Introduction to the General Theory of Singular Perturbations

398 Pages·1992·4.582 MB·English

by  S. A. Lomov

Translations of
MATHEMATICAL
MONOGRAPHS
Volume 112
i
Introduction to
the General Theory
of Singular Perturbations
. S, A. Lomov
H
American Mathematical Society
Introduction to
the General Theory
of Singular Perturbations
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(Continued in the back of this publication)
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 112
Introduction to
the General Theory
of Singular Perturbations
S. A. Lomov
American Mathematical Society
Providence, Rhode Island
Translated from the Russian by J. R. Schulenberger
Translation edited by Simeon Ivanov
1991 Mathematics Subject Classification. Primary 34E 15, 34-02.
ABSTRACT. The book presents in a systematic manner for the first time a general approach to
the integration of singularly perturbed differential equations describing nonuniform transitions
such as the occurrence of a boundary layer, discontinuities, boundary effects, etc. The method
of regularization of singular perturbations presented in the book is applied to the asymptotic
integration of systems of ordinary differential equations (linear and nonlinear) and linear partial
differential equations.
The book is intended for physicists, mathematicians, engineers, and students who come in
contact with applied mathematics.
Library of Congress Cataloging-in-Publication Data
Lomov, S. A.
[Vvedenie v obshchuiu teoriiu singuliarnykh vozmushchenii. English]
Introduction to the general theory of singular perturbations / S. A. Lomov; [translated from
the Russian by J. R. Schulenberger; translation edited by Simeon Ivanov].
p. cm.- (Translations of mathematical monographs; v. 112)
Includes bibliographical references and index.
ISBN 0-8218-4569-1 (alk. paper)
1. Perturbation (Mathematics) I. Ivanov, Simeon. II. Title. III. Series.
QA871.L813 1992 92-26927
515'.35-dc20 CIP
Copyright Oc 1992 by the American Mathematical Society. All rights reserved.
Translation authorized by the
All-Union Agency for Authors' Rights, Moscow
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America
Information on Copying and Reprinting can be found at the back of this volume.
The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
This publication was typeset using .-4MS-TEX,
the American Mathematical Society's TEX macro system.
1098765432 1 979695949392
Contents
Preface to the English Edition
xi
Preface xv
Author's Preface xvii
CHAPTER 1. Introduction. General Survey
1
§1. On perturbations 7
§2. The basic idea of classical perturbation theory 9
§3. Singular perturbations 12
§4. Basic concepts. Terminology 14
1. 0-symbols (14).
2. Asymptotic series (14).
§5. The Schlesinger-Birkhoff theorem 17
§6. The Schlesinger-Birkhoff theorem and asymptotic integration 19
§7. Further development of the theory of singular perturbations 20
§8. Comparison of two types of asymptotic expansions 22
§9. Some notation and auxiliary concepts 23
Part I. Asymptotic Integration of Various Problems for Ordinary
Differential Equations
CHAPTER 2. The Method of Regularization of Singular Perturbations 27
§ 1. The formalism of the regularization method 27
1. Formulation of the problem (27).
2. Regularization of singularities (28).
3. Formal construction of a series for the solution (30).
§2. The space of resonance-free solutions 30
1. The structure of the space (30).
2. Properties of the basic operator in the space of resonance-free
solutions (32).
§3. The theory of resonance-free solutions 32
1. The adjoint operator (32).
2. Normal solvability of the basic operator (33).
3. Uniqueness of the solution (34).
V
vi CONTENTS
§4. Formal regularized series 36
1. Determination of the coefficients of the series of perturbation
theory (36).
2. Uniqueness and other properties of the regularized series (38).
§ 5. Estimation of the remainder term of the asymptotic series for
the fundamental matrix 40
1. Formal construction of the fundamental matrix (40).
2. The asymptotic character of the series (42).
§6. Estimation of the remainder term of the asymptotic series for
the solution of the Cauchy problem 8
1. Auxiliary notation and a lemma (48).
2. Estimation of the remainder term (50).
§7. Convergence of regularized series in the usual sense 58
1. Systems with a diagonal matrix of coefficients (58).
2. Examples (61).
3. Ordinary convergence of the asymptotic series (62).
4. Convergence in a finite-dimensional Hilbert space (62).
5. An example (6 5).
§8. The method of regularization in the case of null points of the
spectrum 6
1. Formulation of the problem (66).
2. The formalism of the regularization method (67).
3. Construction of the adjoint operator in the space of resonance-
free solutions (67).
4. Questions of solvability in the space of resonance-free
solutions (68).
5. A limit theorem (72).
CHAPTER 3. Asymptotic Integration of a Boundary Value Problem 75
§ 1. Special features of boundary value problems 75
1. Characteristic features of boundary value problems (75).
2. Formulation of the problem (78).
3. Stability of the boundary value problem (79).
§2. Construction of an algorithm for asymptotic integration of a
boundary value problem for general systems 82
1. The formalism of the method of regularization for a boundary
value problem (82).
2. Solvability theorems in the space of resonance-free solutions (84).
3. Solvability of the iteration problems (89).
4. Formal asymptotic solution of the original problem (92).
§3. Construction of the Green function 93
1. Reduction of the system to quasi-diagonal form (93).
2. Construction of two fundamental matrices of special form (95).
3. Construction of a fundamental matrix of a singularly perturbed
system with special boundary conditions (97).
4. Construction of the matrix [Pti(0, e) + Qti(1 5 e)]-1 (101).
CONTENTS vii
5. Construction of the matrix Green function (103).
6. A remark on the construction of the Green function for
a more general system (105).
§4. Estimation of the remainder term 107
1. The problem for the remainder term (107).
2. An estimate theorem (107).
CHAPTER 4. Asymptotic Integration of Linear Integro-Differential
Equations 109
§1. Special features of the regularization of singularities in the
presence of integrals of the desired solutions in the oscillatory
case 109
1. Formulation of the problem in the simplest case (109).
2. Partial regularization of the problems (110).
§2. Complete regularization and asymptotic integration 111
1. Regularization and the formalism of the method (111).
2. Solvability of iteration problems (115).
3. Estimation of the remainder term (116).
§3. The Cauchy problem for integro-differential systems 119
1. Formulation of the problem and regularization of
singularities (119).
2. Determination of the coefficients of the formal asymptotic
series (122).
3. Estimation of the remainder term (125).
4. An example (126).
§4. Integro-differential systems of Fredholm type 128
1. Auxiliary propositions (128).
2. Formulation of the problem and regularization of the
operation of differentiation (130).
3. Regularization of the integral term and of the problem for
determining the elements of the asymptotic solution (131).
4. Solvability of the iteration problem (137).
5. Estimation of the remainder term (141).
CHAPTER 5. Some Problems with Rapidly Oscillating Coefficients 143
§1. Construction of the asymptotic series and conditions for the
solvability of the iteration problems 144
1. Formalism of the method (144).
2. The space of solutions (146).
3. The adjoint operator (146).
4. Construction of new recurrent problems (148).
5. Solvability theorems (152).
§2. Justification of asymptotic convergence 155
1. Estimation of the remainder term (15 5).
2. Remark (157).
viii CONTENTS
§3. Solution of the problem of parametric amplification 157
1. An example (157).
2. Solution of the auxiliary system (164).
CHAPTER 6. Problems with an Unstable Spectrum 167
§ 1. The only point of the spectrum has a zero of arbitrary order 167
1. On the problem in the simplest formulation (167).
2. Regularization of the problem (168).
3. Asymptotic integration (168).
4. Passage to the limit (170).
§2. One of the two points of the spectrum has a zero of first order 70
1. Special features of the problem (170).
2. Choice of regularizing functions and regularization (171).
3. Special features of solving the iteration problems (173).
4. The main theorem (176).
§3. The inhomogeneous problem with a turning point 76
1. Preliminary facts regarding the problem (176).
2. Formulation of the problem (178).
3. Regularization of the problem (180).
4. Special features of the asymptotic integration of problems
with turning points (182).
5. Solvability of the iteration problems (186).
6. Estimation of the remainder term (189).
7. Proof of Lemma 18 (191) .
§4. The structure of the fundamental matrix of solutions of
singularly perturbed equations with a regular singular point 194
1. The fundamental system of solutions (196).
2. Obtaining formal solutions (196).
3. Asymptotic convergence of the series (200).
4. The fundamental system in the case of two algebraic
singularities (202).
CHAPTER 7. Singularly Perturbed Problems for Nonlinear Equations 209
§ 1. Weakly nonlinear singularly perturbed problems in the
resonance case 211
1. Formal solutions of weakly nonlinear problems (211).
2. Questions of solvability in the space of resonance-free
solutions (216).
3. The asymptotic character of solutions (223).
4. Examples (225).
§2. Regularized asymptotic solutions of strongly nonlinear
singularly perturbed problems 228
1. Regularization of strongly nonlinear problems (229).
2. Some function classes and their properties (231).
3. Theorems on the solvability of the iteration problems (235).
4. The asymptotic character of formal solutions (249).
5. An example (254).
CONTENTS ix
§3. Connection of the regularization method with the averaging
method 256
1. Regularized asymptotic solutions (257).
2. Asymptotic solutions obtained by the averaging method (259).
3. Global solvability of the truncated equations (264).
Part II. Singularly Perturbed Partial Differential Equations
CHAPTER 8. Asymptotic Integration of Linear Parabolic Equations 271
§ 1. A parabolic singularly perturbed problem with one viscous
boundary 274
1. Few words about the Fourier method (274).
2. Formulation of the problem and basic assumptions (275).
§2. The scheme of the regularization method in the selfadjoint case 276
1. Regularization and the iteration problems (276).
2. The space of resonance-free solutions (278).
3. Solvability of the iteration problems (280).
4. Asymptotic convergence of the series (284).
§3. Connection with the Fourier method and boundary layer theory 285
1. Remarks (285).
2. Example (287).
3. Remarks on the adiabatic approximation in quantum
mechanics (289).
§4. Asymptotic integration of a parabolic equation with two
viscous boundaries 90
1. Formulation of the problem for the linearized one-dimensional
Navier-Stokes equation (290).
2. Regularization of singularities by "viscosity" (292).
3. The iteration problems. The space of resonance-free solutions (294).
4. Theorems on normal and unique solvability (297).
5. Construction of the series of perturbation theory (299).
6. Estimation of the remainder term (304).
§5. Unsolved problems 6
1. Problems without spectrum (306).
2. Problems with two intersecting viscous boundaries (306).
3. Multidimensional problems (307).
CHAPTER 9. Application of the Regularization Method to Some
Elliptic Problems in a Cylindrical Domain 09
§ 1. Formalism of the method for an elliptic problem 309
1. Formulation of the problem (309).
2. Regularization and obtaining iteration problems (310).
§2. Asymptotic well-posedness and convergence of the method 312
1. Unique solvability of the iteration problems (312).
2. A theorem on asymptotic convergence of the series (314).
3. The leading term of the asymptotics (315).