An Efficient Numerical Algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

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 regularization 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.

Automated summary

In this study, we discuss the numerical scheme for a model describing the competition between super- and sub-diffusions driven by fractional Brownian sheet noise. By utilizing the properties of Mittag-Leffler function and Wong-Zakai approximation, we achieve optimal convergence of the regularized solution. The spectral Galerkin method and the Mittag-Leffler Euler integrator are employed for space and time operators, respectively. Error analyses are provided and validated through numerical experiments.