Partial Differential Equations

Description

This book has developed from courses of lectures given by the author over a period of years to the students of the Moscow PhysicoTechnical Institute. It is intended for the students having basic knowledge of mathematical analysis, algebra and the theory of ordinary differential equations to the extent of a university course. Except Chapter I, where some general questions regarding partial differential equations have been examined, the material has been arranged so as to correspond to the basic types of equations. The central role in the book is played by Chapter IV, the largest of all, which discusses elliptic equations. Chapters V and VI are devoted to the hyperbolic and parabolic equations. The method used in this book for investigating the boundary value problems and, partly, the Cauchy problem is based on the notion of generalized solution which enables us to examine equations with variable coefficients with the same ease as the simplest equations: Poisson's equation, wave equation and heat equation. Apart from discussing the questions of existence and uniqueness of solutions of the basic boundary value problems, considerable space has been devoted to the approximate methods of solving these equations: Ritz's method in the case of elliptic equations and Galerkin's method for hyperbolic and parabolic equations.