.5) with

(9.5.6)

the functions vi(x, T) satisfy the recursive set of equations

M0[v0] = 0 i

Mo[vi] - - E Mj[vi_j].

(9.5.7)

j--1 The initial conditons for wi(x, t) read

Wo(X , O) -- g(x), Wi+ 1 (X, O) -- --Vi(X,

1, 2 , . . . , m - 1

0 ) , i --- O,

(9.5.8)

and the boundary conditions for vi(x, T) are given by

Ovo Owo OT (x, O) -- e(h(x) - ~ ( x ,

0)),

lira Vo(X, T) = 0

OVioT(X, O) = --C--~OWi(X, 0), ~-~limvi(x, T) = 0, i = 1, 2 , . . . m .

(9.5.9)

In case the data are all C ~ we may expand Cte(x,t) indefinitely. From this we get the following refinement of Theorem 2. THEOREM 3

Let u~(x, t) satisfy the conditions of Theorem 2 then we have the following approximations of u~(x, t), valid for any n C N t3 {0} and uniformly valid in any compact domain G in t >_0 for e sufficiently small positive: Ue(X , t) : i=0 0U e

n--1 ciwi(x, t) + E ~ivi(x' t ) + E

n-1

n--2

t) + i=0

Ou,ot (x, t) = E r i=0

0(~n+3/4 )

i--0

-) + i=0

-~(x,

+

r -~Ovi(x, ~t) + 0 (~,~) i=0

(9.5.10)

Chapter 9. Perturbations of Hyperbolic Type

226

where empty sums are to be taken as zero. The requirement that all data of our perturbation problem are of class C ~ is of course too strong for an asymptotic expansion up to a certain order. The rule is that the number of terms in the asymptotic expansion may be larger whenever the data are more regular. For n = 1 we get with the aid of (9.5.10) the following result. THEOREM 4 The approximation of the solution u~(x, t) of the initial value problem (9.3.1)-(9.3.3) is for n = 1 given by

~,(~, t) = ~0(~, t) + o(~) Ou~ (~, t) = O~o (~ t) + o(~) oz Ou~

~ Owo

' Owo

t

ot (~' t) = --~ (~, t) + (h(~) - --07- (~, o)) ~xp[-b(~, o) ~] + o(~)

(9.5.11)

uniformly valid in any compact domain G in the half plane t >_ O. 2. The theory as presented in this chapter has been generalized by R. Geel [49] to initial boundary value problems defined in strips {(x, t); 0 < x _< s t >__0} and to so-called characteristic boundary value problems; in the latter case the boundary consists completely or partly of characteristic curves associated with the operator L2. 3. The approximations given in the Theorems 2-4 are valid for finite e-independent values of t; an extension of these results for large time scales is given b y W. Eckhaus and M. Garbey in Ref. [32]. 4. Hsiao and Weihnacht have also investigated linear singularly perturbed hyperbolic equations of another very interesting type [68]. These authors considered the following Cauchy problem

e2utt+ut-u**:O,

u(~, o) = y(~),

t>O,

~ OU(x, O) = g(x),

-c~<x 0

with the initial conditons

~(~, 0)

= g(~),

()U ~ (~, 0) = h(~), - ~

< ~ < +oo;

the coefficients a,b and d are constants. Give minimal conditions on the regularity of the functions that the approximations (9.5.11) hold.

f, g and

2. Investigate the singularly perturbed hyperbolic system

( ) () - ~ q- Ou

e Ovot

1

q- -21(a - b)u q- -~(a q- b)v -- f (x, t), 1

oxOV + 2 1 ( a - b ) u + - ~ ( a + b ) v = f ( x ' t )

-cxD<x [b[.

h in order

This Page Intentionally Left Blank

Chapter 10 SINGULAR PERTURBATIONS IN NONLINEAR INITIAL VALUE PROBLEMS OF S E C O N D O R D E R 1.

Introduction

In this chapter and the next one we will discuss singular perturbations in nonlinear problems. Due to the large variety of possible nonlinear differential equations and the analytical difficulties concerning the integrability and the uniqueness of solutions we are forced to restrict our exposition to some relatively simple types for which we investigate whether the linear theory as treated in the preceding chapters may be generalized. This means that we discuss in this and the next chapter only second order scalar problems; we start with initial value problems since these are more easily to handle than boundary value problems. In the same way as in the preceding Chapters 7 and 9 we construct a formal approximation of the solution and in order to show that this approximation is indeed a good one we should give an a priori estimate of the remainder term that satisfies a singularly perturbed nonlinear initial value problem with the right-hand side of the differential equation and the initial values being small in some order of ~. Functional analysis provides very useful means for getting a priori estimates of solutions of nonlinear differential equations; so several authors, e.g. Hoppenstead [66], [67], Krein [88], van Harten [64], [62], Besjes [9], Geel [49], [48], Geel and de Jager [46, 75], Genet and Madaune [51]-[102], Hsiao, Weihnacht and Esham [69, 34, 36, 39, 40, 35, 38] applied principles from functional analysis to the study of singular perturbation problems. In section 2 we derive a contraction theorem in a Banach space, due to a.o. van Harten [64], [62], which is very useful to obtain the required estimate of the remainder term. This theorem will also be used in the next chapter where we discuss boundary value problems. Section 10.3 is devoted to initial value problems of the type

r

d2u du 2 + a(X, u) ~ + b(x, u) = O,

u(O) -- a,

du ~x (O) = fl,

(10.1.1)

and section 10.4 to the more general nonlinear initial value problem

d2u du du e-~x2 = f ( u , ~xx) , u(O) = a, ~x(O) = j3

(10.1.2)

230

Chapter 10. Perturbations in Nonlinear Initial Value Problems

The generalization to semi-linear initial value problems with a singular perturbation of hyperbolic type

( 02u ot~

02U ) +a(x, t, u) OU Ou o~ Y i + b(~, t, u ) ~ + c(~, t, ~,) : OU(x, O) = g(x), -~

u(x, O) -= f(x),

o, t > o, - ~

< ~
0 twice continuously differentiable with respect to u in (-cr ii. a(x, u) > a0 > 0, Vx E I~,+, Vu E R with a0 some positive constant. First of all we remark that these two conditions are sufficient for the existence of a C2-solution w(x) of the reduced problem (10.3.3) in any segment [0, X] with 0 < x < X < X0; X0 is the smallest value of x where w(x) becomes singular; in case w(x) has no singularity the value of X is an arbitrary positive number. The uniform positivity of a(x, u) is needed because it will appear that the construction of the formal approximation breaks down whenever a(x, u) becomes negative in the segment [0, X], see also Chapter 7, formula (7.1.3). 3.2.

T h e Formal A p p r o x i m a t i o n

Let w(x) be the solution of the reduced problem (10.3.3) with the initial condition w(0) = c~. Using our experience with the linear initial value problem in Chapter 7 we write u(x) = w(x) + ~9(~), (10.3.4) where ~ is the stretched variable defined by x = e~. Inserting (10.3.4) into (10.3.1) we get for ~(~) the initial value problem d2v d~ d~ 2 + a ( ~ , w(6~) + 69(~)) d~

d2w dw = -- e~x 2 - a(x, w(x) + eg(x))-~x -- b(x, w(x) + 69(x))

(10.3.5)

233

3.2. The Formal Approximation _ - ~x- ~ while the initial conditions for 9 read valid for 0 < ~
a 0 > 0 for 0 < x < o o and - c ~ < u < + o o we have

dO

dO

U(X) ~ ~ 6nwn(X) + ~ 6nTlvn(X--) n=O

(10.3.30)

n=O

and

du

dO n=0

dwn

do

dvn x

--~X (X) -~- ~ 6n+1 "-~X ( C , n=0

(10.3.31)

uniformly in every bounded interval 0 < x < X < X0 where X0 is again the upper bound of x values where wo(x) is still regular. The functions w,.,(x) and v,,(~) are obtained after substitution of (10.3.30) into (10.3.1)-(10.3.2) and collecting equal powers of e. For details the reader is referred to Geel [49], pp.29-37; further the remark above is also applicable in this case.

4.1. Introduction 4.

239

A G e n e r a l N o n l i n e a r Initial V a l u e P r o b l e m

4.1.

Introduction

In this section we generalize the quasilinear initial value problem of the preceding section to the more general nonlinear initial value problem

d2u r

du = f ( u , ~xx),

with the initial conditions

x > 0,

du ~xx(0) =/3,

u(0) = a,

(10.4.1)

(10.4.2)

where f is, apart from some conditions to be stated later on, an arbitrary nonlinear function. We have omitted the x-dependence of the function f, because this is not essential in our discussion and further because it simplifies the notation. It is useful to write the initial value problem (10.4.1)-(10.4.2) in the form

du dt = v(t),

dv E-~ = f(u, v),

(10.4.3)

with u(0) = a,

v(0) =/3.

