Integral Equations and Their Applications

The book deals with linear integral equations, that is, equations involving an unknown function which appears under the integral sign. Such equations occur widely in diverse areas of applied mathematics, engineering sciences and physics. They offer a powerful technique for solving a variety of practical problems. One obvious reason for using the integral equations rather than differential equations is that all of the conditions specifying the initial value problems or boundary value problems for a differential equation can often be condensed into a single integral equation. In the case of partial differential equations, the dimension of the problem is reduced in this process so that, for example, a boundary value problem for a partial differential equation in two independent variables transform into an integral equation involving an unknown function of one independent variable. This reduction of what may represent a complicated mathematical model of a physical situation into a single equation is itself a significant step, but there are other advantages to be gained by replacing differentiation with integration. Some of these advantages arise because integration is a smooth process, a feature which has significant implications when approximate solutions are sought.This book contains topics such as Abel's integral equation, Volterra integral equations, Fredholm integral integral equations, singular and nonlinear integral equations, orthogonal systems of functions, Green's function as a symmetric kernel of the integral equations.