Solving Integral Equations with Logarithmic Kernel by Using Periodic Quasi-Wavelet

Abstract

In solving integral equations with logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional Helmholtz equation, we combine the Galerkin method with Beylkin's ([2]) approach, series of dense and nonsymmetric matrices may appear if we use traditional method. By appealing the so-called periodic quasi-wavelet (PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we can find an algorithm with only $O(K(m)^2)$ arithmetic operations where $m$ is the highest level. The Galerkin approximation has a polynomial rate of convergence.