Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations

Abstract

An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach.