Optimal convergence for the regularized solution of the model describing the competition between super- and sub- diffusions driven by fractional Brownian sheet noise

TL;DR

This paper develops an optimal numerical scheme for a model describing the competition between super- and sub-diffusions driven by fractional Brownian sheet noise, achieving high convergence rates through regularization and spectral methods.

Contribution

It introduces a novel regularization approach and combines spectral Galerkin and Mittag-Leffler Euler methods for efficient, accurate simulation of complex anomalous diffusion models.

Findings

Achieved optimal convergence rates for the regularized solution.

Validated the numerical scheme with comprehensive error analysis.

Demonstrated efficiency of the fast Mittag-Leffler Euler integrator.

Abstract

Super- and sub- diffusions are two typical types of anomalous diffusions in the natural world. In this work, we discuss the numerical scheme for the model describing the competition between super- and sub- diffusions driven by fractional Brownian sheet noise. Based on the obtained regulization result of the solution by using the properties of Mittag-Leffler function and the regularized noise by Wong-Zakai approximation, we make full use of the regularity of the solution operators to achieve optimal convergence of the regularized solution. The spectral Galerkin method and the Mittag-Leffler Euler integrator are respectively used to deal with the space and time operators. In particular, by contour integral, the fast evaluation of the Mittag-Leffler Euler integrator is realized. We provide complete error analyses, which are verified by the numerical experiments.