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Two finite difference schemes for time fractional diffusion-wave equation

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Abstract

Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in temporal direction and second-order accuracy in spatial direction. Numerical experiments are carried out to demonstrate the theoretical analysis.

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References

  1. Agrawal, O.P.: Response of a diffusion-wave system subjected to deterministic and stochastic fields. Z. Angew. Math. Mech. 83(4), 265–274 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P.: Limit theorem for continuous-time random walks with two time scales. J. Appl. Prob. 41, 455–466 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C., Liu, F., Turner, I., Anh, V.: A Fourier analysis method for the fractional diffusion equation describing sub-diffusion. J. Comput Phys. 227(2), 886–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, C., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81, 345–366 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  6. Du, R., Cao, W.R., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34(10), 2998–3007 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finance. III, The diffusion limit. In: Mathematical Finance, Trends in Math, pp. 171–180. Birkhäuser, Basel (2001)

    Google Scholar 

  8. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 229–307 (1984)

    Article  MathSciNet  Google Scholar 

  10. 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 

  11. Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional differential equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. 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 

  15. Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27(1), 20–31 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mainardi, F.: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals. Springer, Wien (1997)

    Google Scholar 

  18. Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Physica A 370, 114–118 (2006)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  20. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749–755 (2002)

    Article  MATH  Google Scholar 

  21. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivative, Theory and Applications. Gordon and Breach, New York (1993)

    Google Scholar 

  22. Sanz-Serna, J.M.: A numerical method for a partial integro-differential equations. SIAM J. Numer. Anal. 25(2), 319–327 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meerschaert, M.M., Zhang, Y., Baeumerc, B.: Particle tracking for fractional diffusion with two time scales. Comput. Math. Appl. 59, 1078–1086 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11, 309–319 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49(6), 2302–2322 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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  MathSciNet  MATH  Google Scholar 

  29. Zhuang, P., Liu, F., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the nonlinear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74(5), 645–667 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jianfei Huang, Yifa Tang & Jiye Yang

  2. Departamento de Matemática Aplicada, Facultad de Informática, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040, Madrid, Spain

    Luis Vázquez

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  1. Jianfei Huang
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  2. Yifa Tang
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  3. Luis Vázquez
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  4. Jiye Yang
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Correspondence to Jianfei Huang.

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Huang, J., Tang, Y., Vázquez, L. et al. Two finite difference schemes for time fractional diffusion-wave equation. Numer Algor 64, 707–720 (2013). https://doi.org/10.1007/s11075-012-9689-0

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  • DOI: https://doi.org/10.1007/s11075-012-9689-0

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