Numerical Algorithms

, Volume 72, Issue 3, pp 749–767 | Cite as

Finite element method for space-time fractional diffusion equation

  • L. B. Feng
  • P. Zhuang
  • F. Liu
  • I. Turner
  • Y. T. Gu
Original Paper


In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ 2−γ + h 2) and O(τ 2 + h 2), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ 2 + h 2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.


Finite element method Space-time fractional diffusion equation Riesz derivative Caputo derivative Riemann-Liouville derivative 


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  1. 1.
    Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-planck equation. J. Comput. Appl. Math. 16, 209–219 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Liu, F., Zhuang, P., Anh, V., Turner, I.: A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM J. (E) 47, 48–68 (2006)MathSciNetGoogle Scholar
  6. 6.
    Fix, G.J., Roop, J.P.: Least squares finite element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Reports 371, 461–580 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hejazi, H., Moroney, T., Liu, F.: A finite volume method for solving the time-space fractional advection-dispersion equation. In: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14-17, Hohai University, Nanjing, China (MS11, Paper ID 038) (2012)Google Scholar
  9. 9.
    Zhang, H., Liu, F., Anh, V.: Garlerkin finite element approximations of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217, 2534–2545 (2010)MathSciNetMATHGoogle Scholar
  10. 10.
    Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)MATHGoogle Scholar
  11. 11.
    Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in R 2. J. Comput. Appl. Math. 193, 243–268 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chen, M., Deng, W., Wu, Y.: Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation. Appl. Numer. Math. 70, 22–41 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Shen, S., Liu, F., Anh, V.: Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numer. Algorithms 56, 383–403 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Deng, W.: Finite element method for the space and time fractional Fokker-Plank equation. SIAM J. Numer. Anal. 47, 204–206 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhang, Y., Benson, D.A., Meerschaert, M.M., Labolle, E.M.: Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the macrodispersion experiment site data. Water Resour. Res. 43, W05439 (2007)Google Scholar
  20. 20.
    Zeng, F., Li, C., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37(1), 55–78 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space-time spectral method for the time-fractional Fokker-Planck equation. SIAM J. Sci. Comput. 37(2), A701–A724 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zheng, Y., Li, C., Zhao, Z.: A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker-planck equation. Mathematical problems in engineering (2010)Google Scholar
  23. 23.
    Zheng, Y., Li, C., Zhao, Z.: A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • L. B. Feng
    • 1
  • P. Zhuang
    • 1
    • 4
  • F. Liu
    • 2
  • I. Turner
    • 2
  • Y. T. Gu
    • 3
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  3. 3.School of Chemistry, Physics and Mechanical EngineeringQueensland University of TechnologyBrisbaneAustralia
  4. 4.Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputationXiamen UniversityXiamenChina

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