A Galerkin finite element method for time-fractional stochastic heat equation
This work provides a numerical method for solving a specific class of stochastic fractional PDEs, but it is incremental as it applies existing techniques to a new problem combination.
The paper develops a Galerkin finite element method for the time-fractional stochastic heat equation with multiplicative noise, establishing convergence error estimates for semi-discrete and fully discrete schemes, and verifying theoretical results with a numerical example.
In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo-Mainardi-Moretti-Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.