CHAPTER 10 - APPROXIMATE METHODS OF SOLVING PARTIAL DIFFERENTIAL AND INTEGRAL EQUATIONS

Abstract

This chapter discusses the approximate methods of solving partial differential and integral equations. Partial differential and integral equations are encountered in many fields of science. Only in the simplest cases, a solution can be found in implicit form or in the form of a finite formula. Approximate methods are, therefore, of particular importance for the solution of partial differential equations, sets of partial differential equations, and integral equations, for the problems of mathematical physics. The first category consists mainly of Fourier's method of solving boundary value problems in partial differential equations where the exact solution is in the form of a certain series and the approximate solution is the sum of the first few terms. The method of characteristics for solving equations and sets of equations of the hyperbolic type is essentially a finite difference method. Moreover, only in this method the partial differential equation or set of such equations is first reduced to an equivalent set of ordinary differential equations, which is then solved by the difference method.