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. Ma, Jingtang; Jiang, Yingjun: Moving collocation methods for time fractional differential equations and simulation of blowup (2011)
  2. Petráš, Ivo: Fractional-order nonlinear systems. Modeling, analysis and simulation (2011)
  3. Petráš, Ivo: Modeling and numerical analysis of fractional-order Bloch equations (2011)
  4. Su, Lijuan; Wang, Wenqia; Wang, Hong: A characteristic difference method for the transient fractional convection-diffusion equations (2011)
  5. Wang, Wansheng; Li, Dongfang: Stability analysis of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (2011)
  6. Ahmad, Bashir; Alsaedi, Ahmed: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations (2010)
  7. Brunner, Hermann; Ling, Leevan; Yamamoto, Masahiro: Numerical simulations of 2D fractional subdiffusion problems (2010)
  8. Deng, Weihua: Smoothness and stability of the solutions for nonlinear fractional differential equations (2010)
  9. Petráš, Ivo: A note on the fractional-order Volta’s system (2010)
  10. Sun, Kehui; Wang, Xia; Sprott, J. C.: Bifurcations and chaos in fractional-order simplified Lorenz system (2010)
  11. Xin, Baogui; Chen, Tong; Liu, Yanqin: Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control (2010)
  12. Yang, Qianqian; Liu, Fawang; Turner, Ian: Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term (2010)
  13. Ahmad, Bashir; Nieto, Juan J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions (2009)
  14. Deng, Weihua: Finite element method for the space and time fractional Fokker-Planck equation (2009)
  15. Petráš, Ivo: Chaos in the fractional-order Volta’s system: modeling and simulation (2009)
  16. Tavazoei, Mohammad Saleh; Haeri, Mohammad: Describing function based methods for predicting chaos in a class of fractional order differential equations (2009)
  17. Ahmad, Bashir; Sivasundaram, S.: Theory of fractional differential equations with three-point boundary conditions (2008)
  18. Deng, Weihua: Numerical algorithm for the time fractional Fokker-Planck equation (2007)

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