Integral Equations

Abstract

Integral equations in which the unknown function appears in a convolution occur in some important situations. The equation

$$g(x) = f(x) + \lambda \int_a^b {k(x - y)\;g(y)\;dy} $$

(1)

, where f(x) and k(x) are given functions and λ a given constant, is an example of a Fredholm integral equation of the second kind. (An equation of the first kind is one in which the unknown function g does not appear outside the integral.) If the upper limit of integration b is replaced by the variable x, then (1) is said to be of Volterra, rather than Fredholm, type. By the change of variables x′ = x − a, y′ = y − a, (1) may then he written

$$G(p) = F(p) + \lambda \;K(p)\;G(p)$$

(2)

.