(10.4.4)

We replaced the variable x by the variable t, since this is more common in the theory of dynamical systems. We submit the function f ( u , v) to the following hypotheses i. f is infinitely often differentiable with respect to u and v. ii. There exists a unique function ~(u) such that

f ( u , ~o(u)) = 0,

(10.4.5)

with the property that the initial value problem du d---t- = ~(u),

u(0) - a,

(10.4.6)

has a unique solution u = uo(t) on some closed bounded segment, say 0 _< t _< T. iii. There exists a positive constant ~ > 0 such that for 0 _< t _< T

of av( O(t),


5u -u. Taking A, as in the figure, as the starting point of the trajectory, then the path from A to the cubic is analytically represented by the initial layer. A careful analysis of the velocity field yields that the path is continued by first crossing the cubic C and thereafter it follows C within an approximation of O(c); also the derivative of the path follows

249

4.3. The Estimate of the Remainder Terms and the Final Result

the derivative of C within the approximation of O(r see (10.4.52) and (10.4.53). 1 3 Whenever w < 5u - u and A lies at the other side of the cubic there is no crossing and the path follows again the cubic. This is all in accordance with the theory of this and the preceding section, as long as o ( ( 1 - u 2 ) v - U) "-- 1 - u 2 < - n . So the theory holds as long as the path has not reached the "critical" value u = 1, and the problem arises how to pass the neighbourhood of the "critical" point C (1, - 2). Analyzing the location of the trajectory in this neighbourhood it can be shown that the trajectory contains a point (u*,w*) at a d i s t a n c e o f a t least 0(r 1/3) from ( 1 , - 5 )2 and with W* < - 2 / 3 . d~ becomes O (r ~) and so Inserting these coordinates into (10.4.50) one obtains that -~(u(t), w(t)) gets enough impetus to leave the neighbourhood of (1, - 52) and the path is continued nearly horizontally in the direction of D and the story can be repeated again. The velocity field allows to apply the Poincar~-Bendixson theorem, see e.g. [60, Ch.II] and so there exists a periodic solution. The trajectory of this periodic solution lies in a 1 3 - u for 1 _ < u _ < 2 , w = - 5 2 for neighbourhood of a closed curve defined by w = 5u l U3 - 2 < u < 1, w : 5 -u for - 2 < u < - 1 and w = + 52 for - 1 < - u -< 2. The distance of the periodic trajectory to this curve approaches zero for r $ 0 and it is of order O(r along the branches of the cubic w = 5l U 3 - u. There exists much literature on the van der Pol equation and we refer the reader to Wasow [139, p.301] and O'Malley [112, p.62]. Also should be mentioned the work by Troesch and Urlacher [133], who gave a fine analysis of the behaviour of the trajectories in the neighbourhood of the critical 2 This paper is one of the first of many interesting publications, where points (+1, TS). nonstandard analysis is used in singular perturbation theory [101,133, 134, 17, 4, 136]; see also M. Canalis-Durand, Solution formelle Gevrey d'une ~quation singuli~rement perturb~e; Asymptotic Analysis, 8, (1994). Finally we remark that we discussed in section 3.5 of Chapter 5 the equation d2u

dt----~

+r

This equation becomes with t = r

1 d2u r dT----~

du

2-1)--.

+u-0,

t>0,

0 0, 0 < r 0},

(10.5.1)

with

02u L2[u] = Or---~ -

c2(x, t) 02u Ox 2

and Ll[u] = a(x, t, u) Ox Ou + b(x, t, u ) -Ou ~ + d(x, t, u).

(10.5.2)

The initial conditions are given by

u ( x , O ) - f(x),

Ou -~(x) = g(x),

- c r < x < +cr

(10.5.3)

To avoid tedious book-keeping we assume for convenience a(x, t, u), b(x, t, u), d(x,t,u) C C ~ ( D x R), c(x,t) c C~176 f(x), g(x) E C ~ ( R ) and in addition we require similarly as in Chapter 9, section 3 i. c(x, t) > 0 in D, c(x, t) uniformly bounded in any strip {(x, t)} with - cr < x < +cx), 0 < t < T * ii. b(x,t,u) > O in D x R , iii. la(x, t, u)l < b(x, t, u)c(x, t) in D x R. The last condition ensures the timelike character of the subcharacteristics, see Ch.9, section 2. As in all preceding sections we start with the construction of a formal approximation consisting of the sum of the solution of the reduced problem with 6 - 0 and a correction accounting for the second initial condition. Consecutively we formulate the initial value problem for the error R~ and we give an a priori estimate using again the fixed point theorem which yields also the unique existence of R~ and by this the unique existence of the solution u of the initial value problem (10.5.1)-(10.5.3). Related initial value problems with more space dimensions have been studied by Genet and Madaune [51], [102]. They discussed a nonlinear differential equation of the type

02u -- As) + a(x, t)Ou e(-O--~ -~ + En bk(x,t) ~Ou + c(x, t)u + F(u) = f(x,t) k--1

with (x, t) E ~ • ~ C R '~. The a priori estimates and approximations of u obtained by these authors are estimates and approximations in the L v norm. 5.2.

The Formal Approximation

A formal approximation of our initial value problem (10.5.1)-(10.5.3) is given by

~),

~o(~, t) = ~o(~, t) + ~ o ( ~ , t

(10.5.4)

251

5.2. The Formal Approximation where w0(x,t) satisfies the reduced problem

Owo Owo a(x,t, wo)-~x + b(x,t, w o ) - ~ + d(x,t, wo) - O, - o c < x < + o c ,

t;>0

To(x, O)- f(x), - c ~ < x < +c~,

(10.5.5)

t -i)

while evo(x, = evo(x, T) is the boundary layer correction. This term is a solution of the boundary value problem

02vo

Ovo

a~---~ + b(~, o, wo(~, o ) ) - ~

Ovo Owo OT (x, 0) = g(x) -- ~ ( x ,

- o,

~- > o

0), lim Vo(X, T) -- O.

(10.5.6)

Despite the regularity of the coefficients a(x,t, To), b(x,t, To) and d(x,t, To) and of the function f(x) the solution of the reduced problem may tend to infinity in finite time or may become multivalued after some time T. Therefore we make the crucial assumption that To(x, t) belongs to C ~ in some region ~0 = {(x,t), - o e < x < +c~, 0 _< t _< T} . The boundary layer correction evo(x, t) follows easily from (10.5.6) and it reads

g(x) - O~oo~~(x,~Jm

evo(x,-~t), = - e b(x, 0, W--o0(-x:{)~ exp [ - b ( x , 0 , wo(x,O))-~tj, . Substitution of ~0(x,t) -

(10.5.7)

Wo(x,t)+ evo(x, t) into eL2[u] + Ll[u] = 0 gives eL2[r0] + Ll[r0] = O(e),

(10.5.8)

uniformly in any bounded domain in ft0 , with

rio(X, O)- f ( x ) + evo(x, 0), O~Zo ot (~' o) = g(~),

-~

-oc < x < +ec

< ~ < +~.

Therefore ro(x,t) satisfies the initial value problem (10.4.1)-(10.4.3) up to O(e) and so it is a formal approximation of u(x, t). However it will appear in the next subsection that this formal approximation is not sharp enough to prove that the error Ro(x, t) is O(e), uniformly in any bounded domain in ~0. We need a higher order approximation, viz.

t) ft l (x, t) -- Wo(X , t) ~- CWl (X, t) -~- ~Vo(X, -~

+ e2vl (x, -~t ) .

(10.5.9)

Substitution into (10.5.1)-(10.5.3) and expansion into powers of e gives after a tedious calculation complicated but harmless initial value problems for Wl and Vl. Fortunately, it is not necessary to calculate them explicitly; we need only to verify their existence and regular behaviour. The second term eWl(X, t) of the outer expansion satisfies the initial value problem OWl

OWl

a(x, t, Wo)-~x + b(x, t, Wo)--~ + dl (x, t, To)W1 = -

( 02w0

Ot2

02W0)

Ox 2 ,

(10.5.10)

252

Chapter 10. Perturbations in Nonlinear Initial Value Problems

with - c r < x < +cr

0 < t < T, and the initial condition Wl(X, O) = - v o ( x , 0).

The function reads

di (x, t w0)

is obtained with the aid of the mean value theorem and it

Oa Owo Ob Owo Od ( x, t, wo ) . d i ( x, t, wo ) = -~u ( X, t, wo ) ---~x + -~u ( X, t, wo ) - - ~ + -~u

t satisfies the boundary value problem The second initial layer correction e2v.i(x, 7) 02711

OqVl

OT----~ + b(x, O, Wo(X, 0))-~T = r

T),

(10.5.11)

T > 0,

with the initial conditions

Ovi OWl OT (x, 0) = -- - - ~ (x, 0) and 7----+oo lim vi(T) = 0. r

T) is an expression regular in x and T and it has the form

r

Ovo

= p0(~, ~0(~, 0))~0(~, ~)+ p~(~, ~0(~, 0))-g2~ (~,,) Ovo

-4- p2(X, lV0(X, 0))Wl (X, 0) ~

0~o (z, ~)

(X, T) -+- p3(X, W0(X, 0)) T-~T

+ p~(~, ~o(~, 0))~0(~, ~) O~o (~, ~); for details see[48],[49]. It follows that Wl is C ~ in - c r 1 6 2 0O, O<x O.

An application concerns the oscillations of an extensible elastic string. The same authors consider in [40] the case where g[u] is replaced by some function h(u~). The results are similar as those obtained above in (10.5.29). Other variations on this theme are presented in [35, 38]. 3. The modelling of nonlinear waves leads to interesting singular perturbation problems. Consider the following initial value problem for a solution of the well-known transport equation 0--t + ~

o(1u 2) = 0 ,

-c~<x0

(10.5.35) (10.5.36)

with -~-oo

f

uo(x)dx < oo.

To fix ideas suppose uo(x) > 0 with only one maximum. Because the wave velocity equals u(x, t) the top of the wave will travel faster than the b o t t o m and so the

258

Chapter 10. Perturbations in Nonlinear Initial Value Problems solution is in general multivalued which is not acceptable in physics. (see Figure 10.2) This multivaluedness m a y be removed by a d m i t t i n g solutions which m a y be discontinuous across some curve x = s(t) in the (x, t) plane, a so-called shock wave.

!

Fig. 10.2

Shock Wave at t = to

Let us now make the explicit a s s u m p t i o n t h a t the t o t a l "mass" is a conserved quantity. This m e a n s +~

d f u(x,t)dx dt

O,

or

s(t)-0

-dtd f

oo

u(x t)dx + -~d

-oo

I

~(t)-0

u(x, t)dx -

~(t)+0

oo

-~Ou(x, t)dx +

I

-oo

I -~~ (x, t)dx

8(0+0

+ { u ( ( s ( t ) - o),t) - u ( ( ~ ( t ) + o ) , t ) } ~

ds

= o.

S u b s t i t u t i o n of (10.5.35) gives ~1[u2] --[u] d8 and the velocity of the shock wave becomes

ds

l[u] 2 d-t- = 2 [u] '

(10.5.37)

where [f] denotes the j u m p across the shock f((s(t) + 0), t) - f((s(t) - 0), t). On the other h a n d u m a y be considered as a generalized function U; taking U _= 0 for t < 0 and accounting for discontinuities we have the following relations between the distributional and the "ordinary" derivatives of U and u respectively

OU Ot and

B

Ou Ot

ds

[u]~(~ - ~(t)) ~ + uo(~)~(t)

1 2 OsU _ 0~lU2 + Ox - O x

1 -2

[u~]~(~- ~(t))

5.3. The Estimate of the Remainder Term and Final Results

259

and by (10.5.37) we get for the transport equation (10.5.35) together with the initial condition (10.5.36) the equation

Ot + - ~ x

U 2) = u 0 ( x ) 5 ( t ) ,

U=OfortO

--oo

valid for any smooth test fucntion ~ with compact support in the region t >_ 0. u = u(x, t) is the discontinuous solution of (10.5.35)-(10.5.36) where the shock cuts off equal areas from the area below and above the curve u = u(x, t), see Figure 10.2. Also this discontinuous solution is not quite satisfactory and therefore one introduces a small dissipation such that u becomes continuous, however, with a steep gradient across the curve x = s(t). Oleinik [115] considered the singular p e r t u r b a t i o n problem

0u. o--t- + -~x

u~

= e Ox 2 ,

- c o < x < +co,

t > O,

e > O,

with the initial condition

u~(~, 0) = ~0(~).

(10.5.39)

First we remark that this initial value problems is uniquely solvable for all E > 0 and its solution is single valued for e sufficiently small. Therefore the introduction of a generalized solution is not necessary; see [147], pp.96-97, where the Cole-Hopf transformation is treated. Oleinik showed t h a t u~(x, t) converges for c -+ 0 in distributional sense to a certain function u(x, t) in any interval [a, b] of any straightline t - to > 0 and moreover t h a t this function satisfies the relation (10.5.38). E s h a m [37] took instead of the transport equation the Burgers equation

Ou

Ou

o-~ + u ~

= Z~,

Z>0,

and he studied the effect of a p e r t u r b a t i o n with a small term ~utt. In particular he investigated the initial boundary value problem

e2 02u Ou 02u Ou -~+--~-fl-~x2+U-~-~x=f(x,t), u(0, t ) = u ( 1 , t ) = 0 ,

u(x,O) = g(x),

-1 <x O, Ou 0) = g ( x ) , -cx~ < x < +co. where g ( x ) is an u ( x , O ) = sinx, -bT(X, arbitrary C~176 Sketch the solution w ( x , t) of the reduced initial value problem for the values t = 0 , t = t l > 0 and t = t 2 > > t l (seeRef. [83]).

Chapter 11 SINGULAR PERTURBATIONS IN NONLINEAR BOUNDARY VALUE PROBLEMS OF SECOND ORDER 1.

Introduction

In this chapter we turn our attention to singular perturbations in nonlinear boundary value problems of second order. As in the preceding chapters 7 and 8 we first construct a formal solution consisting of a combination of an approximation of the solution of the unperturbed problem with ~ = 0 and a correction due to the loss of a boundary condition. As to the proof of the justification of this procedure we investigate again the error in the formal approximation and show that this error becomes small when the perturbation parameter goes to zero. We treat nonlinear ordinary differential equations and also partial differential equations of elliptic type, the latter less extensive than the former because of complex and tedious calculations. In section 2 we discuss the boundary value problem

d2u du ~-d--~ + a(~ ' ~) ~ + b(~ ' u) - o,

o<x a o > O and one a t x - 1 whenever a(x, w) < a0 < 0. The justification of the construction is established by using again the fixed point theorem of section 10.2; it will appear that the verification of the condition 10.2.2 requires some extra effort. In the next section 3 we drop the condition a(x, w) -71=0 and this may give rise to the occurrence of interior layers (shocks). The theory is illustrated with the well-known Lagerstrom-Cole example

d2u du ~-ff-~x2 + U--~x

u = O,

u(0)=a0

for the cases / 3 > a § and f l < a + l . Another interesting example is the boundary value problem

d2u

du

e~x2 + U ~ xx = 0 ,

u(-1)=a,

u(+l) -/3

262

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

for several values of a and ft. This example shows the "capriciousness" of singular perturbations, a phrase from W. Wasow [144]. Section 4 is devoted to autonomous conservative equations of the type ~2 d2 u

+ f (u) = O,

u(O) - a,

u(1) = ~,

and we use phase-plane representation and some geometric intuition. In section 5 we present the theory of Haber and Levinson [59] concerning the more general boundary value problem

d~ (d~) ~-~2 = / u , ~ ,

u(0) = a and u(1) = ~,

where .f(u, v) is submitted to the condition

0/( dg ) Ov g ( ~ ) ' ~ ( ~ )

>0,

0___x a0 > 0, see (7.2.12)-(7.2.14). The formal approximation is constructed again with the aid of a composite expansion N

N+I

Z

Z

k=0

k=0

c

+

(11.2.4)

To obtain an estimate of RN(X;---c) which is uniformly small in [0, 1] it will appear that it is sufficient to take N = 1. The function To(X) is defined by the unique solution of the reduced problem (11.2.3) and the function Wl(X) satisfies the equation

e --d-~x2 +

d x 2 + . . . + a (x , w o + e W l + . . . ) + b(z, w0 + 6Wl + ' " )

( -~xdwo+ e --d--xx + " " ) = 0,

264

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

or in a first approximation

dwo dwl {a(x, Wo + ewl) - a(x, w0)}--~x + ea(x, wo) dx d2wo .-~{b(x, w 0 --~ ~ W l )

