Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs

W W Mohammed,K A Elnagar,A H El-Bassiouny,M A Sohaly

doi:10.4236/ajcm.2014.44024

Abstract

Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.

Highlights

Stochastic partial differential equations (SPDEs) frequently arise from applications in areas such as physics, engineering and finance
In many cases it is difficult to derive an explicit form of their solution
It is well known that explicit time discretization via standard methods leads to a time step restriction due to the stiffness originating from the discretization of the diffusion operator (e.g. the Courant-Friedrichs-Lewy (CFL) ∆t ≤ C (∆x)2, where Δt and Δx are the time and space discretization, respectively)

Summary

Introduction

Stochastic partial differential equations (SPDEs) frequently arise from applications in areas such as physics, engineering and finance. Some of the main numerical methods for solving stochastic partial differential equations (SPDEs), like finite difference and finite element schemes, have been considered [1]-[9], e.g. Mohammed [8] discussed stochastic finite difference schemes by three points under the following condition of stability:. (2014) Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs. American Journal of Computational Mathematics, 4, 280-288. Our aim of this paper is to use the stochastic finite difference schemes by seven points that are strong convergences to our problem and much better stability properties than three and five points.

Preliminaries

Applications

Conclusion and Future Works