Discrete product integration methods for eigen-problems of a class of non-compact integral operators

In this paper, we develop discrete product integration methods for the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a weakly singular integral operator. Our discrete product integration scheme is a product integration scheme in which the weakly singular integrals appeared in the weight functions are computed numerically by Gauss–Legendre-type quadrature formulas. We show that our methods are convergent and their convergence rates are max{hr+1,m-2d}max{hr+1,m-2d}, which is dependent on the accuracy of both the numerical quadrature and approximate solution of the product integration scheme. Our numerical results confirm that discrete product integration methods are particularly efficient for solving this kind eigenvalue problem.