Numerical approximation for fractional diffusion equation forced by a tempered fractional Gaussian noise

TL;DR

This paper develops a numerical scheme for a fractional diffusion equation driven by tempered fractional Gaussian noise, establishing regularity, error estimates, and confirming results through numerical experiments.

Contribution

It introduces a novel numerical approach combining spectral Galerkin and semi-implicit Euler methods for this complex stochastic PDE, with rigorous error analysis.

Findings

Error estimates in mean-squared L2 norm are derived.

Numerical experiments validate the theoretical error bounds.

The scheme effectively approximates solutions to the fractional stochastic diffusion equation.

Abstract

This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method is used for space approximation; after that the system is transformed into an equivalent form having better regularity than the original one in time. Then we use the semi-implicit Euler scheme to discretize the time derivative. In terms of the temporal-spatial error splitting technique, we obtain the error estimates of the fully discrete scheme in the sense of mean-squared $L^2$-norm. Extensive numerical experiments confirm the theoretical estimates.