Numerical Study of Volterra Integral Equations with Delay

Abstract

This thesis is devoted to the development of Taylor collocation method for a class of Volterra-type problems, including linear delay Volterra integral equations, and Volterra integro-differential equations with two constant delays. Such models naturally arise in many applied fields, including population dynamics, epidemiology, iscoelasticity, control theory, and systems with memory effects, where hereditary phenomena and time delays play a crucial role. The solution is approximated by piecewise Taylor polynomials on each subinterval, leading to an explicit and direct numerical scheme. A key advantage of the method is that it avoids the solution of large algebraic systems, which significantly reduces computational cost and enhances numerical stability. The coefficients of the local Taylor expansions are computed recursively using the governing equations, ensuring a consistent and globally defined approximation. A rigorous convergence analysis is carried out under ppropriate smoothness assumptions on the data. Using discrete Grönwall-type inequalities, it is shown that the proposed method converges. Several numerical experiments are presented to validate the theoretical results and to illustrate the accuracy, and efficiency of the method for different delay configurations.