Numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points

Abstract

An integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points (ACP). As a first step a Gaussian quadrature of order n (hence of polynomial accuracy 2n− 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x̄k, determined as roots of appropriate transcedental functions, in order to retain the 2n − 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that xk may be arbitrary and yet the high (2n − 1) accuracy of the Gaussian quadrature is retained.