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 323 articles , 1 standard article )

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  1. Chen, Yanping; Lin, Xiuxiu; Huang, Yunqing: Error analysis of spectral approximation for space-time fractional optimal control problems with control and state constraints (2022)
  2. Colbrook, Matthew J.; Ayton, Lorna J.: A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations (2022)
  3. Jia, Jinhong; Yang, Zhiwei; Zheng, Xiangcheng; Wang, Hong: Analysis and numerical approximation for a nonlinear hidden-memory variable-order fractional stochastic differential equation (2022)
  4. Lee, Eunjung; Na, Hyesun: Dual system least-squares finite element method for a hyperbolic problem (2022)
  5. Liu, Huan; Zheng, Xiangcheng; Wang, Hong; Fu, Hongfei: Error estimate of finite element approximation for two-sided space-fractional evolution equation with variable coefficient (2022)
  6. Liu, Juan; Zhang, Juan; Zhang, Xindong: Semi-discretized numerical solution for time fractional convection-diffusion equation by RBF-FD (2022)
  7. Lyu, Liyao; Chen, Zheng: Local discontinuous Galerkin methods with novel basis for fractional diffusion equations with non-smooth solutions (2022)
  8. McLean, William; Mustapha, Kassem: Uniform stability for a spatially discrete, subdiffusive Fokker-Planck equation (2022)
  9. Sun, Jing; Deng, Weihua; Nie, Daxin: Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion (2022)
  10. Wang, Yibo; Du, Rui; Chai, Zhenhua: Lattice Boltzmann model for time-fractional nonlinear wave equations (2022)
  11. Yang, Zongze; Wang, Jungang; Yuan, Zhanbin; Nie, Yufeng: Using Gauss-Jacobi quadrature rule to improve the accuracy of FEM for spatial fractional problems (2022)
  12. Zada, Laiq; Nawaz, Rashid; Alqudah, Mohammad A.; Sooppy Nisar, Kottakkaran: A new technique for approximate solution of fractional-order partial differential equations (2022)
  13. Zhao, Xuan; Li, Xiaoli; Li, Ziyan: Fast and efficient finite difference method for the distributed-order diffusion equation based on the staggered grids (2022)
  14. Zheng, Minling; Jin, Zhengmeng; Liu, Fawang; Anh, Vo: Matrix transfer technique for anomalous diffusion equation involving fractional Laplacian (2022)
  15. Zhu, Xiaogang; Li, Jimeng; Zhang, Yaping: A local RBFs-based DQ approximation for Riesz fractional derivatives and its applications (2022)
  16. 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)
  17. Bai, Zhong-Zhi; Lu, Kang-Ya: Optimal rotated block-diagonal preconditioning for discretized optimal control problems constrained with fractional time-dependent diffusive equations (2021)
  18. 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)
  19. 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)
  20. Fu, Hongfei; Zhu, Chen; Liang, Xueting; Zhang, Bingyin: Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations (2021)

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