Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule

Abstract

This paper discusses the use of Product Simpson's rule to solve the integral equation eigenvalue problem $\lambda f(x) = \int_{-1}^1k(|x - y|)f(y)dy$ where $k(t) = \ln|t|$ or $k(t) = t^{-1}, 0 < < 1,\lambda, f$ and are unknowns which we wish to obtain. The function $f(y)$ in the integral above is replaced by an interpolating function $L^f_n(y) = \sum_{i=0}^n f(x_i)\phi_i(y),$ where $\phi(y)$ are Simpson interpolating elements and $x_0, x_1,...,x_n$ are the interpolating points and they are chosen to be the appropriate non-uniform mesh points in $[-1, 1].$ The product integration formula $\int_{-1}^1 k(y)f(y)dy\approx \sum_{i=0}^n w_if(x_i)$ is used, where the weights wi are chosen such that the formula is exact when $f(y)$ is replaced by $L^f_n(y)$ and $k(y)$ as given above. The five eigenvalues with largest moduli of the two kernels $K(x, y) = \ln|x-y|$ and $K(x, y) = |x- y|^{-n}, 0 < \alpha < 1$ are given. Keywords: igenvalue; product integration; singular kernel; integral eequation.