Convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type

In this paper, we consider the discrete Legendre spectral Galerkin and discrete Legendre spectral collocation methods to approximate the solution of mixed type Hammerstein integral equation with smooth kernels. The convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained. We have shown that, when the quadrature rule is of certain degree of precision, the rates of convergence for the Legendre spectral Galerkin and Legendre spectral collocation methods are preserved. We obtain superconvergence rates for the iterated discrete Legendre Galerkin solution. By choosing the collocation nodes and quadrature points to be same, we also obtain superconvergence rates for the iterated discrete Legendre collocation solution.