# Eigenvalue problem for Fredholm integral

I have trouble solving this task to find eigenvalues and eigenvectors for the Fredholm integral. I could not find an analytic solution to the equation. That is why I am trying to solve it numerically in Wolfram Mathematica.

b and c are constants = 0.1 I saw a similar task for another kernel function (Fredholm Integral Equation of the 2nd Kind with a Singular Difference Kernel), but the Gauss-Legendre quadrature formula code doesn't allow solving this task because of singularity (division by zero are constantly happening).

Code to find 2 eigenvalues and eigenfunctions:

points = 100;
integrand[x_] =(exp[-0.1*Abs[x-y]]*cos[0.1*Abs[x - y]]) f[x];
domain = {0,1};
{nodes, weights} = Most[NIntegrateGaussRuleData[points, MachinePrecision]];
midgrid = Rescale[nodes, {0, 1}, domain];
grid = Flatten[{domain[[1]], midgrid, domain[[-1]]}];
int = -Subtract @@ domain weights.Map[integrand, midgrid];
{b, m} = CoefficientArrays[int, f /@ grid];
mat = Table[m, {y, grid}];
{val, vec} = Eigensystem[mat, 2];
ListLinePlot[vec[[;; 2]], PlotRange -> All]

• Welcome to MSE. Please edit your question and share the code you have tried. Nov 4, 2021 at 16:40
• Thanks. Yes, please. Nov 5, 2021 at 13:12
• It is not clear what is your problem. Do you try to solve Fredholm integral equation with method proposed in the post linked or do you try to get eigensystem itself? Nov 6, 2021 at 5:14

There are several typos in the code, and after small correction we have

points = 100;
integrand[x_,y_] := (Exp[-0.1*Sqrt[(x - y)^2]]*Cos[0.1*Sqrt[(x - y)^2]]);
domain = {0, 1};
{nodes, weights} =
Most[NIntegrateGaussRuleData[points, MachinePrecision]];
midgrid = Rescale[nodes, {0, 1}, domain];

mm = Table[
integrand[midgrid[[i]], midgrid[[j]]] weights[[i]], {i,
Length[midgrid]}, {j, Length[midgrid]}];

{val, vec} = Eigensystem[mm, 2];
ListLinePlot[vec[[;; 2]], PlotRange -> All]


• Thanks for this. Nov 6, 2021 at 22:05