# How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

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.

• What regularity have $L$ resp. $K$? Are they $L^2$, continuous? – Daniel Fischer Nov 4 '15 at 20:11
• @DanielFischer Hi, thanks for the comment! $K$ is very well-behaved. It's a sinc function of an algebraic function of $p$ and $q$ times a 2-D gaussian centered at the origin. As for $L$, I'm not sure how to answer, can we say anything about it at all since it's an unknown function? – quantum_loser Nov 6 '15 at 9:43
• On what space does the operator live? Does it map $L^2([a,b])\to L^2([a,b])$, $C([a,b]) \to C([a,b])$? – Daniel Fischer Nov 6 '15 at 9:46
• @DanielFischer It's $L^2 \rightarrow L^2$, I'm fairly certain. Edit: And everything should stay real-valued. – quantum_loser Nov 6 '15 at 10:52
• Then you might have some success with a Fourier expansion. Take an ONB $\{ \varphi_n : n \in \mathbb{N}\}$ of $L^2([a,b])$. Then $\psi_{mn}(x,y) = \varphi_m(x)\cdot \varphi_n(y)$ is an ONB of $L^2([a,b]\times[a,b])$. Expanding $K$ in terms of $\psi_{mn}$ and an ansatz $L = \sum c_n\varphi_n$ might be feasible. – Daniel Fischer Nov 6 '15 at 11:01