Projection and multi-projection methods for second kind Volterra-Hammerstein integral equation

In this article, we discuss the piecewise polynomial based Galerkin method to approximate the solutions of second kind Volterra-Hammerstein integral equations. We discuss the convergence of the approximate solutions to the exact solutions and obtain the orders of convergence O(hr) and O(h2r), respectively, for Galerkin and its iterated Galerkin methods in uniform norm, where h, r denotes the norm of the partition and smoothness of the kernel, respectively. We also obtain the superconvergence results for multi-Galerkin and iterated multi-Galerkin methods. We show that iterated multi-Galerkin method has the order of convergence O(h3r) in the uniform norm. Numerical results are provided to demonstrate the theoretical results.