Numerical algorithm for the time fractional Fokker-Planck equation. Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error $O(k^{min{1+2alpha ,2}})+O(h^{2})$, and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for $alpha =1.0$ with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for $alpha =0.8$ the convergent order in space is confirmed and the numerical results with different time step sizes are shown.

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  1. Wei, Leilei; He, Yinnian; Zhang, Yan: Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method (2013)
  2. Xin, Baogui; Li, Yuting: 0-1 test for chaos in a fractional order financial system with investment incentive (2013)
  3. Yang, Yong-Ju; Baleanu, Dumitru; Yang, Xiao-Jun: Analysis of fractal wave equations by local fractional Fourier series method (2013)
  4. Yan, Liang; Yang, Fenglian: A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations (2013)
  5. Yu, YanYan; Deng, WeiHua; Wu, YuJiang: Positivity and boundedness preserving schemes for the fractional reaction-diffusion equation (2013)
  6. Zhou, Zhiqiang; Wu, Hongying: Finite element multigrid method for the boundary value problem of fractional advection dispersion equation (2013)
  7. Cao, Xue-Nian; Fu, Jiang-Li; Huang, Hu: Numerical method for the time fractional Fokker-Planck equation (2012)
  8. Choi, Y. J.; Chung, S. K.: Finite element solutions for the space fractional diffusion equation with a nonlinear source term (2012)
  9. Gao, Guang-Hua; Sun, Zhi-Zhong; Zhang, Ya-Nan: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions (2012)
  10. Huang, Fenghui: A time-space collocation spectral approximation for a class of time fractional differential equations (2012)
  11. Li, Can; Deng, Weihua; Wu, Yujiang: Finite difference approximations and dynamics simulations for the Lévy fractional Klein-Kramers equation (2012)
  12. Vanani, S. Karimi; Aminataei, A.: A numerical algorithm for the space and time fractional Fokker-Planck equation (2012)
  13. Zhang, Na; Deng, Weihua; Wu, Yujiang: Finite difference/element method for a two-dimensional modified fractional diffusion equation (2012)
  14. Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren: The extended fractional subequation method for nonlinear fractional differential equations (2012)
  15. Chen, Liping; Chai, Yi; Wu, Ranchao: Control and synchronization of fractional-order financial system based on linear control (2011)
  16. Deng, Weihua; Li, Can: Finite difference methods and their physical constraints for the fractional Klein-Kramers equation (2011)
  17. Garrappa, Roberto; Popolizio, Marina: On the use of matrix functions for fractional partial differential equations (2011)
  18. Li, Can; Deng, Weihua; Wu, Yujiang: Numerical analysis and physical simulations for the time fractional radial diffusion equation (2011)
  19. Lin, Yumin; Li, Xianjuan; Xu, Chuanju: Finite difference/spectral approximations for the fractional cable equation (2011)
  20. Magdziarz, Marcin; Orzeł, Sebastian; Weron, Aleksander: Option pricing in subdiffusive Bachelier model (2011)

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