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

Document Type : Research Paper


1 Assistant Professor, Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan-342037, India.

2 Department of Mathematics Indian Institute of Technology Kharagpur, Kharagpur - 721 302, India


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 $\mathcal O(h^{2r}),$ 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 $\mathcal O(h^{3r})$ in the uniform norm. Numerical results are provided to demonstrate the theoretical results.