Numerical Solution of Integral Equations of the First Kind with Nonsmooth Kernels

Abstract

It is well known that first kind integral equations having nonsmooth kernels can sometimes be solved satisfactorily without regularization. Our objective here is to quantify the relationship between the smoothness of the kernel and the amenability of the problem to standard numerical methods. To this end, we analyze least squares and Galerkin’s method in a Sobolev space setting geared to integral equations of the first kind with nonsmooth kernels. Error estimates are obtained, as well as condition number bounds for the coefficient matrix of the discretized problem. The key parameter in these estimates is a smoothing index which gauges the number of derivatives the integral transformation adds to $L^2$ functions to which it is applied.