Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

TLDR: It is proved that the exact solution of Volterra-Urysohn integral equations of second kind with a smooth kernel is approximated with the order of convergence \(\mathcal {O}(h^{r})\) in uniform norm for iterated multi-Galerkin method.

Abstract: In this paper, we consider the Galerkin method to approximate the solution of Volterra-Urysohn integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We show that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{r})\) for the Galerkin method, whereas the iterated Galerkin solutions converge with the order \(\mathcal {O}(h^{2r})\) in uniform norm, where h is the norm of the partition and r is the smoothness of the kernel. For improving the accuracy of the approximate solution of the integral equation, the multi-Galerkin method is also discussed here and we prove that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{3r})\) in uniform norm for iterated multi-Galerkin method. Numerical examples are given to illustrate the theoretical results.