Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm–Hammerstein integral equations

Abstract

In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O ( n − r ) , whereas the iterated Legendre Galerkin method converges with the order O ( n − r − α + 1 2 ) for the algebraic kernel, and order O ( log n n − r − 1 2 ) for logarithmic kernel in both L 2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O ( ( 1 + c l o g n ) n − r − 2 α + 1 2 ) for the algebraic kernel, and order of convergence O ( ( log n ) 2 ( 1 + c l o g n ) n − r − 3 2 ) for logarithmic kernel in both L 2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results.