Fredholm integral equation

Summary

A Fredholm integral equation is a linear integral equation whose integral runs over a fixed set, such as $[a, b]$. The upper limit is not the variable $x$. It comes in a first kind, $\int_a^b K(x, s) \varphi(s) \, ds = f(x)$, and a second kind, $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$. For the second kind with a continuous or square-integrable kernel, Fredholm theory gives a complete picture of solvability. This picture is much as for square systems of linear equations. Boundary value problems for differential equations lead to Fredholm equations through Green's functions.