# Solving homogeneous Fredholm Equation of the second kind

I am trying to solve a homogeneous Fredholm integral equation of the second kind, i.e. $\lambda y(x) = \int\limits_a^b e^{i[\phi(t)+k(t-x/M)^2]} y(t)\,dt$

where $\lambda$ is the eigenvalue (to be determined), and $\phi(t)$ is a polynomial in $t$. Has anyone come across a Mathematica routine to do this? Thanks! Mark

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If k is large, one obvious thing that comes to mind is to use the saddle point approximation. The general equation does not look like the one you can solve exactly. Numerically, you can try some finite difference scheme, or may be something similar to what I described here. –  Leonid Shifrin May 14 '13 at 21:25
Could you give a particular $a,b,k,M$ and $\phi(t)$ you're studying? –  Ｊ. Ｍ. May 15 '13 at 8:44
Leonid: Thanks for the comments. The problem with stationary-phase is that past the first term, succeeding terms depend on the value at the end points, which doesn't really help. I also looked at your link - it's not clear to me how one gets the eigenvalue using that method. –  Mark May 15 '13 at 17:45
@J.M.: let a=-1, b=1, k~500, M=3, phi = c2 t^2 - c4 t^4 with c2 and c4 constants of order 100 and 10 respectively –  Mark May 15 '13 at 17:59
Hmm, that's big. Your kernel's quite oscillatory too; There is a modification of Clenshaw-Curtis by Piessens and Branders for finite oscillatory integrals, which can be useful if you're taking the discretization route. You might want to look into that. –  Ｊ. Ｍ. May 15 '13 at 18:01