Error estimates of finite element methods for nonlinear fractional stochastic differential equations

Yanpeng Zhang,Xiaocui Li,Xiaoyuan Yang

doi:10.1186/s13662-018-1665-0

Abstract

In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes.

Highlights

In the last few years, fractional calculus has attracted lots of attention
To the authors’ knowledge, no result has been reported on the error estimation of nonlinear fractional stochastic partial differential equations with multiplicative noise based on the form of mild solutions proposed in [24], so the motivation of this paper is to fill this gap
4.2 Error estimates By using the GMMP scheme (4.1) we indicate the approximation of u(tn) by un ≈ u(tn)

Summary

Introduction

In the last few years, fractional calculus has attracted lots of attention. The increasing interest in fractional equations is motivated by their applications in various fields of science such as fluid mechanics, heat conduction in materials with memory, physics, chemistry, and engineering [1,2,3,4,5]. We consider the following initial value problem for the nonlinear fractional stochastic partial differential equation (SPDE) with multiplicative noise:. The existence of mild solutions for a class of nonlinear fractional stochastic partial differential equations has been discussed in [24]. Since the random effects on transport of particles in medium with thermal memory can be exactly modeled by fractional stochastic differential systems, it is important and necessary to discuss numerical schemes and error estimation for stochastic fractional equations. To the authors’ knowledge, no result has been reported on the error estimation of nonlinear fractional stochastic partial differential equations with multiplicative noise based on the form of mild solutions proposed in [24], so the motivation of this paper is to fill this gap.

Mittag–Leffler function The Mittag–Leffler function is defined by

Space discretization

L02 ds

The GMMP scheme

Conclusions and discussions