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Finite element method for space-time fractional diffusion equation

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-planck equation. J. Comput. Appl. Math. 16, 209–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Reports 371, 461–580 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  10. Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deng, W.: Finite element method for the space and time fractional Fokker-Plank equation. SIAM J. Numer. Anal. 47, 204–206 (2008)

    Article  MathSciNet  Google Scholar 

  18. Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zheng, Y., Li, C., Zhao, Z.: A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker-planck equation. Mathematical problems in engineering (2010)

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    L. B. Feng & P. Zhuang

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001., Australia

    F. Liu & I. Turner

  3. School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia

    Y. T. Gu

  4. Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen, 361005, China

    P. Zhuang

Authors
  1. L. B. Feng
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  2. P. Zhuang
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  3. F. Liu
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  4. I. Turner
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  5. Y. T. Gu
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Correspondence to F. Liu.

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Feng, L.B., Zhuang, P., Liu, F. et al. Finite element method for space-time fractional diffusion equation. Numer Algor 72, 749–767 (2016). https://doi.org/10.1007/s11075-015-0065-8

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  • DOI: https://doi.org/10.1007/s11075-015-0065-8

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