Kantorovich Methods for Urysohn Integral Equations

In this paper, the Kantorovich method for the numerical solution of nonlinear Urysohn equations with a smooth kernel is considered. The approximating operator is chosen to be either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials of degree \(\le r-1\) . This method have asymptotic series expansions and the orders of convergence can be further improved by the Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by calculating the integrals with a numerical quadrature formula. Numerical examples are given to illustrate the theoretical estimates.