Convergence analysis of Galerkin and multi-Galerkin methods for linear integral equations on half-line using Laguerre polynomials

Abstract

This paper considers the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. Here we are able to prove that approximate solution in Galerkin method converges to the exact solution with order $\mathcal {O}(n^{-\frac{r}{2}})$ in weighted $L^{2}-$norm. Also the approximate solution in iterated-Galerkin method converges with order $\mathcal {O}(n^{-r})$ in both infinity and weighted $L^{2}-$norm, where r is the smoothness of the solution and n is the highest degree of the Laguerre polynomials employed in the approximation. We also show that multi-Galerkin and iterated multi-Galerkin methods gives superconvergence results using Laguerre polynomials. In fact, we are able to establish that the approximate solutions in multi-Galerkin and iterated multi-Galerkin methods converges to the exact solution with orders $\mathcal {O}(n^{-\frac{3r}{2}})$ and $\mathcal {O}(n^{-2r}),$ respectively, in weighted $L^{2}-$norm. Numerical results are presented to confirm the theoretical results.