-

b(x, Wo) } - - c

dx 2

or with the aid of the mean value theorem

dwl{Oa dwoOb } a(x, wo)--~ x + ~u(X, w o ) ~ + ~u(X, wo) wl =

d2wo dx 2 ,

(11.2.5)

valid for 0 < x < 1, while the boundary condition is w1(1) = 0 . We remark that" this boundary value problem for Wl and generally all the boundary value problems for the terms wk of the so-called outer expansion are linear, so that no complications in the construction of the outer expansion arise. To meet the boundary condition w(O) - a we should introduce the boundary layer terms ~kvk(~c) with ~ - - ~ . Putting

u(~) ~ ~ k ~ k ( ~ ) + ~k~k(~)"= W + Y and assuming that the outer expansion W is known we get for the boundary layer part V the differential equation

d2W l d2V ( dW l dV ) ~--s + - - ~ + a(x, W + Y) ~ + -~7 ( + b(x, W + Y) = O, or

1 2V1

{

dW + b(x, W + V) - ~d~ 2 -~- a(x, W + v) dV + a(x, W + V)-~x

} = -~

,

or

dzV dV dW d~ 2 -+-a(x, W + V) ~ + 6{a(x, W + V) - a(x, W ) } - ~ x + r ( d2W dW "1 =- r + a(x, W) ~ + b(x, W) ~ := d(x,r In case we take N = 1 we have d(x, r - O(r equation as follows ~r d2vk _~_ + a{e~, ~r (wk(r

W + V) - b(x, W)} (11.2.6)

uniformly in [0, 1]. We write this

vk(~)}~r kdvk d~

+ ~[a{~, r ~ ( ~ (~) + ,~(~))} - a{~, r ~ ~ ( ~ ) } ] r ~ - dwk ~ (~r + ~[b{~, ~ ( ~ ( ~ ) + ~(~))} - b{~r ~ ~ ( ~ ) } ]

= d(~, ~).

265

2.1. The Formal Approximation

Using the regularity of the functions a(z, u) and b(x, u) and expanding this expression into powers of e we obtain after straightforward calculations the following differential equations for the boundary layer terms v0, vl, v2"

d2vo

dvo

d~C-Z--- 4. a (0, wo(O)+ vo(~))-~__ = O,

d2Vld~ + ----~

dVl 4a(O, wo(O) + vo(~r --~

"(11.2.7)

-~uOa(0,

w0(0) + v0(~)) -d-~Vl dvo (~r =

dvo

el(~, V o ) T ( ~ ) 4. fl(vo) -- fl(O)"-- gl(~, Vo) a;

(11.2.8)

and d2v2 d~---7 4. a(0, w0(0) + v0(~)) ~

dvl

~- e2(~, VO, V l ) - ~ ( ~ ) ,

dvo

Oa (0, wo(O) + vo(~C))-d-~ v2(~) + Ouu dvo

+ e3(~, VO, V l ) ' - ~

4. f2(~, VO, Vl) -- f2(~, O, O)

J

(11.2.9)

:= g2(~, vo, ~l)

where el, e2, e3, fl and f2 are smooth differentiable functions. The equations (11.2.7)(11.2.9) are valid for 0 < ~ < c~ and we have as boundary conditions Vo(O ) -- OL -- Wo(O),

Vl(O) -- --Wl(O),

v2(O ) -- 0

(11.2.10)

and lim Vo(~)- lim v~(~)- lim v2(~)= O.

(11.2.11)

The equation (11.2.7) is a nonlinear equation while the other two are linear. The solution of (11.2.7) with the conditions (11.2.10)-(11.2.11) is readily obtained with the aid of the inverse of the function v 0 - v0(~); putting ~ = ~ ( v 0 ) w i t h ~ ( v 0 ( 0 ) ) - ~ ( a - w 0 ( 0 ) ) 0 and ~(0) - c~ we get for ~ the equation

---+

a(O, ~ o ( 0 ) + ~o) = O,

where the prime denotes differentiation with respect to v0. Hence already anticipating limcx~ ~d~ = 0, we obtain ~--+ vo

1

~'(~o)

_- - / a(O, wo(O) + r/)dr/:=

Q(vo),

0

and therefore

Vo

~ - v(~o) - +

/

a-wo(O)

de Q(~)"

(11.2.12)

266

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

Because a(0, w 0 ( 0 ) + y) > a0 > 0 we have ~'(v0) negative or positive according to v0 being respectively positive or negative and it follows that v0(~) decreases or increases whenever v0(~c) is respectively positive or negative. Moreover

~0

d~ 2 ~ - a

(0, w0(0)) dvo

for v0 small and so we have negative exponential behaviour of v0(~) for ~r large which proves the boundary layer character of the function v0(~r Using the smoothness of the right-hand side of (11.2.8) and the boundary layer character of vo it is not difficult to solve consecutively the boundary value problems (11.2.8), (11.2.9), (11.2.10), (11.2.11). The equation (11.2.8) is equivalent with the equation

d2Vl~-d~----I- ~d { a (0, wo(O) +

vo(~)) 9Vl (~) } = gl(~, Vo)

(11.2.13)

or

dvl d----( +

a(0, w0(0)

+

v0(~))

9V l - -

/

gl (r/, v0(r/))dr/,

(11.2.14)

oo

where we have already taken care of the condition lim Vl (~) = 0. Due to the exponential behaviour of v0 and the form of gl(r/, v0(r/)) as given in (11.2.8) the integral in the right-hand side of (11.2.14) converges. A solution of the homogeneous equation corresponding with (11.2.14) is according to (11.2.7) given by d,o d~ and so we get with the boundary condition Vl(0) = -Wl (0) and by variation of constants after a straightforward calculation the result

v~(() -

-

[

W 1 (0) ~,

d~ (0)

dvo

+ \ d~ (7"1) { f gl 07', vo(rl/))drl/}drl --~-(~r

(11.2.15)

o

From the boundary layer behaviour of v0 we get immediately from (11.2.15) that also vl(~) has the right boundary layer behaviour. The boundary value problem (11.2.9) with (11.2.10)-(11.2.11) may be solved in a similar way and the result reads

v~(~) =-

-~(~) o

) { / g~(~', v0(~'), v~(~'))a~'}e~

-~-(~1.

(11.2.16)

~?

Summarizing all these results we have constructed the following attempt for an asymptotic approximation of the solution u(x) of our boundary value problem (11.2.1)(11.2.2)

G

:= ~(~1 + R(~, ~1.

(11.2.17)

267

2.2. The Estimate of the Remainder Term and Final Results

From (11.2.6) and the boundary conditions for w~ and v~ we get that ~(x) satisfies the boundary value problem d2u

e--~x2 + a(x, ft)

dfi

+ b(x ~) = O(e 2)

uniformly in [0, 1], with ~(0) = c~, ~(1) = ~ + o ( e N ) , (N arbitrarily large and O(e N) means asymptotically zero). Therefore the function g(x) is a formal approximation of the order 2 for the solution u(x) of the boundary value problem (11.2.1)-(11.2.2). Moreover, because

r with

d2ttd2R 2+r

(dftdR) 2 + a(x, fi + R ) ~x + ~

g ( 0 ) + R(0, e) - c~ and

f i ( 1 ) + R(1, e) = ~,

+ b(x, g + R ) - O, we have for the remainder term

R(x, e) the boundary value problem d2R . e~+a(x,

dR d~ ft + R ) ~ + {a(x, ft + R) - a(x, ft) }--~x + {b(x, ft + R) - b(x, ~) } -

O(e2), uniformly in [0, 1], with R(0) - 0 and R(1) - o(eN). To estimate this remainder term it is useful to have homogeneous boundary conditions. Therefore we change R(x) by an amount o(eN), which has of course no influence on the estimate of R(x). We introduce

R(x) - R ( x ) - R(1). r

and ~(x) = g ( x ) + R(1)r

where r C C~[0, 1] with e.g. r - 1 for 3/4 _< x _< 1 and r 1 0 _< x _< 5" Due to the regularity of the coefficients a and b we get finally d2R

d/~

- 0 for

d~

c--~x2 + a(x, ~t + R) -~x + {a(x, ~ + [~) - a(x, u)} d---x + {b(x, ~ + it) - b(x, ~)} = O(e2), uniformaly in [0, 1],

(11.2.18)

with the boundary conditions R(0) = R(1) = 0. It is now our task to derive from this boundary value problem an estimate of the /~(x). This will be carried out in the next section with the aid of the fixed point treated in the preceding chapter. Then it will also become clear why we had a formal approximation up to O(~ 2) and why a formal approximation up to not sufficient to produce an appropriate estimate of the remainder term. 2.2.

T h e E s t i m a t e of t h e R e m a i n d e r

(11.2.19) function theorem to make O(e) is

Term and Final Results

The boundary value problem for the remainder term /~(x) is given by the equations (11.2.18)-(11.2.19).

