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.
Highlights
Stochastic Volterra-integral equations arise in mathematical modeling of many physical phenomena such as mechanics, medical, finance, reactor dynamics, etc
Since most stochastic differential equations cannot be solved analytically, a number of numerical method are applied to obtain the approximate solutions of stochastic ordinary differential equations, such as the stochastic Runge–Kutta method, the stochastic linear multistep method and the stochastic Taylor method
3 Error analysis we provide the error analysis of the spectral collocation method for SVIDE
Summary
Stochastic Volterra-integral equations arise in mathematical modeling of many physical phenomena such as mechanics, medical, finance, reactor dynamics, etc. Gauss–Lobatto points for the approximate solution of SVIDEs. For this purpose, let us consider the stochastic Volterra integro-differential equation with convolution kernels of the form [18] Since most stochastic differential equations cannot be solved analytically, a number of numerical method are applied to obtain the approximate solutions of stochastic ordinary differential equations, such as the stochastic Runge–Kutta method, the stochastic linear multistep method and the stochastic Taylor method.
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