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
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. $\endgroup$ – Leonid Shifrin May 14 '13 at 21:25k
is so large, I would still start with a stationary phase, and then perhaps use some iterative scheme. The scheme I linked to will be very hard to apply directly, given the highly oscillatory kernel of your equation - one would need to discretize on a very fine grid, and likely also use extended precision arithmetic - and even then I don't think this will be robust. To my mind, you first have to factor out the highly oscillatory part, which stationary phase should do for you - and then you could perhaps apply some finite difference scheme to the corrections. $\endgroup$ – Leonid Shifrin May 15 '13 at 19:58