Equations of mathematical physics

The present book consists of an introduction and six chapters. The introduction discusses basic notions and definitions of the traditional course of mathematical physics and also mathematical models of some phenomena in physics and engineering. Chapters 1 and 2 are devoted to elliptic partial differential equations. Here much emphasis is placed on the Cauchy-Riemann system of partial differential equations, that is on fundamentals of the theory of analytic functions, which facilitates the understanding of the role played in mathematical physics by the theory of functions of a complex variable. In Chapters 3 and 4 the structural properties of the solutions of hyperbolic and parabolic partial differential equations are studied and much attention is paidj to basic problems of the theory of wave equation and heat conduction equation. In Chapter 5 some elements of the theory of linear integral equations are given. A separate section of this chapter is devoted to singular integral equations which are frequently used in applications. Chapter 6 is devoted to basic practical methods for the solution of partial differential equations. This chapter contains a number of typical examples demonstrating the essence of the Fourier method of separation of variables, the method of integral transformations, tho fi è He-difference method, the method of asymptotic expansions and also the variational methods. To study the book it is sufficient for the reader to be familiar with an ordinary classical course on mathematical analysis studied in colleges. Since such a course usually does not involve functional analysis, the embedding theorems for function spaces are not included in the present book. A.V. Bitsadze"