Computing the singular behavior of solutions of Cauchy singular integral equations with variable coefficients

A numerical technique to determine the singular behavior of the solution of Cauchy singular integral equations (CSIE) with variable coefficients is proposed. The fundamental solution, which is a solution of the corresponding homogeneous equation, is constructed by a quadrature-collocation scheme. This leads to a system of nonlinear equations to approximate the exponents of singularity. Newton's method has been found to yield a useful approximation to these exponents. Once we have numerically obtained the weight function which determines the behavior of the solution of Cauchy singular integral equation, it can be used to solve a variety of nonhomogeneous CSIE with variable coefficients.