CHAPTER 1 - INTRODUCTION

Abstract

An integral equation is an equation in which an unknown function appears under one or more integral signs. Integral equations occur naturally in many fields of mechanics and mathematical physics. They also arise as representation formulas for the solutions of differential equations. Indeed, a differential equation can be replaced by an integral equation that incorporates its boundary conditions. As such, each solution of the integral equation automatically satisfies these boundary conditions. Integral equations also form one of the most useful tools in many branches of pure analysis, such as the theories of functional analysis and stochastic processes. An integral equation is called linear if only linear operations are performed in it upon the unknown function. Many interesting problems of mechanics and physics lead to an integral equation in which the kernel K(s, t) is a function of the difference (s—t) only: K(s,t) = k(s-t), where K is a certain function of one variable.