Matching of Asymptotic Expansions of Solutions of Boundary Value Problems

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Matching of Asymptotic Expansions of Solutions of Boundary Value Problems

Copyright

     ©1992 by the American Mathematical Society

     ISBN 0-8218-4561-6

     QA379.I4 1992 515'.35-dc20

     LCCN 92-12324

Contents

Preface

Interdependence of Chapters

Introduction

CHAPTER I  Boundary Layer Functions of Exponential Type

     §1. Boundary value problems for ordinary differential equations

     §2. Partial differential equations

CHAPTER II  Ordinary Differential Equations

     §1. A simple bisingular problem

     §2. Matching procedure for asymptotic expansions

     §3. Nonlinear equation. Intermediate boundary layer

CHAPTER III  Singular Perturbations of the Domain Boundary in Elliptic Boundary Value Problems

     §1. Three-dimensional problem in a domain with a small cavity

          1. The Laplace equation

          2. An elliptic equation with variable coefficients

     §2. Flow past a thin body

     § 3. Two-dimensional boundary value problem in a domain with a small hole

     §4. Analysis of the asymptotics in the case where the limit problem has no solution

     §5. Example of solving a boundary value problem with a complex asymptotics

CHAPTER IV  Elliptic Equation with Small Parameter at Higher Derivatives

     §1. The case where a characteristic of the limit equation coincides with a part of the boundary

     §2. Asymptotics of the solution in a domain with nonsmooth boundary

     §3. The case of a singular characteristic tangent to the boundary of the domain from the outside

     §4. The case of a characteristic tangent to the boundary of the domain from the inside

     § 5. Remarks

CHAPTER V  Singular Perturbation of a Hyperbolic System of Equations

     § 1. Construction of the inner expansion

     §2. Construction of an f.a.s. in the outer domain (under discontinuity lines)

     §3. Construction of f.a.s. in the vicinity of singular characteristics

     §4. Construction of an f.a.s. in the outer domain (above discontinuity curves)

     §5. Justification of the asymptotic expansion

CHAPTER VI  Cauchy Problem for Quasilinear Parabolic Equation with a Small Parameter

     §1. Outer expansion. Asymptotics of the solution near the discontinuity curve

     §2. Shock wave caused by discontinuity of the initial function

     §3. Breaking of waves. Smoothness of the discontinuity curve. Asymptotics of the outer expansion coefficients

     §4. Asymptotics of solutions near the origin

     §5. Construction of asymptotics in the vicinity of the discontinuity curve

     §6. Construction of the uniform asymptotic expansion

     §7. Asymptotics of the flame wave

Notes and Comments on Bibliography

     Chapter I

     Chapter II

     Chapter III

     Chapter IV

     Chapter V

     Chapter VI

Bibliography

Subject Index

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