Superconvergent scheme for a system of green Fredholm integral equations

In this study, a numerical scheme to a system of second-kind linear Fredholm integral equations featuring a Green's kernel function is proposed. This involves introducing Galerkin and iterated Galerkin (IG) methods based on piecewise polynomials to tackle the integral model. A thorough analysis of convergence and error for these proposed methods is carried out. Firstly, the existence and uniqueness of solutions for the Galerkin and iterated Galerkin methods are established. Later, the order of convergence is derived using tools from functional analysis and the boundedness property of Green's kernel. The Galerkin scheme has $O\left(h^{\alpha }\right)$ order of convergence. Next, the superconvergence of the iterated Galerkin (IG) method is established. The IG method exhibits $O\left(h^{\alpha +{\alpha }^{⁎}}\right)$ order of convergence. Theoretical findings are validated through extensive numerical experiments.