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

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 O(n-r2)in weighted L2-norm. Also the approximate solution in iterated-Galerkin method converges with order O(n-r)\in both infinity and weighted L2-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 O(n-3r2norm. Numerical results are presented to confirm the theoretical results.