Error analysis of a fully discrete method for time-fractional diffusion equations with a tempered fractional Gaussian noise

Abstract

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α∈(0,1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis.