Preliminaries

Abstract

An integral equation is an equation in which the unknown function u(x) appears under an integral sign [1&#sx20s13;7]. A standard integral equation in u(x) is of the form:

$$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(1.1)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a function of two variables x and t called the kernel or the nucleus of the integral equation. The function u(x) that will be determined appears under the integral sign, and it appears inside the integral sign and outside the integral sign as well. The functions f(x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed.