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

  • Xing Liu
  • Weihua DengEmail author
Original Paper


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 \(\partial \left (t^{2H}\right )=2Ht^{2H-1}\partial t\) avoids the singularity of numerical computation at t = 0, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For H > 0.5, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. The stability and convergence of the numerical schemes are demonstrated by using Fourier method. 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.


Singularity Nonuniform discretization Computational efficiency Mean squared displacement 


Funding information

This work was financially supported by the National Natural Science Foundation of China under Grant No. 11671182, and the Fundamental Research Funds for the Central Universities under Grants No. lzujbky-2018-ot03.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhouPeople’s Republic of China

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