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 f(x) = R 1−1 k(|x − y|)f(y)dy where k(t) = ln|t| or k(t) = t−, 0 < < 1, , f and are unknowns which we wish to obtain. The function f(y) in the integral above is replaced by an interpolating function Lfn(y) = Pni=0 f(xi)i(y), where i(y) are Simpson interpolating elements and x0, x1, . . . ,xn are the interpolating points and they are chosen to be the appropriate non-uniform mesh points in [−1, 1]. The product integration formula R 1−1 k(y)f(y)dy Pni=0 wif(xi) is used, where the weights wi are chosen such that the formula is exact when f(y) is replaced by Lf 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|−, 0 < < 1 are given. Keywords eigenvalue, product integration, singular kernel, integral equation.