268

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

In order to apply the fixed point theorem we introduce the nonlinear map F ~ ] from the normed space N into the Banach space B. The space N is defined as the linear space N := { P I P e C2[0, 1], p(O) = p(1) = O} (11.2.20) with norm

I pl

d2p ~(~)1 + e 2 ~O,al:~I~1,

- max Ip(x)l + ~ max I dp to,~ tO, l~

(11.2.21)

while the Banach space B is given as B := { q l q e C[0, 1]}

(11.2.22)

Ilqll = max Iq(x)l. [0,1]

(11.2.23)

with norm

The map F[p] is defined by

d2p dp d~ F[p] - e--~x2 + a(x, fL + P)-~x + [{a(x, fL + p) -- a(x, u)}~xx + {b(~, ~ + p) - b(~, ~)}],

(11.2.24)

where fi is the given formal approximation of the solution of the boundary value problem (11.2.1)-(11.2.2). F ( 0 ) = 0 and the linearization of F(p) in p = 0 reads

d2p dp { Oa dfz Ob } Lip] - e --~-~x2 + a ( x , u) -~x + ~ ( x , ~ ) -~x + ~ ( x , ~ ) p

(11.2.25)

and hence

dp {a(x ~ p) a(x, ~) Oa (x, fL) } d~ q2[p] = F[p] - LiP] - { a(x, fi + p) - a(x, fi) } ~x + ' + - P--~u dx + {b(~, ~ + p) - b(~, ~1 - p ~Ob (x, fi)}

(11.2.26)

To apply the fixed point theorem of Chapter 10 we should verify the conditions (10.2.2) and (10.2.3).

The condition (10.2.2) This condition concerns the bijectivity of the operator L and the continuity of L - I , i.e. ] L-l[q] ] < l-lllqll,

Vq E B,

(11.2.27)

where I is some number independent of q. Assuming bijectivity one might at first sight try to prove this relation with the aid of the maximum principle and the barrier function technique as has been used in Chapter

269

2.2. The Estimate of the Remainder Term and Final Results

7, subsections 7.2.3 and 7.2.5. However we meet here a rather unpleasant situation since for the application of the maximum principle we should require

Oa

dfi

--a~(~' ~) " ~

Ob

+ ~u (~' ~) < ~ < o

(11.2.28)

Because ~da = O ( 1 ) in the neighbourhood of x = 0 and because we did not fix the sign of oa (x, ~) in this neighbourhood we are not sure that we can fulfil this condition. The trick to introduce i5 = pe -~ does not help either since in that case t5 satisfies the equation d2i~

L[~] =e-~x2 + {a(x, ~t) +

2sO} di6

--~x+

[ Oa d~t Ob ] ~02 + ~a(x, ~t) + { -~u (X, {t) -~x + Ou (x, Ct)} ~ = e -ox 9q(x). Due to the regularity of a(x,~), b(x,~), a(x, fi) > a0 > 0 and the boundary layer behaviour of ~ there exist positive constants A,B, C and K, independent of e, such that

0 < a(x,~z) __ao > O, Vx E [0, 1], Vu C 1R and with ao some positive constant. Under the assumption that the solution of the reduced boundary value problem

a(x, w) --dTx dw +

b(x, w) - O, - 0 < x < l,

w(1)=/~

exists uniquely in [0, 1] we have the result that also the solution of (11.2.40)-(11.2.41) exists uniquely for e sutticiently small. Moreover the following approximations hold, uniformly in [0, 1], Ue(X) -- W O ( X ) +

and

du~

CWl(X)-~- VO(x) + gVl (X) "~-0(C2 )

d~o ~

d~ (~) = ~

+

~)+0(~) ~~

"

(11.2.42)

(11.2.43)

The outer solution terms Wo and 6W 1 a r e given by (11.2.3) and (11.2.5) and the boundary layer terms vo and CV1 b y (11.2.12) and (11.2.15).

3.

Transition Layers We consider again the boundary value problem r

d2u du 2 + a(x, u) -~x + b(x, u) = 0,

with the boundary conditions

0 < x < 1,

(11.3.1)

274

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

u(0) - a,

u(1) =/3.

(11.3.2)

In contrast with the preceding section we drop the conditions (i) and (ii) of subsection 11.2.1. This means t h a t a(x, u) may become zero for some value of u and x with 0 < x < 1 and there is the risk t h a t the reduced equation

a(x, w) dw + b(x, w) = O,

(11.3.3)

becomes singular; take e.g. a(x, u) = u and aft < O. In this case we are in a situation similar as t h a t of a turning point, discussed in section 7.3, with the difference t h a t this point is not a priori given by some value of x, but its location depends in general on the differential equation and the b o u n d a r y values. Instead of the conditions i) and ii) we suppose now t h a t there exist functions UL (x) and uR(x) which satisfy the requirements duL a ( x , UL ) ~ nt- b(x, UL ) = O,

O < x < &,

duR a(x, u R ) - ~ + b(x, uR) = O,

_< x < 1,

uL(O) =c~ and

(11.3.4)

uR(1) -- fl,

(11.3.5)

while a(x, UL) < O,

0 < X < ~ and a(x, uR) > 0,

~ < x < 1.

(11.3.6)

Assuming t h a t our b o u n d a r y value problem has a solution we may infer from T heorem 2 of section 10.3 t h a t UL(X) and uR(x) are good approximations of u(x) up to O(e) in the intervals 0 < x < ~ and ~ < x < _ 1 respectively. du (x) in x - 0 and x - 1 are We remark t h a t according to this t h e o r e m the values of ~-~x of no relevance in an O(E)-approximation of u(x). In the supposition t h a t UL(X) and uR(x) are known we have now to construct an interior layer connecting u ,.~ UL(X) and u ~ uR(x); it will appear t h a t this layer has the appearance of a shock wave whenever UL(5:) =/=uR(~). This transition is analyzed by stretching the x-variable around the still unknown value of ~ by x-~ -- ~ ,

(11.3.7)

and we represent the transition layer by a function v(~) that should satisfy in approximation the relation U(X) ,~ UL(X ) nt- V(~) -- UL(X) + V

E

with lim v(~) = 0 and lim v(~) = uR(~ + 0) - UL(X, -~- 0). ~-~-~ ~+oo

(11.3.9)

The latter limits implies t h a t the solution UL(X) of (11.3.3) can be continued into an interval 0 < x < ~ + (f, where 6 is arbitrarily small positive but independent of e.

275

3. Transition Layers Substitution of (11.3.8)into (11.3.1) gives

r

+ -d2v ~ + ca(g: + ~-~, UL "+"V)~ dui 1 dv ) \ dx +-e-~ +r - d=~ ~ a (~ + ~ , ~ ( ~ + ~ ) + ~(~))-~dv + 6{a(& + r

UL(X + c~) + v({)) -- a(5; + r

UL(X + r

duL (~c+ e{)

+ ~{b(~ + ~, u~(~ + ~) + ~(~)) - ~(~ + ~, u~(~ + ~)} = 0. Assuming sufficient regularity of the functions a and b we get in first approximation the following boundary value problem for v(~)"

d2v dv d{-----5 + a(~, UL(5:) + v)--~ = O,

(11.3.10)

with lim v({) = 0 and lim v({) = ~-+-c~ ~+m Integration of (11.3.10) with

UR(~C)-- UL(g:).

(11.3.11)

lim dv {-+-m 7~(~) = 0 yields

dv f a(~, UL('X) + v(~')) ~-~(~')d~' dv d~c + = O, --(20

or

uL(~)+v d---~+ ,,L (i:)

a(~:, rl)drl = O, with ( ~lim v(() - UR(~:) -- UL(~:). +~

(11.3.12)

uR(~:) > UL(~:) the function v(() is monotonically increasing from zero to uR(~) - UL(~) when

In case

w

f

a(~, ~)d~ < O,

for all values of w between UL('Yc) and uR(&); quite similarly in case uR(~c) < ui(~,) the function v(~c) is monotonically decreasing from zero to uR(x) UL(~C) when -

a(bc, rl)dr] > O, ,,L(~) for all values of w between

un(~c) and UR(~C).

276

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

So it follows that we obtain a monotone transition whenever w

-

f

(11.3.13)

< 0,

UL(~,) for all values of w between UL(YS) and uR(&). Finally, because ~d" approaches zero for ~ -+ +c~ we obtain from (11.3.12) the following equation for the location of the transition layer

/

(11.3.14)

a(&, rl)drl = 0.

We illustrate this discussion with a few simple but interesting examples. Examples 1. A well-known example has been provided by Lagerstrom and Cole; it is given by the boundary value problem

d2u du e-d~x2 + U-~x

u = O,

O < x < 1,

u(O) = c~, u(1) =/3.

