The Eigenvalue Problem for a Pair of Coupled Integral Equations Arising in the Numerical Solution of Laplace’s Equation

Abstract

Using Green’s third formula one can reduce the mixed boundary value problem for Laplace’s equation in two dimensions to a pair of coupled integral equations. This pair of integral equations can be solved numerically, and in many cases of interest this is an efficient way to solve the boundary value problem. Here we investigate the spectrum for the associated eigenvalue problem. It follows from our results that it is always feasible to solve numerically the boundary value problem by the above method.