On the Weighted Galerkin Method of Numerical Solution of Cauchy Type Singular Integral Equations

Abstract

The well-known weighted Galerkin method for the direct numerical solution of Cauchy type singular integral equations of the first and the second kind, based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials, is shown to be identical to the corresponding method based on the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind and the application to the latter of the weighted Galerkin method. This result permits the direct transfer of the whole set of convergence and rate-of-convergence results for the weighted Galerkin method of numerical solution of Fredholm integral equations with regular kernels to the case of Cauchy type singular integral equations. An application to the case where Chebyshev polynomials are used is also made, and the available convergence results for the direct weighted Galerkin method are rederived with essentially no analysis.