On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions

Abstract

The main intention of the present work is to develop a numerical scheme based on radial basis functions (RBFs) to solve fractional stochastic integro-differential equations. In this paper, the solution of fractional stochastic integro-differential equation is approximated by using strictly positive definite RBFs such as Gaussian and strictly conditionally positive definite RBFs such as thin plate spline. Then, the quadrature methods are used to approximate the integrals which are appeared in this scheme. When we use thin plate spline to approximate the solution of mentioned equation, we encounter logarithm-like singular integrals which cannot be computed by common quadrature formula. To overcome this difficulty, we introduce the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integral appeared in this case. This method transforms the solution of linear fractional stochastic integro-differential equations to the solution of linear system of algebraic equations which can be easily solved. We also discuss the error analysis of the proposed method and demonstrate that the rate of convergence of this approach is arbitrary high for infinitely smooth RBFs. Finally, the efficiency and accuracy of the proposed method are checked by some numerical examples.

Introduction

Fractional stochastic integro-differential equation is a generalization of the fractional Fokker–Planck equation which describes the random walk of a particle [1]. This model has the following form

$$D^{\alpha }f\left(t\right)=g\left(t\right)+{\int }_0^t{k}_1(t,s)f\left(s\right)ds+{\int }_0^t{k}_2(t,s)f\left(s\right)dB\left(s\right),t\in [0,1],$$

where g(t), k₁(t, s) and k₂(t, s) for t, s ∈ [0, 1] are given functions, f(t) is the unknown function which should be approximated, B(t) denotes standard Brownian motion process defined on the probability space (Ω, A, P), D^α denotes fractional derivative operator of order α and ${\int }_0^t{k}_2(t,s)f\left(s\right)dB\left(s\right)$ is the Itô integral. Furthermore, Lebesgue and Itô integrals in Eq. (1) are well defined.

Integro-differential equations have strong physical background and also have many practical applications in scientific fields such as population and polymerrheology. These problems are often dependent on a noise source that is ignored due to the lack of powerful computing devices and are modeled by deterministic integro-differential equations. In recent years by increasing computational power, it has become apparent that real phenomena, classically modeled by deterministic integro-differential equations, can be more satisfactorily modeled by stochastic integro-differential equations. Numerical solution of various kinds of stochastic integral equations due to randomness has its own difficulties. In many situations, such equations can be rarely solved in analytic form and therefore design and analysis of special algorithms to solve them is an important topic of numerical analysis. Recently, operational matrix methods are extensively used to solve stochastic integral equations [2], [3] and Stratonovich integral equations [4], [5]. Special algorithms for solving fractional stochastic integro-differential equation are presented in some papers which are published recently. Asgari [6] provides a numerical method to solve fractional stochastic integro-differential equation by using block-pulse functions. Mirzaee and Samadyar [7] applied the operational matrix method based on orthonormal Bernstein polynomials to solve fractional stochastic integro-differential equations. Also, spectral collocation method [8] and Galerkin method [9], [10] have been used to solve this equation. However, there are still very few papers discussing the numerical solution of fractional stochastic integro-differential equations.

In this paper, we develop a new meshless approach based on radial basis functions to solve fractional stochastic integro-differential equations. Unlike traditional methods such as finite difference method, finite element method and finite volume method, in meshless method we should discrete the domain of the problem. The meshless techniques are introduced to avoid mesh generation. In a meshless method, a set of scattered data is applied instead of meshing the domain of the problem. So, meshfree methods are often better suited to cope with changes in the geometry of the domain than classical discretization methods. Also, meshless method to solve problems of high dimensions is more convenient than other methods. Some of the meshfree methods are the element-free Galerkin method, kernel methods, moving least squares method [11], [12], meshless local Petrov–Galerkin methods [13], radial point interpolation method [14], etc.

This paper is organized as follows. In Section 2, we review some essential preliminaries on fractional calculus, stochastic calculus and RBFs which are used in the next sections. In Section 3, we present a new numerical method to solve fractional stochastic integro-differential equations of the form (1). Our method is based on the approximation of the solutions of these equations by using RBFs. For this aim, we utilize Gaussian and Thin plate spline RBFs to approximate the solution of Eq. (1) and then use the quadrature formula to approximate the appeared integrals. Our main discussion on stability and convergence analysis of the numerical method is presented in Sections 4 and 5, respectively. The efficiency of the proposed method has been checked by various examples in Section 6. Finally, conclusion of the present paper is given in Section 7.