Spectral approximated superconvergent methods for system of nonlinear Volterra Hammerstein integral equations

In this article, we develop the Jacobi spectral multi-Galerkin method alongside the Kumar-Sloan technique to approximate systems of non-linear Volterra Hammerstein integral equations. We conduct a comprehensive superconvergence analysis for both smooth and weakly singular kernels in both infinity and weighted-$L^2$ norms. Our findings include the derivation of superconvergence rates for the multi-Galerkin method without resorting to iterated versions. Notably, our conclusions highlight the enhanced performance of multi-Galerkin approximation compared to Jacobi spectral Galerkin methods, while maintaining the same system size for both Jacobi spectral multi-Galerkin and Galerkin methods. To validate the robustness and efficiency of our theoretical results, numerical examples are provided.