CAS Wavelet Method for the Numerical Solution of Boundary Integral Equations with Logarithmic Singular Kernels

In this paper, we present a computational method for solving boundary integral equations with logarithmic singular kernels which occur as reformulations of a boundary value problem for Laplace's equation. The method is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis. This approach utilizes the nonuniform Gauss-Legendre quadrature rule for approximating logarithm-like singular integrals and so reduces the solution of boundary integral equations to the solution of linear systems of algebraic equations. The properties of CAS wavelets are used to make the wavelet coefficient matrices sparse, which eventually leads to the sparsity of the coefficient matrix of the obtained system. Finally, the validity and efficiency of the new technique are demonstrated through a numerical example.