Legendre spectral projection methods for Urysohn integral equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n12−r) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n−2r) in both L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n−r) in both L2-norm and infinity norm, n being the highest degree of the Legendre polynomial employed in the approximation and r being the smoothness of the kernel. We are able to obtain similar superconvergence rates for the iterated Galerkin solution for Urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions.