I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite useful for physicists, but all I have read is "these are useful in physics but we don't have time for that here . . . blah, blah, blah compact operators blah blah blah existence and uniqueness blah blah blah Banach spaces blah blah blah". (I love all these topics, but the point stands.)
So my question is: Is there a simple problem in physics, perhaps suitable for advanced undergraduates, that contorts itself into a Fredholm integral equation? I reject the Poisson equation as an example, because it is too simple and there is no reason to learn Fredholm theory to solve it.