A Galerkin finite element method for time-fractional stochastic heat equation

TL;DR

This paper develops a Galerkin finite element method for solving a time-fractional stochastic heat equation with multiplicative noise, providing convergence analysis and numerical verification for heat transport in porous media with memory effects.

Contribution

It introduces a novel combination of Galerkin finite element spatial discretization with a specific temporal scheme for the fractional stochastic heat equation, including error estimates.

Findings

Convergence error estimates for semi-discrete and fully discrete schemes.

Numerical results confirm theoretical convergence rates.

Method effectively models heat transport with memory and randomness.

Abstract

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.