High-order collocation methods for nonlinear delay integral equation

The classical collocation methods based on piecewise polynomials have been studied for delay Volterra integral equations of the second-kind in Brunner (2004). These collocation methods have uniform order m for any choice of the collocation parameters and can achieve local superconvergence in the grid points by choosing the suitable collocation parameters. In this paper with the aim of increasing the order of classical collocation methods, we use a general class of multistep methods based on Hermite collocation methods and prove that this numerical method has uniform order 2m+2r for r previous time steps and m collocation points. Some numerical examples are given to show the validity of the presented method and to confirm our theoretical results.