Legendre Kantorovich methods for Uryshon integral equations

In this paper, the Kantorovich method for the numerical solution of nonlinear emph{Uryshon} equations with a smooth kernel is considered. The approximating operator is chosen to be either the orthogonal projection or an interpolatory projection using a Legendre polynomial basis. The order of convergence of the proposed method and those of superconvergence of the iterated versions are established. We show that these orders of convergence are valid in the corresponding discrete methods obtained by replacing the integration by a quadrature rule. Numerical examples are given to illustrate the theoretical estimates.