Superconvergent Nyström method for Urysohn integral equations (Q512842)

A superconvergent Nyström method is proposed for solving Urysohn nonlinear integral equations, i.e., \[ x(s) + \int\limits^1_0 k(s,t,x(t))dt = f(s), \qquad k, \frac{\partial u}{\partial x} \in C^{2r}, \quad r=1,2,\ldots . \] Using an interpolatory projection onto the set of \(r\) Gauss points, it is shown that the proposed method has an order of \(3r\) and one step of iteration improves the convergence order up to \(4r\). The size of the nonlinear system of equations that must be solved to calculate the approximate solution using this method remains the same as the range of the interpolatory projection. Numerical results are given to illustrate the improvement of the order.