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 265 articles , 1 standard article )

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  1. Yu, Bo; Jiang, Xiaoyun; Xu, Huanying: A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation (2015)
  2. Yu, Yanyan; Deng, Weihua; Wu, Yujiang: Positivity and boundedness preserving schemes for space-time fractional predator-prey reaction-diffusion model (2015)
  3. Zhang, Lu; Sun, Hai-Wei; Pang, Hong-Kui: Fast numerical solution for fractional diffusion equations by exponential quadrature rule (2015)
  4. Zhao, Lijing; Deng, Weihua: A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives (2015)
  5. Zheng, Minling; Liu, Fawang; Turner, Ian; Anh, Vo: A novel high order space-time spectral method for the time fractional Fokker-Planck equation (2015)
  6. Zhong, Suchuan; Ma, Hong; Peng, Hao; Zhang, Lu: Stochastic resonance in a harmonic oscillator with fractional-order external and intrinsic dampings (2015)
  7. Atangana, Abdon; Kilicman, Adem: On the generalized mass transport equation to the concept of variable fractional derivative (2014)
  8. Atangana, Abdon; Tuluce Demiray, Seyma; Bulut, Hasan: Modelling the nonlinear wave motion within the scope of the fractional calculus (2014)
  9. Bu, Weiping; Tang, Yifa; Yang, Jiye: Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations (2014)
  10. Caballero, Josefa; Darwish, Mohamed Abdalla; Sadarangani, Kishin; Shammakh, Wafa M.: Existence results for a coupled system of nonlinear fractional hybrid differential equations with homogeneous boundary conditions (2014)
  11. Chen, J.; Liu, F.; Liu, Q.; Chen, X.; Anh, V.; Turner, I.; Burrage, K.: Numerical simulation for the three-dimension fractional sub-diffusion equation (2014)
  12. Chen, Liping; He, Yigang; Chai, Yi; Wu, Ranchao: New results on stability and stabilization of a class of nonlinear fractional-order systems (2014)
  13. Chen, Minghua; Deng, Weihua: A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation (2014)
  14. Dai, Huiya; Wei, Leilei; Zhang, Xindong: Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the fractional diffusion-wave equation (2014)
  15. Huang, Jianfei; Nie, Ningming; Tang, Yifa: A second order finite difference-spectral method for space fractional diffusion equations (2014)
  16. Li, Can; Zhao, Tinggang; Deng, Weihua; Wu, Yujiang: Orthogonal spline collocation methods for the subdiffusion equation (2014)
  17. Lin, Fu-Rong; Yang, Shi-Wei; Jin, Xiao-Qing: Preconditioned iterative methods for fractional diffusion equation (2014)
  18. Ma, Jingtang; Liu, Jinqiang; Zhou, Zhiqiang: Convergence analysis of moving finite element methods for space fractional differential equations (2014)
  19. Ren, Jincheng; Sun, Zhi-Zhong: Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations (2014)
  20. Shen, Yongjun; Yang, Shaopu; Sui, Chuanyi: Analysis on limit cycle of fractional-order van der Pol oscillator (2014)

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