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 56 articles )

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  1. Yang, Yin; Huang, Yunqing; Zhou, Yong: Numerical solutions for solving time fractional Fokker-Planck equations based on spectral collocation methods (2018)
  2. Asl, Mohammad Shahbazi; Javidi, Mohammad: An improved PC scheme for nonlinear fractional differential equations: error and stability analysis (2017)
  3. Deng, Jingwei; Zhao, Lijing; Wu, Yujiang: Fast predictor-corrector approach for the tempered fractional differential equations (2017)
  4. Jahanshahi, S.; Babolian, E.; Torres, D. F. M.; Vahidi, A. R.: A fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives (2017)
  5. Nguyen, Thien Binh; Jang, Bongsoo: A high-order predictor-corrector method for solving nonlinear differential equations of fractional order (2017)
  6. Zeng, Fanhai; Li, Changpin: A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation (2017)
  7. Zhao, Yue; Bu, Weiping; Zhao, Xuan; Tang, Yifa: Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation (2017)
  8. Angstmann, C. N.; Donnelly, I. C.; Henry, B. I.; Jacobs, B. A.; Langlands, T. A. M.; Nichols, J. A.: From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations (2016)
  9. Bhrawy, A. H.; Zaky, M. A.; Van Gorder, R. A.: A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation (2016)
  10. Angstmann, C. N.; Donnelly, I. C.; Henry, B. I.; Nichols, J. A.: A discrete time random walk model for anomalous diffusion (2015)
  11. Deng, Kaiying; Chen, Minghua; Sun, Tieli: A weighted numerical algorithm for two and three dimensional two-sided space fractional wave equations (2015)
  12. Deng, Weihua; Hesthaven, Jan S.: Local discontinuous Galerkin methods for fractional ordinary differential equations (2015)
  13. Fairweather, Graeme; Zhang, Haixiang; Yang, Xuehua; Xu, Da: A backward Euler orthogonal spline collocation method for the time-fractional Fokker-Planck equation (2015)
  14. Hafez, Ramy M.; Ezz-Eldien, Samer S.; Bhrawy, Ali H.; Ahmed, Engy A.; Baleanu, Dumitru: A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations (2015)
  15. Vong, Seakweng; Wang, Zhibo: A high order compact finite difference scheme for time fractional Fokker-Planck equations (2015)
  16. Yu, Bo; Jiang, Xiaoyun; Xu, Huanying: A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation (2015)
  17. Zhao, Lijing; Deng, Weihua: A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives (2015)
  18. Zhong, Suchuan; Ma, Hong; Peng, Hao; Zhang, Lu: Stochastic resonance in a harmonic oscillator with fractional-order external and intrinsic dampings (2015)
  19. Chen, Liping; He, Yigang; Chai, Yi; Wu, Ranchao: New results on stability and stabilization of a class of nonlinear fractional-order systems (2014)
  20. Ren, Jincheng; Sun, Zhi-Zhong: Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations (2014)

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