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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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  • $\begingroup$ 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. $\endgroup$ – Leonid Shifrin May 14 '13 at 21:25
  • $\begingroup$ Could you give a particular $a,b,k,M$ and $\phi(t)$ you're studying? $\endgroup$ – J. M. will be back soon♦ May 15 '13 at 8:44
  • $\begingroup$ 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. $\endgroup$ – Mark May 15 '13 at 17:45
  • $\begingroup$ @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 $\endgroup$ – Mark May 15 '13 at 17:59
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    $\begingroup$ @Mark Since your value of k 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
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Rahbar and Hashemizadeh's approach

You could use the code provided by PlatoManiac in response to a similar question which implements the method from Rahbar and Hashemizadeh's paper A Computational Approach to the Fredholm Integral Equation of the Second Kind.

NISolve.nb

The Mathematica library includes code for Numerical Solution of One-Dimensional Linear Integral Equations of the Second Kind.

To get the code to work in recent versions of Mathematica you'll need to remove the line

<<Utilities`FilterOptions`;

and change the two occurrences of

fintopt=FilterOptions[FunctionInterpolation,opt];

to

fintopt=Sequence@@FilterRules[{opt},FunctionInterpolation];

(these instructions were detailed in this previous post for a different function).

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