Difference methods for stochastic space fractional diffusion equation driven by additive space-time white noise via Wong-Zakai approximation

摘要

The fractional diffusion equations are needed to describe the diffusion process in an inhomogeneous medium; further incorporating the noise term in the fractional diffusion equations leads to stochastic fractional diffusion equation. In this work, our motivation is to design stochastic finite difference schemes for understanding the behavior of the solution of modified stochastic space fractional diffusion equations. Initially, the noise term is approximated by a piecewise constant random process that leads to a modified stochastic differential equation. We applied a shifted Grunwald-Letnikov formula (one shift) for discretizing the spatial derivative involved in the space fractional diffusion equation. The principle results of difference schemes, that is, consistency, stability, and convergence, are proved in a mean-square sense. Further, the accuracy and efficiency of the proposed schemes are investigated through numerical experiments.