Introductory Concepts of Integral Equations

As stated in the previous chapter, an integral equation is the equation in which the unknown function u(x) appears inside an integral sign [1–5]. The most standard type of integral equation in u(x) is of the form 2.1 $$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a known function, of two variables x and t, called the kernel or the nucleus of the integral equation. The unknown function u(x) that will be determined appears inside the integral sign. In many other cases, the unknown function u(x) appears inside and outside the integral sign. 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.