Product Integration for Singular Integrals and Singular Integral Equations

Abstract

Integral equations with weakly singular kernels often have solutions which have derivative singularities at the end points of the range of integration. The error analysis of a product integration method for such integral equations depends on the error analysis of the product integration method applied to integrals of the form $\int\limits_0^1 {g\left( t \right)f\left( t \right)dt}$ where g(t) is “weakly singular” in [0,1] and f(t) is continuous but has derivative singularities in [0,1]. The error analysis of de Hoog & Weiss [3] does not hold for such f(t). In this paper the analysis of de Hoog & Weiss is modified to deal with such integrals and the results applied to a product integration method for integral equations. Numerical examples are given.