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 242 articles , 1 standard article )

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  1. Abbaszadeh, Mostafa; Dehghan, Mehdi: A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation (2020)
  2. Aboelenen, Tarek: Discontinuous Galerkin methods for fractional elliptic problems (2020)
  3. Bouharguane, Afaf; Seloula, Nour: The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations (2020)
  4. Bradji, Abdallah: A new gradient scheme of a time fractional Fokker-Planck equation with time independent forcing and its convergence analysis (2020)
  5. Cao, Junying; Wang, Ziqiang; Xu, Chuanju: A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative (2020)
  6. Carrer, J. A. M.; Solheid, B. S.; Trevelyan, J.; Seaid, M.: The boundary element method applied to the solution of the diffusion-wave problem (2020)
  7. Fu, Yayun; Cai, Wenjun; Wang, Yushun: Structure-preserving algorithms for the two-dimensional fractional Klein-Gordon-Schrödinger equation (2020)
  8. Ghaffari, Rezvan; Ghoreishi, Farideh: Error analysis of the reduced RBF model based on POD method for time-fractional partial differential equations (2020)
  9. Huang, Xipei; Lin, Lifeng; Wang, Huiqi: Generalized stochastic resonance for a fractional noisy oscillator with random mass and random damping (2020)
  10. Huang, Yun-Chi; Lei, Siu-Long: Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations (2020)
  11. Hu, Xindi; Zhu, Shengfeng: Isogeometric analysis for time-fractional partial differential equations (2020)
  12. Jiang, Tao; Wang, Xing-Chi; Huang, Jin-Jing; Ren, Jin-Lian: An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry (2020)
  13. Li, Meng; Huang, Chengming; Ming, Wanyuan: A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations (2020)
  14. Lin, Zeng; Wang, Dongdong; Qi, Dongliang; Deng, Like: A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations (2020)
  15. Liu, Huan; Cheng, Aijie; Wang, Hong: A parareal finite volume method for variable-order time-fractional diffusion equations (2020)
  16. Liu, Xing; Deng, Weihua: Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion (2020)
  17. Maurya, Rahul Kumar; Devi, Vinita; Singh, Vineet Kumar: Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation (2020)
  18. Nie, Daxin; Sun, Jing; Deng, Weihua: Numerical scheme for the Fokker-Planck equations describing anomalous diffusions with two internal states (2020)
  19. Ren, Jincheng; Liao, Hong-lin; Zhang, Zhimin: Superconvergence error estimate of a finite element method on nonuniform time meshes for reaction-subdiffusion equations (2020)
  20. Shao, Xin-Hui; Zhang, Zhen-Duo; Shen, Hai-Long: A generalization of trigonometric transform splitting methods for spatial fractional diffusion equations (2020)

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