The Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Collocation and Iterated Collocation Methods

The subject of this paper is a variable stepsize one-step method of collocation type for solving general nonlinear second kind Volterra integral equations. We extend the iterated collocation method corresponding to polynomial spline collocation to nonlinear Volterra integral equations of the second kind. The resulting superconvergence properties of either the collocation approximation or the iterated collocation approximation are used to obtain (local and global) error estimates which in turn form the basis of a variable stepsize code. The performance of this code is illustrated by means of numerous test problems.