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 279 articles , 1 standard article )

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  1. Ameen, Ibrahem G.; Zaky, Mahmoud A.; Doha, Eid H.: Singularity preserving spectral collocation method for nonlinear systems of fractional differential equations with the right-sided Caputo fractional derivative (2021)
  2. Bai, Zhong-Zhi; Lu, Kang-Ya: Optimal rotated block-diagonal preconditioning for discretized optimal control problems constrained with fractional time-dependent diffusive equations (2021)
  3. 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)
  4. Chen, Xiaoli; Yang, Liu; Duan, Jinqiao; Karniadakis, George Em: Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks (2021)
  5. Fu, Yayun; Cai, Wenjun; Wang, Yushun: A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach (2021)
  6. Islam, Tarikul; Akter, Armina: Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics (2021)
  7. Jia, Jinhong; Wang, Hong; Zheng, Xiangcheng: A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions (2021)
  8. Jia, Jinhong; Wang, Hong; Zheng, Xiangcheng: A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis (2021)
  9. 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)
  10. Li, Buyang; Wang, Hong; Wang, Jilu: Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order (2021)
  11. Lin, Fu-Rong; Wang, Qiu-Ya; Jin, Xiao-Qing: Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations (2021)
  12. Liu, Huan; Zheng, Xiangcheng; Chen, Chuanjun; Wang, Hong: A characteristic finite element method for the time-fractional mobile/immobile advection diffusion model (2021)
  13. Liu, Xinfei; Yang, Xiaoyuan: Mixed finite element method for the nonlinear time-fractional stochastic fourth-order reaction-diffusion equation (2021)
  14. Patel, Vijay Kumar; Bahuguna, Dhirendra: An efficient matrix approach for the numerical solutions of electromagnetic wave model based on fractional partial derivative (2021)
  15. Qu, Haidong; She, Zihang; Liu, Xuan: Neural network method for solving fractional diffusion equations (2021)
  16. She, Zi-Hang; Qu, Hai-Dong; Liu, Xuan: Stability and convergence of finite difference method for two-sided space-fractional diffusion equations (2021)
  17. 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)
  18. Uz Zaman, Tareq; MacLachlan, Scott; Hossain, M. Enamul: Numerical modeling of a memory-based diffusivity equation and determination of its fractional order value (2021)
  19. Yuan, Minjuan; Wang, Liang; Jiao, Yiyu; Xu, Wei: An improved cell mapping method based on dimension-extension for fractional systems (2021)
  20. Zhang, Guoyu; Huang, Chengming; Fei, Mingfa; Wang, Nan: A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equations (2021)

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