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

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 a E (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 Gr & uuml;nwald- 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.