(11.3.15)

The reduced initial value problems are

duL

UL--dx

UL -- O,

duR

un---dx- - - u R = 0 ,

UL(O ) = 5,

O < X < ~ and

UR(1)--~,

&<x 0 and 6 independent of E; therefore the equation for the transition layer is determined by (11.3.12), which becomes in our example ~+ad-v

d---~+

or

dv

d~

~drl = 0,

1 2 { (~ + ~ + ~)~ - (~ + ~) ~}'

3. Transition Layers

277

with lim v ( ~ ) = ( ~ + / 3 - 1 ) - ( ~ : + a ) = / 3 - a - 1 . The transition point x = :~ is determined by the condition (11.3.14), which reads &+fl-1

/

1 r/dr/= ~{ (2 4- fl - 1) 2 - (2 + a) 2 } = O,

and therefore ~+~-

1- +(&+a).

In case of the + sign we get a = f l - 1 and the solution of the boundary value problem (11.3.15) is simply u = x 4 - a , 0 < x < 1 and there is no shock. The other possibility is 1 ~- ~(1-a-~), (11.3.18) and this puts another restriction on the boundary values of a 0 ce 4- 1 Interior Shock

Shock layer region in a -

fl plane

It follows that a and fl must satisfy the inequalities - 1 < a + fl < +1.

(11.3.19)

278

Chapter 11. Perturbations in Nonlinear Boundary Value Problems Moreover UL(Z) and u R ( z ) should be negative respectively positive in 0<x 0 and with the aid of (11.3.18) we get besides the conditions (11.3.17) and (11.3.19) the condition /3 > a + 1.

(11.3.20)

It follows t h a t the shock layer gives a positive j u m p from v = 0 to v = r SO

~+(~-t-v

I rldrl= i, +o,

and

(l+ot--,6)-I-v 89 O 0, ~ < 9 < 1;

therefore the discussion of the case with ~ > c~ + 1 breaks down. However the reduced equation has also the trivial solution u ( ~ ) = ~ 0 ( ~ ) -= 0,

3. Transition Layers

279

and so it is quite natural to expect that in the case approximation for u ( x ) is valid

/3 < c~ + 1 the following

u ( x ) = x + ol for 0 < x _< - a u ( x ) -- O f o r - c ~ _ < x _ < 1 - / 3 u ( x ) -- x + ~ -

l for l - /3 _< x _< l.

(11.3.22)

This approximation is called an angular approximation. The example with all possible values for c~ and /3 has been treated extensively in [24], where also the proof of the correctness of the approximation is given by application of the maximum principle; this principle may be generalized for quasilinear differential equations, see [119]. Another approach to analyze quasilinear boundary value problems is by using some geometric intuition applicable to autonomous equations and therefore also to our example. The independent variable will be denoted by t, which is more common in a discussion such as follows. The boundary value problem is of the form u(O)=c~O

(11.3.23)

and we introduce the so-called Lidnard transformation du

z = e--~ + F ( u ) .

(11.3.24)

The differential equation becomes the "fast-slow" system du

e-d-[ = z - F ( u ) dz d--[ = - g ( u ) .

(11.3.25)

In case of the Lagerstrom-Cole example we have du

~--J-i = zdz d-t - u,

1

-~u 2, (11.3.26)

and it is easy to sketch the trajectories of this dynamical system in the (u, z)-plane; see Figure 11.3. d~t = O ( ~ ) As long as z - S u1 2 is significantly different from zero -&and dz u = e z _ ~ 2 -O(e). Therefore the time used to move along finite stretches of d-~

the horizontal parts of the trajectories is to be neglected in a first approximation.

280

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

D \\\

t

9 t/

\ A~8

""

>

ts

Fig. 11.3

Trajectories in the (u, z)-plane

Further it is clear t h a t the trajectories with neighbourhood of the curve

z > 0 are most of the time in the

1U2 , Z = F(U) = -~ and along this curve we have

dZ ~-V-dt

udU dr'

which is nothing else as the reduced equation with ~ - 0. The first case /~ > c~ 4- 1 corresponds with the curve A B C D ; the part A B with UL(t), the part C D with uR(t) and the part B C with the shock layer. The second case ~ < c~ 4- 1 is represented by the situation t h a t the trajectory passes t h r o u g h an e-neighbourhood of the singular point (u, z) - (0, 0). Since dz -- u -- O(~) the solution u - u(t) will stay some time in the d~dt ---- O(1) and ~neighbourhood of u - - 0 , see Figure 11.3. The Li~nard transformation is very useful to obtain quickly a good image of the behaviour of the solution. For m a n y other interesting examples the reader is referred to O'Malley [112], Lutz-Goze e.a. [101, 133, 134, 17, 4, 136]. The latter authors use n o n s t a n d a r d analysis and they have obtained m a n y interesting results, as well old as new. A n o t h e r example is taken from an interesting and stimulating paper "The capriciousness of singular perturbations" by W. Wasow [144]. The b o u n d a r y value problem reads

d2u

~-~ +

du

u~

= 0,

-1 < ~ < +1,

u(-1)

= ~,

~(1) = ~.

(11.3.27)

3. Transition Layers

281

In case a < 0 and ~ > 0 the reduced equation has solutions

UL(X) = O~, --l O . So let us assume that U L ( X ) - - a < O approximates u(x) for - - l _ < x < 1 and further that the boundary value u(1) = ~ is, for the time being, arbitrary; it may be positive, zero or negative. Stretching the x-coordinate in the neighbourhood of x = 1 by lmx C

gives for the boundary layer term v(~c) the boundary value problem d 2 (c~ + v) d~ 2

-

d(c~ + v) d~

( c ~ + v ) ~

= 0,

0 0, V((, r/) 9 ]R2, (~, 77) # (0, 0).

(11.6.3)

The differential expression is elliptic in 1) with respect to u = u(x, y) iff (11.6.3) holds in each point (x, y) C l). To formulate a maximum principle for the nonlinear elliptic differential equation of the type (11.6.2) we consider a function w(x, y) that satisfies the inequality

(

OW OW 02W 02W 02W)

F x , y , w , Ox' Oy' Ox 2' OzOy' Oy 2

< f(x,y)

and we get

F(x, y, u, u~, uy, u ~ , uxy, uyy) - F(x, y, w, wx, wy, wx~, w~v, wyy) > O. Introducing the difference v(x, y) = u(x, y ) - w(x, y) and using the mean value theorem we obtain for v(x, y) the linear differential inequality

no[v]-

~

o-~-~-~x2+ ~

N 0~ +

ooxoy +

~

o-~y2 +

oN + ~u oV>~

(11.6.4)

where the coefficients are evaluated at the arguments uo -- w + O ( u - w), uox wx + O ( u x - wx), etc. with O= O(x,y) and 0 < O(x,y)< 1. If we now make the crucial assumption that the functions u(x, y) and w(x, y) exist, that L0 is elliptic in l) with respect to u0 and finally that (-b-4~)0~_< 0, V(x, y) C ~9, then it follows from (11.6.4) that v(x,y) cannot have a positive maximum in an interior point of :D. From this simple statement one obtains the following lemma which is a generalization of the maximum principle for linear elliptic equations, see Chapter 8, section 2. LEMMA 1

Let the twice continuously differentiable functions ~(x, y), ~ l ( x , y) and ~2(x, y) satisfy within a domain ~) the inequalities L[~I] < L [ r

< L[@2], V(x,y) E :D,

(11.6.5)

where L is the differential operator defined in (11.6.2), elliptic in l) with respect to the functions ~ + 0 1 ( g 2 1 - ~ ) and ~ + 0 2 ( ~ 2 - ~ ) and OF < 0 for all ( x , y ) E T ~ and for all functions u(x, y) C C2(:D); 01,02 are defined as 0 above. If moreover

9 ~(~, y) < ~(~, y) < ~,(~, y), then this relation holds also for all points (x, fl) C f).

v(~, y) e or,

(11.6.6)

293

6.3 Elliptic Equations without First Derivatives

The proof is simple; it is similar as the proof of the maximum principle for ordinary differential equations, as given in Lemma 2 of Chapter 7, section 2.2. Finally, we remark that the inequality (11.6.5) may be replaced by the sharper condition L[~I] _< L[O] < L[ql2],V(x,y) e 7),

(11.6.5")

see [119], Chapter 2, section 16 and Chapter 1, section 1. If we want to apply this lemma to estimate the error in an asymptotic approximation of the solution of a singularly perturbed nonlinear boundary value problem we are hampered by the condition

OF _< 0, Ou

V(x,y) e T~,

Vu e C2(D).

