Approximation for stochastic time-space fractional cable equations driven by rough noise

The time-space fractional cable equation arises from extending the generalized fractional Ohm's law to model anomalous diffusion processes. In this paper, we develop and analyze a numerical approximation for stochastic nonlinear time-space fractional cable equation driven by rough noise. The model features both two nonlocal terms in time and one in space. By an operator theoretic approach, we establish the existence, uniqueness and regularity of solutions. To regularize the rough noise, we introduce a spectral Wong-Zakai approximation and derive its convergence rate. For the fully discrete scheme, we employ the spectral Galerkin method for spatial discretization and the backward Euler convolution quadrature for temporal discretization, and we derive error estimates under explicit parameter conditions. Finally, numerical experiments are presented to validate the theoretical convergence rates.