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

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 2 n − 1) is employed for the numerical approximation of the integral. Until now the collocation, which follows, was performed on special points x̄ k , determined as roots of appropriate transcedental functions, in order to retain the 2 n − 1 degree of polynomial accuracy of the Gaussian quadrature. In this paper an appropriate interpolatory technique is proposed, so that x k may be arbitrary and yet the high (2 n − 1) accuracy of the Gaussian quadrature is retained.