(11.6.6)

This is rather serious as may already become clear from the following semilinear example

02u ~ Ou u) = O. aij(xl,x2, u, u~,, u~.2) OxiOxj + bi(Xl,X2, U)~xi + d(xl,x2, i,j=l

i=l

To be sure that OF < 0 one needs the restrictions that aij and bi do not depend on u and that Od +5 KV~. Hence R - - O ( v / ~ )

uniformly in ~ and we have the following theorem.

THEOREM 2

Let u(x, y) be the solution of the boundary value problem

h with u(x, y) - ~o(x, y) along Of~ and suppose that the reduced equation

f(~, v, ~ ) = 0 has a solution in f~. Under the assumptions that all data are infinitely often differentiable and that

, o f v,~) < -~ < o, o~(~

v(~,v) e ~ ,

WER

297

6.3 Elliptic Equations without First Derivatives

with 5 independent of e, then the following approximation for the solution holds u(x,y) - w(x,y) + r

a) + O(x/~), uniformly in ~.

(11.6.21)

The function w(x, y) is the unique solution of the reduced equation f (x, y, w) = 0 and -e- a) is the boundary layer term with exponential behaviour. The function v/~ v(T,a) is given by the boundary value problem (11.6.15)-(11.6.16) and r is an appropriate C~ off function.

r

Remark

As to the proof of the validity of the approximation (11.6.18) it is also possible to use the fixed point theorem of Chapter 10, section 2. We consider again the boundary value problem (11.6.20) for the remainder term R(x, y)

02R

F(R) "= ~ ~

02R)

+ ~

+ f ( x , y , ~ + R) - f ( x , y , ~ ) - g(x,y) = O(v/~),

(11.6.22)

uniformly in Ft, with R(x,y)log t = O. The linearized differential operator L reads Lip]- r

02p

~

02p~

+ ~]

Of

+ ~-~u(x, y, ~)p

and therefore

~o] = f(x, y, gz + p ) - f(x, y , ~z) - ~u Of (x, Y, u)P : ~-U-gU2 02f (x, Y, u + Op)p2. The space N is chosen as

N = { p i p ( x , y) C C2(Ft), p(x, Y) logt = 0}, with norm

I P(x,Y) [ = m a x Ip(x,y)l, and the space B as

B - {qlq(x,y) e C(h)} with norm [[q(x, y)[[ =

max_ [q(x, Y)I.

Because O--4(x, of y, 5) < - 5 < 0 we may apply the maximum principle to the linear boundary value problem Lip] -- q, Ploa - 0; choosing a suitable barrier function one easily shows that the condition (10.2.2) of the fixed point theorem is fulfilled, i.e.

]L-~[g]l ~ ~-lllgll,

Vg ~ B,

298

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

where g-1 is independent of r The Lipschitz condition (10.2.3) becomes

IIr

r

02 f

= m~x~ ~

02 f

= max~t ~

02.1'

(~, y, ~ + e~V~)V~ - ~

(~, y, ~ + 0~p~)Vl~

(x, y, ~t + 02P2)(P22 _ p2) +

< C 1 m a x {([Pl[ + [P2[)IP2 - Pll} +

f~

02 f (x ft + Olpi) } p 2 ~u2t.x ' y, (t + 02P2) - ~u 2 , Y,

{02f,

C2max {]pl[ 2- IP2 - Pll} < Cp IP2 - il91 I F~

,

where C1, C2 and C are constants independent of e and this inequality is valid for all pl, p2 in a ball ~N(P) with [ p] < 1. Applying finally the fixed point theorem to the boundary value problem (11.6.22) we obtain the result that the remainder term uniquely exists and moreover max IR(x, Y)I = O(v/~ ). From the unique existence of R(x, y) we get also the unique existence of the solution u(x, y) of our boundary value problem, provided the reduced equation has a solution. Therefore we may omit in Theorem 2 the assumption concerning the unique existence of the full boundary value problem (11.6.8)-(11.6.9); however the condition that f ( x , y, w) = 0 has a solution in ~ should be retained. We illustrate the theory with two examples. Examples 1. Consider the boundary value problem

~Au - u - g2(x,y)u 3 = O,

(x,y) e ~,

u]ogt = ~0]o~,

(11.6.23)

where g(x, y) is a positive Coo-function and ~t a bounded domain with 0~t of class Coo. By our theorem we get

u(x,y) = r

a) + O(v/~), uniformly in

where v(7,a) is the boundary layer function determined by (11.6.15)and (11.6.16) and r a suitable C~176 off function. Constructing an infinite asymptotic series for u(x, y) one obtains that u(x, y) is asymptotically equal to zero uniformly in any closed e-independent interior domain of ~. 2. Another boundary value problem is from the often cited paper by Berger and Fraenkel [6]; it reads as follows

~

+ ~ - g~(~, y ) ~ = 0,

(~, y) e ~,

~]o~ = 0,

(11.6.24)

299

6.3 Elliptic Equations without First Derivatives

with the same assumptions as in the example above. Berger and Fraenkel showed that there exists besides the trivial solution also a unique 1 positive solution u(x, y) that tends to g-(E~,y) as e $ 0 outside a narrow neighbourhood of width O(x/~) concentrated along 0Ft. Just as before a formal solution is easily constructed, viz.

s where r ~Ou

v(T,a)

y) = g(x, y-------~+ r

, a) "= ~

g(~,y)

is the boundary layer function with

a C~ off function. The condition o~ y, ~) < - 6 < 0 and we have

+ ~(p, a; c),

(11.6.25)

v ( O , a ) - - { g ( x , y ) } - l ] o ~ and

y, u) < - 6 < 0 implies

ol Ou (x,y,~) - 1 - 3{1 + g(x,y)~(p,a; ~)}2. The function ~(p,a; ~) is monotone in

~

and it increases from

- { g ( x , y ) } -1 to

zero and so we have ~(p, a; ~) _< 0. It follows that ~ (x, y, ~) is certainly not negative in a neighbourhood of the boundary 0Ft and we cannot use Theorem 2. Nevertheless the formal solution (11.6.25) can be justified and this is possible by a modification of the proof of the fixed point theorem of section 10.2. The remainder term R(x,y) = u ( x , y ) - ~t(x,y) satisfies according to (11.6.25) the boundary value problem

02R ~)2R~ FIR] = e -~x2 + -~y2 ] + ( 1 - 3g2~t2)R- 3g2(tR2 - g 2 R 3 = h(x, y) = O(x/~), (11.6.26) uniformly in Ft, with RIo a = O. The linearization of F is n[p]=e

02p

~ + ~

02p )

+(1-3g2~2)p,

and therefore

V~[p] = _3g2~tp 2 _ g2p3. The coefficient (1 - 3g2g 2) is not negative in all of ~t and so we cannot apply the maximum principle to obtain an a priori estimate of the solution of

L[p] - q E C(~t) in the space N = {PIP E C2(~t), Ploct - 0}. As remedy one uses instead of the spaces C ( ~ ) and N Sobolev space H~(Ft) respectively, see next chapter.

the spaces L2(Ft) and the

300

Chapter 11. Perturbations in Nonlinear Boundary Value Problems

Berger and Fraenkel show by using the known expression of fi(x, y) that

cOp) 2 s

cOp 2

~2

p2 dxdy] s

:= C~llpll~, vp e H,~(r~). This means that L is a positive operator (strongly coercive) in H0~(f~) and it follows from the Lax-Milgram theorem that L is invertible and one has the estimate

Ilpll~ -~ IlqllL_____~,

(11.6.27)

(see next chapter). Therefore the constant ~ in (10.2.2) equals e. To apply the fixed point theorem we need also condition (10.2.3) and the requirement that the right-hand side of (11.6.26) satisfies the estimate

1 IlhllL2 _~ ~ p o -

O(~p0),

with p0 -+ 0 for e --~ 0. However h(x,y) -- O(vf~) and so in order to use our fixed point theorem we need an expansion of the formal solution into more than one term. Taking a sufficient number of terms such that Ilhlli2 = O(~ 2) we get according to (10.2.6)

IIRII~ _< 2~-~llhllL~ = O(~), and so

u(x, y) -+ ~(x, y)

for e $ 0 in the norm of H~(n), which means

f{ ( ~a (,., _ ,~))= + ( ~a (._ ~))~ + (~- ~1~} e~dy= o(~1.

(11.6.28)

~2

As to some definitions we refer the reader to the next chapter and for details of the calculations to the paper by Berger and Fraenkel [6].

