Numerical Analysis of an Integral Equation with a δ Function in the Kernel

Abstract

An integral equation which has a two-part kernel, where one part contains a δ function, is analyzed with respect to its analytic structure in the λ plane and with respect to numerical approximations to it. The analytic structure of the approximate solution is also investigated. It is found that there is no difficulty in approximating the Dirac δ function by a Kronecker δ; even though the approximate kernel does not approach the true kernel, the solution corresponding to the approximate kernel does approach the true solution. In keeping with the fact that the kernel does not meet the Fredholm conditions, we find that the solution has branch cuts in the λ plane. A form for the solution analogous to Fredholm's solution which emphasizes the analytic structure (i.e., a branch cut) is obtained.