I have the equation
\begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray}
Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous Fredholm integral equation of the second kind, apparently. $K$ is symmetric, that is $K(p,q)=K(q,p)$, if that helps anything. It is also not separable, not even approximately.
Everything I've found online just handles the separable case. One reference even said solving such non-separable equations was "usually impossible". Another said that "Once the eigenfunctions are known the equation can be solved." But this does not help since the eigenfunctions of this integral operator are exactly what I'm looking for.
Any ideas? I'd be happy (albeit less) with a good way to find good approximations.