6.4 Elliptic Equations with First Derivatives We should like to discuss in this final section the two dimensional generalization of the boundary value problem treated in section 2. To avoid nonessential complications we take the perturbing operator EL2 as eA and the domain f~ is a circular disc in R2. The quasilinear boundary value problem analogous to that of section 2 then reads

( 02~

02~'~

\ o~ + ~

O~

Ou

] + a(~, V; ~) ~ + b(~, V; u) ~ + c(~, y, ~) = O,

(11.6.29)

valid for 0 < x 2 + y2 < 1, while the boundary condition is

u(x, Y)lan = (p(x, Y)lon,

x 2 + y2 -- 1.

(11.6.30)

6.4 Elliptic Equations with First Derivatives

301

Despite these simplifications there are still many complications which make this boundary value problem very unattractive. We have already experienced in section 2 the difficulties to overcome in the proof of the validity of the formal procedure. Besides these we have now the extra complication that the solution of the reduced problem becomes singular in the points A and B where the subcharacteristics are tangent to the boundary 0ft. This complication was already met in Chapter 8 where the linear case has been treated; an acceptable solution was found by presenting an approximation of the solution valid outside arbitrarily small neighbourhoods of the points A and B. The same will be done in this section but not without making a further serious simplification. The boundary value problem we shall now discuss is given by

~

+ 0y ~ ] - N + f(~' y; ~1 = 0,

0 < x 2 + y2 < 1

(11.6.31)

with the boundary condition

u(x, y) = ~(x, y) for x 2 + y2 = 1.

(11.6.32)

The essential simplification lies in the fact that the coefficients of the first derivatives in (11.6.29) are independent of x, y and u. (11.6.31)-(11.6.32) become in polar coordinates

~02u e\~+rN

l Ou

1 02u~ +~-~002]-sin

OOu Or

cos 0 0 u r 00

~-f(r, O, u) = O

(11.6.33)

valid for 0 _< r < 1, 0 < 0 < 27r, with the boundary condition u(1, 0) = ~o(cos 0, sin 0 ) : = ~o(0), 0 _< 0 < 2~r.

(11.6.34)

We denote ~(0) by ~+(0) for 0 < 0

r

and if there exists a positive

H~(~t)

w

e Hg(f~)

(12.3.16)

then the generalized Dirichlet problem B[~a, u] = (~, f),

V~ E C~(~t),

u e H~(f~)

(12.3.17)

has for any f E L2(f~) a unique solution u with

I1~11,-,,< 1-11filL=.

(12.3.18)

C

The existence of the generalized solution follows from the Lax-Milgram theorem by taking F[v] = (v, f). The estimate (12.3.18) is obtained from (12.3.16) and the existence of a sequence { ~ } C C ~ ( f t ) with lim ~ = u in H ~ ( f t ) ; we have the inequalities n-~oo

cllull~ ___ Re B[u, u] < IB[u, ~]1 = n -lim IB[~o., ~]1 - - # or = n -lim [(So,-,,f ) [ - - (u, f) < [[UiiL=- II/IIL= _< Ilu~ll. II/IIL= ' - + O0 and so the estimate (12.3.18) is obvious. A bilinear form that satisfies the inequality (12.3.16) is called strongly coercive in constant with coercive as defined by the Gs inequality (12.3.10). Regarding applications to elliptic boundary value problems it is important to give some examples of strongly coercive bilinear forms. 1. Let 12 be a bounded domain and L = - A + k with A the Laplace operator and k some positive constant. We have B[v, u] = E (DPv' DPu) + k(v, u) Ipl=l -

~-5-&~_.dz+ k _ _ , _ _ .

Svudx,

[1

and therefore

B[u, u] ~ rnin(1, k)llul[ 2.

3.2 Elliptic Partial Differential Equations

321

It follows t h a t the b o u n d a r y value problem

-~(~)

+ k~(~) - f ( ~ ) ,

~ e a,

~1o~ - 0

has a unique generalized solution in H 1 for all f C L2(fl). This solution is also a classical solution if 0~t is of class C 1, u C C 2 ( ~ ) N C ( ~ ) and f C C(Ft) A L2(~t). 2. Let ~t be b o u n d e d and L = - A . To show t h a t we have also in this case a strongly coercive bilinear form we need the Poinca% inequality which may be applied to domains only b o u n d e d in one direction. This inequality reads n

i1~11~ < d 2 -

I 112 ; d 2 ~ - ~ '

[lO~

V~ C C~(Ft)

(12.3.19)

i--1 where d is the m a x i m a l diameter of ft. Suppose ft C {x C 1[~ []x~I < c}, then we have for ~ C C ~ ( f t ) Xn

f ~-:~ O~ (x 1, x2 , . . . . , x,~

~(~)=

~, ()d(.

~C

Using Cauchy-Schwarz we get +c

I~(x)l~ _< 2,

+c

Ib--~-~-(xl,x~.,...,-,~_l,e)l'd~=d --C

Ib--~ 1

,...,x,_l,

--C

therefore

+c

+c

_< --C

~C

integrating once again with respect to XliX2,...,xn_ 1 we obtain (12.3.19). The bilinear form B[u, u] associated with L - - A satisfies according to (12.3.19) Re

B[u,

n/

Ou]2 1 2 1 ~ d~ = I-I~ > ~1-1~ + 7~ I"~

u] = E i=0

E

1 1 _> min (~, ~-d--ff)ilull2,

Vu e

and so - A is strongly coercive. We remark t h a t coercive bilinear form.

L2

H~(f~), +A

3. Let f~ be b o u n d e d and L = A 2. For any qo E C~(f~)

i

a

n

"= J = l a n [

i=1 j = l f~

02 ~

2

~

does not yield a strongly we have the relation

~ 2

322

Chapter 12. Perturbations of Higher Order Applying the Poincar~ inequality to ~cOx l we have 2

_< d 2 L2

and hence n

0~O ]2

n < d2 ~"~ --ixi L2 -i=1

i=1

2

1

n

n

02~

=d2EE OxiOxj i=l j=l

L2

Therefore we get for any q0 9 C ~ (f~) I1 11 =

+

+

-< (d~+ x)l~~ ~1+ I1,~,~. 2

_< {(d 2 + l)d 2 + l}l~,l~ = {(d ~- + 1)d 2 + 1}B[~o, ~o] and we obtain by u = lim ~0,~ in Ho2(f~) the inequality n--~ r

Re B[u, u] = (Au, Au) > cllul[ 2,

Vu 9 Ho2(f~),

from which the strong coercivity follows. To give an example of a coercive but not a strongly coercive bilinear form we consider for U(Xl,X2) the boundary value problem Au+#u--0,

0_r=v/x2+x~0,

withu(ro)=0.

This problem has a nontrivial solution

= g0(v~), if V/-fir0 is a zero of the Bessel function J0. It follows that the boundary value problem - - A u - - #u = f (xl,X2), 0 ~ r < to, u(ro) = 0,

# > 0

does not have a unique solution if v/-fir0 is a zero of J0 and therefore the strongly elliptic operator - A - # with J0 (v/-fi r0) - 0 cannot have a strongly coercive bilinear form. On the other hand it has a coercive bilinear form; this follows from Theorem A or by direct computation. As to existence of generalized solutions of coercive elliptic operators we have the following Fredholm alternative THEOREM

D

I f the differential operator L satisfies the conditions 1, 2, 3 of Theorem A and f E L2(~) then we have the next two possibilities 1. B[~o, u]-- (~o, f ) with ~o e C~(f~) has a unique solution u 9 H~(12), or

323

4.1 The Boundary Value Problem 2. there exists a finite number of linear independent solutions the homogeneous equation

U[v, ~] = o, In this case B[qo, u] - (qo, f) This solution is not unique.

vj, j - 1, 2 , . . . g

of

v~ e c g (a).

has a solution if and only if (f, vj) = O, j - 1, 2 , . . . ,g.

4 Elliptic Singular P e r t u r b a t i o n s

of Higher Order

4.1 The Boundary Value Problem We study the following Dirichlet Problem for a real valued function u that satisfies the partial differential equation L~[u(x)] = ~L2m[u(x)] + L2k[U(X)] = h(x),

x e Ft C

(12.4.1)

with the boundary conditions

~gu On t (x) = O,

g = O, l, . . . m - 1 ,

x e OFt,

(12.4.2)

where Ft is a bounded domain with boundary of class C ~176The differential operators are given by L2k--

E (-1)IplDP(aP~(x)D~) ' o 0 is still free and we make now an optimal choice such that the left-hand side becomes as small as possible; this is the case when u = 1 + 2/2(k - m + 1), L

or

(12.4.22) 1

/]---

2 ( m - k)" So we have finally obtained the a priori estimate eli5]] 2 + 1]2,]]~ -- O(e2(:-k) ).

(12.4.23)

The estimate for z follows from

~llzll~m + Ilzll~ __