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. Bai, Zhong-Zhi; Lu, Kang-Ya: Optimal rotated block-diagonal preconditioning for discretized optimal control problems constrained with fractional time-dependent diffusive equations (2021)
  2. Carrer, J. A. M.; Solheid, B. S.; Trevelyan, J.; Seaid, M.: A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem (2021)
  3. Fu, Yayun; Cai, Wenjun; Wang, Yushun: A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach (2021)
  4. Jia, Jinhong; Wang, Hong; Zheng, Xiangcheng: A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions (2021)
  5. Jia, Jinhong; Wang, Hong; Zheng, Xiangcheng: A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis (2021)
  6. Jia, Junqing; Jiang, Xiaoyun; Zhang, Hui: An L1 Legendre-Galerkin spectral method with fast algorithm for the two-dimensional nonlinear coupled time fractional Schrödinger equation and its parameter estimation (2021)
  7. Liu, Xinfei; Yang, Xiaoyuan: Mixed finite element method for the nonlinear time-fractional stochastic fourth-order reaction-diffusion equation (2021)
  8. Srivastava, Nikhil; Singh, Aman; Kumar, Yashveer; Singh, Vineet Kumar: Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrix (2021)
  9. 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)
  10. Aboelenen, Tarek: Discontinuous Galerkin methods for fractional elliptic problems (2020)
  11. Bouharguane, Afaf; Seloula, Nour: The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations (2020)
  12. Bradji, Abdallah: A new gradient scheme of a time fractional Fokker-Planck equation with time independent forcing and its convergence analysis (2020)
  13. Cao, Junying; Wang, Ziqiang; Xu, Chuanju: A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative (2020)
  14. 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)
  15. Dehestani, Haniye; Ordokhani, Yadollah; Razzaghi, Mohsen: Fractional-order Genocchi-Petrov-Galerkin method for solving time-space fractional Fokker-Planck equations arising from the physical phenomenon (2020)
  16. Fu, Yayun; Cai, Wenjun; Wang, Yushun: Structure-preserving algorithms for the two-dimensional fractional Klein-Gordon-Schrödinger equation (2020)
  17. Ghaffari, Rezvan; Ghoreishi, Farideh: Error analysis of the reduced RBF model based on POD method for time-fractional partial differential equations (2020)
  18. Huang, Xipei; Lin, Lifeng; Wang, Huiqi: Generalized stochastic resonance for a fractional noisy oscillator with random mass and random damping (2020)
  19. Huang, Yun-Chi; Lei, Siu-Long: Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations (2020)
  20. Hu, Xindi; Zhu, Shengfeng: Isogeometric analysis for time-fractional partial differential equations (2020)

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