An Efficient Solution for Stochastic Fractional Partial Differential Equations with Additive Noise by a Meshless Method

TLDR: In this paper, a meshless method based on the radial basis functions was proposed to solve one-dimensional stochastic heat and advection-diffusion equations, where the spatial derivatives were approximated by Kansa approach.

Abstract: With respect to wide range of applications of stochstic partial differential equation (SPDE) and high ability of meshless methods to solve complicated problems, in this paper, an efficient numerical method for the time fractional SPDE, formulated with Caputo’s fractional derivative, based on meshless methods is presented. This article presents a meshless method based on the radial basis functions to solve one-dimensional stochastic heat and advection–diffusion equations. In here, first, we approximate the time fractional derivative of the mentioned equations by a scheme of order $$ \mathsf {O}(\tau ^{2-\alpha }) $$ , $$ 0<\alpha <1 $$ then the spatial derivatives are approximated by Kansa approach. Numerical examples are presented to show the efficiency and effectiveness of the proposed method in solving fractional SPDEs.