Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

Abstract

In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When $0<H<0.5$, a change of variables $\partial \left(t^{2H}\right)=2H{t}^{2H-1}\partial t$ avoids the singularity of numerical computation at $t=0$, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For $0.5<H<1$, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.

arXiv comment10 pages, 3 figures

References in corpus (7)

• On an explicit finite difference method for fractional diffusion equations
• Boundary problems for the fractional and tempered fractional operators
• A Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations
• Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
• Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times
• Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator
• Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian