This book deals with the solution of singularly perturbed boundary value problems for differential equations. It presents, for the first time, a detailed and systematic treatment of the version of the matching method developed by the author and his colleagues. A broad class of problems is considered from a unified point of view, and the procedure for constructing asymptotic expansions is discussed in detail. The book covers formal constructions of asymptotic expansions and provides rigorous justifications of these asymptotics. One highlight is a complete asymptotic analysis of Burger's equation with small diffusion in the neighborhood of the gradient catastrophe point. The book is suitable as a text for graduate study in asymptotic methods in calculus and singularly perturbed equations.
Readership: Graduate students and researchers specializing in differential equations.
Author(s): A. M. Ilin
Series: Translations of Mathematical Monographs, Vol. 102
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+X+281+B
Cover
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
Back Cover
