FODE

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.


References in zbMATH (referenced in 258 articles , 1 standard article )

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  1. Yan, Shao-Hong; Chen, Xiao-Hong; Xie, Gong-Nan; Cattani, Carlo; Yang, Xiao-Jun: Solving Fokker-Planck equations on Cantor sets using local fractional decomposition method (2014)
  2. Yan, Yubin; Pal, Kamal; Ford, Neville J.: Higher order numerical methods for solving fractional differential equations (2014)
  3. Zhang, Haixiang; Yang, Xuehua; Han, Xuli: Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation (2014)
  4. Zhang, Ya-Nan; Sun, Zhi-Zhong: Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation (2014)
  5. Zhang, Ya-nan; Sun, Zhi-zhong; Liao, Hong-lin: Finite difference methods for the time fractional diffusion equation on non-uniform meshes (2014)
  6. Zhang, Yuxin: ([3, 3]) Padé approximation method for solving space fractional Fokker-Planck equations (2014)
  7. Zhao, Lijing; Deng, Weihua: Jacobian-predictor-corrector approach for fractional differential equations (2014)
  8. Zhao, Xuan; Xu, Qinwu: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient (2014)
  9. Atangana, Abdon; Oukouomi Noutchie, S. C.: Stability and convergence of a time-fractional variable order Hantush equation for a deformable aquifer (2013)
  10. Jia, Hong-Yan; Chen, Zeng-Qiang; Qi, Guo-Yuan: Topological horseshoe analysis and circuit realization for a fractional-order Lü system (2013)
  11. Li, Limei; Xu, Da: Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation (2013)
  12. Li, Limei; Xu, Da; Luo, Man: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation (2013)
  13. Liu, Jian; Liu, Shutang; Yuan, Chunhua: Modified generalized projective synchronization of fractional-order chaotic Lü systems (2013)
  14. Li, Yajing; Wang, Yejuan: Uniform asymptotic stability of solutions of fractional functional differential equations (2013)
  15. Ren, Jincheng; Sun, Zhi-zhong: Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with von Neumann boundary conditions (2013)
  16. Ren, Jincheng; Sun, Zhi-Zhong; Zhao, Xuan: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions (2013)
  17. Su, Lijuan; Cheng, Pei: A high-accuracy MOC/FD method for solving fractional advection-diffusion equations (2013)
  18. Uchaikin, Vladimir V.: Fractional derivatives for physicists and engineers. Volume I: Background and theory. Volume II: Applications (2013)
  19. Wang, Peiguang; Hou, Ying: Generalized quasilinearization for the system of fractional differential equations (2013)
  20. Wang, Yuan-Ming: Maximum norm error estimates of ADI methods for a two-dimensional fractional subdiffusion equation (2013)

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