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. Cheng, Xiujun; Duan, Jinqiao; Li, Dongfang: A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations (2019)
  2. Dehghan, Mehdi; Abbaszadeh, Mostafa: Error estimate of finite element/finite difference technique for solution of two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives (2019)
  3. Fu, Yayun; Song, Yongzhong; Wang, Yushun: Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation (2019)
  4. Jia, Jinhong; Wang, Hong: A fast finite volume method for conservative space-time fractional diffusion equations discretized on space-time locally refined meshes (2019)
  5. Jiang, Yingjun; Xu, Xuejun: A monotone finite volume method for time fractional Fokker-Planck equations (2019)
  6. Li, Binjie; Luo, Hao; Xie, Xiaoping: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data (2019)
  7. Li, Can; Deng, Weihua; Zhao, Lijing: Well-posedness and numerical algorithm for the tempered fractional differential equations (2019)
  8. Li, Meng; Zhao, Jikun; Huang, Chengming; Chen, Shaochun: Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data (2019)
  9. Marques Carrer, José Antonio; Seaid, Mohammed; Trevelyan, Jon; dos Santos Solheid, Bruno: The boundary element method applied to the solution of the anomalous diffusion problem (2019)
  10. Ma, Weiyuan; Li, Changpin; Deng, Jingwei: Synchronization in tempered fractional complex networks via auxiliary system approach (2019)
  11. Mohamed, Amany S.; Mokhtar, Mahmoud M.: Spectral tau-Jacobi algorithm for space fractional advection-dispersion problem (2019)
  12. Ren, Lei; Liu, Lei: A high-order compact difference method for time fractional Fokker-Planck equations with variable coefficients (2019)
  13. Saberi Zafarghandi, Fahimeh; Mohammadi, Maryam; Babolian, Esmail; Javadi, Shahnam: Radial basis functions method for solving the fractional diffusion equations (2019)
  14. Salehi Shayegan, Amir Hossein; Zakeri, Ali: Quasi solution of a backward space fractional diffusion equation (2019)
  15. Sidi Ammi, Moulay Rchid; Jamiai, Ismail; Torres, Delfim F. M.: A finite element approximation for a class of Caputo time-fractional diffusion equations (2019)
  16. Sun, HongGuang; Chang, Ailian; Zhang, Yong; Chen, Wen: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications (2019)
  17. Sun, Jing; Nie, Daxin; Deng, Weihua: Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative (2019)
  18. Wang, Tingting; Song, Fangying; Wang, Hong; Karniadakis, George Em: Fractional Gray-Scott model: well-posedness, discretization, and simulations (2019)
  19. Wang, Yanyong; Yan, Yubin; Hu, Ye: Numerical methods for solving space fractional partial differential equations using Hadamard finite-part integral approach (2019)
  20. Yépez-Martínez, H.; Gómez-Aguilar, J. F.: A new modified definition of Caputo-fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM) (2019)

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