Synopses & Reviews
This volume is devoted to the further development of the asymptotic theory for analysing solutions of a wide range of nonlinear periodic boundary value problems. It suggests a systematic approach to constructing asymptotic methods for solving wave equations, a particular ordinary differential equation of the second order, hyperbolic differential equations and partial differential equations with small parameters. Audience: This book will be of interest to researchers and postgraduate students whose work involves partial differential equations, mathematical physics, or approximations and expansions.
Synopsis
The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and that for general two- dimensional equations were investigated by Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last decade the theory of solvability on the whole of boundary value problems for nonlinear differential equations has received intensive development. Significant results for nonlinear elliptic and parabolic equations of second order were obtained in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in general of nonlinear hyperbolic equations, which are connected to the theory of local and nonlocal boundary value problems for hyperbolic equations, there are only partial results obtained by Bronshtein, Pokhozhev, Nakhushev.
Table of Contents
Preface.
1. Existence Theorems for Hyperbolic Equations.
2. Periodic Solutions of the Wave Ordinary Differential Equations of Second Order.
3. Periodic Solutions of the First Class Systems.
4. Periodic Solutions of the Second Class Systems.
5. Periodic Solutions of the Second Order Integro-Differential Equations of Hyperbolic Type.
6. Hyperbolic Systems with Fast and Slow Variables and Asymptotic Methods for Solving Them.
7. Asymptotic Methods for the Second Order Partial Differential Equations of Hyperbolic Type. Bibliography. Index.