Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type

Abstract

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equations of mixed type. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, $\mathcal {O}(n^{-r})$ in $L^{2}$-norm and $\mathcal {O}(n^{\frac{1}{2}-r})$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $\mathcal {O}(n^{-2r})$ in both $L^{2}$-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $\mathcal {O}(n^{-r})$ in both $L^{2}$-norm and infinity norm, n being the highest degree of Legendre polynomial employed in the approximation and r being the smoothness of the kernels.