Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method

In this paper, we present a Taylor-series expansion method for a class of Fredholm singular integro-differential equation with Cauchy kernel. This method uses the truncated Taylor-series polynomial of the unknown function and transforms the integro-differential equation into an nth order linear ordinary differential equation with variable coefficients. By Galerkin method we use the orthogonal Legendre polynomials as a basis for finding the approximate solution of nth order differential equation. By the property of odd or even function we reduce the singularity of the integrals to the one point. Some numerical examples are also given to illustrate the efficiency and accuracy of the method.