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 \mathcal O(h^{r}) and \mathcal O(h^{۲r}), 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 \mathcal O(h^{۳r}) in the uniform norm. Numerical results are provided to demonstrate the theoretical results.

Keywords:

Volterra-Hammerstein integral equations , Galerkin method , Multi-Galerkin method , Piecewise polynomials , Superconvergence rates