Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces

Abstract

For the numerical solution of Fredholm integral equations on [−1,1] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods.