Hybrid collocation methods for Fredholm integral equations with weakly singular kernels

A hybrid collocation method for Volterra integral equations with weakly singular kernels was developed in [Y. Cao, T. Herdman, Y. Xu, A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal. 41 (2003) 364–381. [5]]. The main purpose of this paper is to study the numerical solution of Fredholm integral equations of the second kind with weakly singular kernels by using the hybrid collocation method. The optimal order of convergence for this method for Fredholm integral equations is proved. We also apply the hybrid collocation method to Fredholm integro-differential equations. The superconvergence property of the iteration of the hybrid collocation solution is established. Numerical examples are presented to confirm the theoretical results.