Numerical approximation for a stochastic time-fractional cable equation

An efficient numerical method is proposed to address a stochastic time-fractional cable equation driven by fractionally integrated additive noise. Under the reasonable assumptions, we rigorously establish for the first time, the existence, uniqueness, and regularity of the mild solution for this equation. For spatial discretization, a semi-discrete scheme is constructed employing the Galerkin FEM, and the optimal spatial error estimate is derived based on the semigroup approach. In temporal discretization, a piecewise constant function is introduced to approximate the noise, leading to the formulation of a regularized stochastic time-fractional cable equation. A detailed proof of the temporal error estimates is provided via the semigroup approach. Numerical experiments demonstrate that the temporal convergence order attains $O\left({\tau }^{1/2}\right)$ for initial data of either smooth or non-smooth type. The order is independent of the parameters ${\alpha }_1\in (0,1)$, ${\alpha }_2\in (0,1)$, and $\beta \in (0,1)$ in the equation. These results perfectly align with the theoretical predictions.