دانلود کتاب Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Encyclopaedia of Mathematical Sciences, 34)

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Cover
Copyright Page
List of Editors, Authors and Translators
Contents
I. Equations with Rapidly Oscillating Solutions
Foreword
§1. Local Asymptotics
§2. Lagrangian Manifolds
§3. Passing to the p-Representation
§4. The Maslov Canonical Operator
§5. Some Applications of the Canonical Operator
§6. The WKB Method for Nonlinear Equations
References
II. Asymptotic expansion as t ->+8 of the solutions of exterior boundary value problems for hyperbolic equations
Part I. The asymptotic expansion of solutions to exterior mixed boundary value problem
§1. Analytic Continuation of the Resolvent for Exterior Elliptic Problems and the Short Wave Approximation
§2. The long-wave approximation and the asymptotic expansion as t->+8 of solutions to mixed boundary value problems
Part II. The Scattering Problem
§1. Quasidassical Asymptotic Expansion of the Solution to the Scattering Problem
§2. Asymptotics of the Scattering Amplitude
Part III. The Parametrix and the Full Asymptotic Expansion of the Spectral Function of Differential Operators in Rn
§1. The Parametrix for Hyperbolic Equations and Systems
§2. Asymptotics of the Spectral Function
References
III. The Higher-Dimensional WKB Method or Ray Method. Its Analogues and Generalizations
Introduction
Chapter 1. Fundamental Non-Local Short-Wave Expansions
§1. The Classical Ray Method
§2. Point Source of Vibrations in an Inhomogeneous Medium
§3. Short-Wave Expansion in a Neighborhood of a Nonsingular Piece of a Caustic
Chapter 2. Some Modifications of the Ray and Caustic Expansions
§1. Asymptotics of Vibrations of Whispering Gallery Type
§2. Surface Wave Propagated Along an Impedance Surface
Chapter 3. Gaussian Beams and Their Applications
§1. Solutions Concentrated in a Neighborhood of a Fixed Ray
§2. The Case of a Closed Ray
§3. Summation of Gaussian Beams
Chapter 4. On Other Short-Wave Diffraction Problems
§1. The Case of Smooth Reflecting Boundaries
§2. Various Problems
References
IV. Semiclassical Asymptotics of Eigenfunctions
§1. Introduction
§2. Quasimodes and Spectrum
§3. A Classical Dynamical System and Principles for Constructing Quasimodes
§4. Quasimodes Corresponding to Kolmogorov Tori
§5. History of the Problem and Some of the Problems That Have Been Studied
References
V. The Boundary Layer
§l. The Exponential Boundary Layer
§2. The Method of Matching Asymptotic Expansions
§3. An Elliptic Equation with a Small Parameter in the Highest Derivatives
§4. Singular Perturbations of the Domain Boundary
§5. A Quasilinear Parabolic Equation
Comments on the Literature
References
VI. The Averaging Method for Partial Differential Equations (Homogenization) and Its Applications
Foreword
Chapter 1. Problems of the Mechanics of Nonhomogeneous Structures Described by Partial Differential Equations with Rapidly Oscillating Coefficients
§1. Media with Periodically Arranged Inhomogeneities
§2. Strongly Nonhomogeneous Media
Chapter 2. Asymptotic and Numerically Asymptotic Methods of Solving Problems of the Mechanics of Nonhomogeneous Structures
§1. Separation of the Fast and Slow Variables
§2. Averaged Equation of Infinite Order
§3. Expansion with Respect to Two Parameters
§4. The Boundary Layer Method in Averaging Problems
§5. Description of Processes in Periodic Media by Means of Functions Depending on the Fast Variables
Chapter 3. Numerically Asymptotic Methods for Weakly Nonlinear Problems
References
Author Index
Subject Index