Abstract

This article discusses Galerkin, multi-Galerkin methods, and their iterated versions to approximate the solution of Urysohn-type integral equations on the half-line using Laguerre polynomials as basis functions. Here, we consider that the kernel function is sufficiently smooth. We have shown that the approximate solution in Galerkin and iterated Galerkin methods converges to the exact solution with the order 𝒪(𝑛−𝑟2) and order 𝒪(𝑛−𝑟), respectively. Here, r represents the smoothness of the solution, and n denotes the degree of the polynomials. We improve the result using the multi-Galerkin method and its iterative method. In fact, we prove that multi-Galerkin and iterated multi-Galerkin methods converge to the exact solution with the order 𝒪(𝑛−3𝑟2) and 𝒪(𝑛−2𝑟), respectively. Numerical results are presented to support theoretical results.

Keywords

Multi-Galerkin method, Superconvergence results, Laguerre polynomials, Urysohn integral equation.

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