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

TL;DR

This paper develops a numerical scheme for stochastic nonlinear time-space fractional cable equations driven by rough noise, establishing convergence and regularity results using operator theory and spectral methods.

Contribution

It introduces a novel numerical approximation combining spectral Galerkin and backward Euler methods for complex fractional equations with rough noise, with proven convergence.

Findings

Established existence, uniqueness, and regularity of solutions.

Proved convergence of the numerical scheme via Wong-Zakai approximation.

Provided error estimates for the numerical approximation.

Abstract

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 involves both two nonlocal terms in time and one in space. By an operator theoretic approach, we establish the existence, uniqueness, and regularities of solutions. We also obtain a convergence result for the regularized equation via Wong-Zakai approximation to regularize the rough noise. The numerical scheme approximates the model in space by the standard spectral Galerkin method and in time by the backward Euler convolution quadrature method. After that, error estimates are established.