Convergence analysis for the system of linear integral equations on the half-line

This paper develops projection and multi-projection methods, for the numerical approximation of systems of linear Fredholm integral equations of the second kind with smooth kernels, where each equation is defined on the half-line. Specifically, we analyze the Galerkin, iterated Galerkin, multi-Galerkin, and iterated multi-Galerkin methods employing piecewise polynomial basis functions. A rigorous theoretical framework is established for these methods, and we derive superconvergence results analogous to those for a single linear Fredholm integral equation of the second kind. It is shown that the iterated Galerkin method achieves higher convergence rates than the standard Galerkin approach. The analysis is further extended to the multi-Galerkin and iterated multi-Galerkin formulations, where the iterated multi-Galerkin method improves the superconvergence results of the iterated Galerkin and standard multi-Galerkin methods. The theoretical findings are validated through numerical results.