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. Jia, Jinhong; Zheng, Xiangcheng; Fu, Hongfei; Dai, Pingfei; Wang, Hong: A fast method for variable-order space-fractional diffusion equations (2020)
  2. 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)
  3. Li, Binjie; Wang, Tao; Xie, Xiaoping: Analysis of a temporal discretization for a semilinear fractional diffusion equation (2020)
  4. Li, Meng; Fei, Mingfa; Wang, Nan; Huang, Chengming: A dissipation-preserving finite element method for nonlinear fractional wave equations on irregular convex domains (2020)
  5. Li, Meng; Huang, Chengming; Ming, Wanyuan: A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations (2020)
  6. Lin, Zeng; Wang, Dongdong; Qi, Dongliang; Deng, Like: A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations (2020)
  7. Lin, Zi-Fei; Li, Jiao-Rui; Wu, Juan; Pham, Viet-Thanh; Kapitaniak, Tomasz: Effect of the policy and consumption delay on the amplitude and length of business cycle (2020)
  8. Liu, Huan; Cheng, Aijie; Wang, Hong: A parareal finite volume method for variable-order time-fractional diffusion equations (2020)
  9. 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)
  10. Maurya, Rahul Kumar; Devi, Vinita; Singh, Vineet Kumar: Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation (2020)
  11. Nie, Daxin; Sun, Jing; Deng, Weihua: Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states (2020)
  12. Nie, Daxin; Sun, Jing; Deng, Weihua: Numerical scheme for the Fokker-Planck equations describing anomalous diffusions with two internal states (2020)
  13. Pougkakiotis, Spyridon; Pearson, John W.; Leveque, Santolo; Gondzio, Jacek: Fast solution methods for convex quadratic optimization of fractional differential equations (2020)
  14. Ren, Jincheng; Liao, Hong-lin; Zhang, Zhimin: Superconvergence error estimate of a finite element method on nonuniform time meshes for reaction-subdiffusion equations (2020)
  15. Shao, Xin-Hui; Zhang, Zhen-Duo; Shen, Hai-Long: A generalization of trigonometric transform splitting methods for spatial fractional diffusion equations (2020)
  16. Sheng, Changtao; Shen, Jie; Tang, Tao; Wang, Li-Lian; Yuan, Huifang: Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains (2020)
  17. Singh, Brajesh Kumar; Kumar, Anil: Numerical study of conformable space and time fractional Fokker-Planck equation via CFDT method (2020)
  18. Zhu, X. G.; Nie, Y. F.; Ge, Z. H.; Yuan, Z. B.; Wang, J. G.: A class of RBFs-based DQ methods for the space-fractional diffusion equations on 3D irregular domains (2020)
  19. Alzahrani, S. S.; Khaliq, A. Q. M.; Biala, T. A.; Furati, K. M.: Fourth-order time stepping methods with matrix transfer technique for space-fractional reaction-diffusion equations (2019)
  20. Bai, Zhong-Zhi; Lu, Kang-Ya: On banded (M)-splitting iteration methods for solving discretized spatial fractional diffusion equations (2019)

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