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Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

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Abstract

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

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References

  1. Compte, A.: Continuous time random walks on moving fluids. Phys. Rew. E 55, 6821–6831 (1997)

    Article  Google Scholar 

  2. Compte, A., Caceres, M.O.: Fractional dymamics in random velocity fields. Phys. Rev. Lett. 81, 3140–3143 (1998)

    Article  Google Scholar 

  3. Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000)

    Article  MathSciNet  Google Scholar 

  4. Baeumer, B., Meerschaert, M.M., Benson, D.A., Wheatsraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543–1550 (2001)

    Article  Google Scholar 

  5. Chen, C.-M., Liu, F., Turner, I., Anh, V.: A new Fourier analysis method for the Galilei invariant fractional advection diffusion equation. ANZIAM J. 48(CTAC2006), C775–C789 (2007)

    MathSciNet  Google Scholar 

  6. Chen, C.-M., Liu, F., Turner, I., Anh, V.: Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, C.-M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, C.-M., Liu, F., Anh, V.: Numerical analysis of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl. Math. Comput. 204, 340–351 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, C.-M., Liu, F., Anh, V.: A Fourier method and an extrapolation technique for Stokes first problem for a heated generalized second grade fluid with fractional derivative. J. Comput. Appl. Math. 223, 777–789 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Enson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-despersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)

    Article  Google Scholar 

  12. Ervin, V.S., Roop, J.P., Paul, J.: Variational solution of fractional advection dispersion equations on bounded domains in R-d. Numer. Meth. Part. D. E. 23, 256–281 (2007)

    Article  MATH  Google Scholar 

  13. Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)

    MATH  Google Scholar 

  14. Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46(E), 488–504 (2005)

    MathSciNet  Google Scholar 

  15. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)

    Article  MathSciNet  Google Scholar 

  16. Cushman, J.H., Cinn, T.R.: Fractional advection-disperision equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res. 36, 3763–3766 (2000)

    Article  Google Scholar 

  17. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2. J. Comput. Appl. Math. 193, 243–268 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165–2174 (2003)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Meerschaert, M.M., Nane, E., Xiao, Y.: Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432–445 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in randam media. Physica A 185, 87–97 (1992)

    Article  Google Scholar 

  22. Zhuang, P., Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22(3), 87–99 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46(2), 1079–1095 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Metzler, R., Klafter, J., Sokolov, I.M.: Anomalous transport in external fields: continuous time random walksand fractional diffusion equations extended. Phys. Rev. E 58, 1621–1633 (1998)

    Article  Google Scholar 

  25. Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)

    Article  Google Scholar 

  26. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)

    Article  MathSciNet  Google Scholar 

  28. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284, 521–541 (2002)

    Article  Google Scholar 

  29. Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von neumman-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shen, S., Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion. ANZIAM J. 46(E), 871–887 (2005)

    MathSciNet  Google Scholar 

  31. Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205, 719–736 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Balakrishnan, V.: Anomalous diffusion in one dimension. Physica A 132, 569–580 (1985)

    Article  MATH  Google Scholar 

  33. McLean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. (2009). doi:10.1007/s11075-008-9258-8

    MathSciNet  Google Scholar 

  34. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhang, Y., Meerschaert, M.M., Baeumer, B.: Particle tracking for time-fractional diffusion. Phys. Rev. E 78, 036705 (2008)

    Article  Google Scholar 

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Authors and Affiliations

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

    Chang-Ming Chen

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

    Fawang Liu, Ian Turner & Vo Anh

Authors
  1. Chang-Ming Chen
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  2. Fawang Liu
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  3. Ian Turner
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  4. Vo Anh
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Correspondence to Fawang Liu.

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Chen, CM., Liu, F., Turner, I. et al. Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer Algor 54, 1–21 (2010). https://doi.org/10.1007/s11075-009-9320-1

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  • DOI: https://doi.org/10.1007/s11075-009-9320-1

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