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Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave Equation

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Abstract

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton–Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection–reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.

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References

  1. Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S.: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. (2014, in press)

  2. Cao, J.Y., Xu, C.J.: A high order schema for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., Liao, C.: The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219, 1737–1748 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockburn, B., Mustapha, K.: A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. (2014). doi:10.1007/s00211-014-0661-x

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Diethelm, K., Ford, N.J.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diethelm, K., Ford, N.J., Freed, A.D., Weilbeer, M.: Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math. 186, 482–503 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, H.F., Li, C.P.: Numerical algorithms for the fractional diffusion-wave equation with reaction term. Abstr. Appl. Anal. 2013, 493406 (2013)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Hanygad, A.: Multidimensional solutions of time-fractional diffusion-wave equations. Proc. R. Soc. Lond. A 458, 933–957 (2002)

    Article  Google Scholar 

  11. Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64, 707–720 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jafari, M.A., Aminataei, A.: An algorithm for solving multi-term diffusion-wave equations of fractional order. Comput. Math. Appl. 62, 1091–1097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jafari, H., Momani, S.: Solving fractional diffusion and wave equations by modified homotopy perturbation method. Physics Letters A 370, 388–396 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jin, B., Lazarov, R., Zhou, Z.: On two schemes for fractional diffusion and diffusion-wave equations. arXiv:1404.3800 (2014)

  15. Li, C.P., Zeng, F.H.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, L.M., Xu, D., Luo, M.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471–485 (2013)

    Article  MathSciNet  Google Scholar 

  17. Lin, R., Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear Anal. 66, 856–869 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9–25 (2013)

  19. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lubich, C.: A stability analysis of convolution quadratures for Abel-Volterra integral equations. IMA J. Numer. Anal. 6, 87–101 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luchko, Y., Mainardi, F., Povstenko, Y.: Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation. Comput. Math. Appl. 66, 774–784 (2013)

    Article  MathSciNet  Google Scholar 

  22. Mainardi, F.: The time fractional diffusion-wave equation. Radiophys. Quantum Electron. 38, 13–24 (1995)

    Article  MathSciNet  Google Scholar 

  23. Mao, Z., Xiao, A.G., Yu, Z.G., Shi, L.: Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations. Sci. World J. 2014, 143983 (2014)

    Google Scholar 

  24. McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Metzler, R., Nonnenmacher, T.F.: Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation. Chem. Phys. 284, 67–90 (2002)

    Article  Google Scholar 

  26. Murillo, J.Q., Yuste, S.B.: An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form. J. Comput. Nonlinear Dyn. 6, 021014 (2011)

    Article  Google Scholar 

  27. Murillo, J.Q., Yuste, S.B.: A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations. Eur. Phys. J. Spec. Top. 222, 1987–1998 (2013)

    Article  Google Scholar 

  28. Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Podlubny, I.: Fractional Differential Equations. Acdemic Press, San Dieg (1999)

    MATH  Google Scholar 

  30. Ren, J.C., Sun, Z.Z.: Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions. J. Sci. Comput. 56, 381–408 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. 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 

  33. Sweilam, N.H., Khader, M.M., Adel, M.: On the stability analysis of weighted average finite difference methods for fractional wave equation. Fract. Differ. Calc. 2, 17–29 (2012)

    Article  MathSciNet  Google Scholar 

  34. Vázquez, L.: From Newton’s equation to fractional diffusion and wave equations. Adv. Differ. Equ. 2011, 169421 (2011)

    Article  Google Scholar 

  35. Yang, J.Y., Huang, J.F., Liang, D.M., Tang, Y.F.: Numerical solution of fractional diffusion-wave equation based on fractional multistep method. Appl. Math. Model. 38, 3652–3661 (2014)

    Article  MathSciNet  Google Scholar 

  36. Zeng, F.H., Li, C.P., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zeng, F.H., Li, C.P., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. (2014, in press)

  38. Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the anonymous referees for their constructive comments and suggestions which led to an improved presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Fanhai Zeng

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Fanhai Zeng

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  1. Fanhai Zeng
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Correspondence to Fanhai Zeng.

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This research is supported by National Natural Science Foundation of China (No. 11171256).

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Zeng, F. Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave Equation. J Sci Comput 65, 411–430 (2015). https://doi.org/10.1007/s10915-014-9966-2

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  • DOI: https://doi.org/10.1007/s10915-014-9966-2

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