A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Abstract

Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [ − 1 , 1 ]. For the evaluation of the integral term efficiently a Legendre–Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed.