Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson’s Rule - Razali, Mohamad Rashidi; S. Nasser, Mohamed M. (MATEMATIKA, 2002)
This paper discusses the use of Product Simpson's rule to solve the integral equation eigenvalue problem λ f(x) = _-1^1k(|x - y|)f(y)dy where k(t) = |t| or k(t) = t^-1, 0 < < 1,λ, 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) = _i=0^n f(x_i)φ_i(y), where φ(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 _-1^1 k(y)f(y)dy≈ _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) = |x-y| and K(x, y) = |x- y|^-n, 0 < α < 1 are given. Keywords: igenvalue; product integration; singular kernel; integral eequation.