Legendre Superconvergent Degenerate Kernel and Nyström Methods for Nonlinear Integral Equations

We study polynomially based superconvergent collocation methods for the approximation of solutions to nonlinear integral equations. The superconvergent degenerate kernel method is chosen for the approximation of solutions of Hammerstein equations, while a superconvergent Nyström method is used to solve Urysohn equations. By applying interpolatory projections based on the Legendre polynomials of degree ≤ n, we analyze the property of superconvergence of these methods and their iterated versions. The numerical data are presented to validate the theoretical results.