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

This paper discusses the use of Product Simpson’s rule to solve the integral equation eigenvalue problem lf(x)=ò1 1 k(|x y|)f(y)dy where k(t) = ln|t| or k(t) = t-a, 0 < a < 1, l, 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) = ni=0 f(xi)fi(y), where fi(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 ò1 1 k(y)f(y)dy ni=0 wif(xi) is used, where the weights wi are chosen such that the formula is exact when f(y) is replaced by Lfn(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|-a, 0 < a < 1 are given.