Collocation Methods for General Volterra Functional Integral Equations with Vanishing Delays

Abstract

We analyze the existence, uniqueness, and regularity properties of solutions for general Volterra functional integral equations with the delay function $\theta(t)$ vanishing at the initial point of the given interval $[0,T]$ (with $\theta(t)=qt, \; 0<q<1$, representing an important special case). The focus of the paper is then on the the attainable order of convergence, and the question of possible superconvergence, for collocation solutions in certain piecewise polynomial spaces. Numerical experiments complement the theoretical convergence results.