Energy estimates for two-dimensional space-Riesz fractional wave equation

Original Paper


The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first use the energy method to estimate the one-dimensional space-Riesz fractional wave equation. The stiff matrices are proved to be commutative for two-dimensional case, which ensures to carry out of the priori error estimates and the energy method. Then, the unconditional stability and convergence with the global truncation error \(\mathcal {O}(\tau ^{2}+h^{2})\) are theoretically proved with the constant coefficients and numerically verified.


Riesz fractional wave equation Nonlocal wave equation Priori error estimates Energy method Numerical stability and convergence 


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We would like to thank the anonymous referees for their input to Example 4.2, and for several suggestions and comments that led to much better results and an improved presentation.


  1. 1.
    Bhrawy, A.H., Zaky, M.A., Van Gorder, R.A.: A space-time Legendre spectral tau method for the two-sided space Caputo fractional diffusion-wave equation. Numer. Algorithm 71, 151–180 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, M.H., Deng, W.H.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, M.H., Deng, W.H.: High order algorithm for the time-tempered fractional Feynman-Kac equation. J. Sci. Comput.
  4. 4.
    Chen, M.H., Deng, W.H.: Convergence proof for the multigrid method of the nonlocal model. SIAM J. Matrix Anal. Appl. 38, 869–890 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, C., Thomée, V., Wahlbin, L.B.: Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 198, 587–602 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cuesta, E., Lubich, Ch., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Deng, K.Y., Chen, M.H., Sun, T.L.: A weighted numerical algorithm for two and three dimensional two-sided space fractional wave equations. Appl. Math. Comput. 257, 264–273 (2015)MathSciNetMATHGoogle Scholar
  8. 8.
    Dougls, J.: On the numerical integration of \(u_{xx}+u_{yy}=u_{tt}\) by implicit methods. J. Soc. Ind. Appl. Math. 3, 42–65 (1955)Google Scholar
  9. 9.
    Dougls, J.: Alternating direction methods for three space variables. Numer. Math. 6, 428–453 (1964)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012)MathSciNetMATHGoogle Scholar
  11. 11.
    Garg, M., Manohar, P.: Matrix method for numerical solution of space-time fractional diffusion-wave equations with three space variables. Afr. Mat. 25, 161–181 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hao, Z.P., Lin, G., Sun, Z.Z: A high-order difference scheme for the fractional sub-diffusion equation. Int. J. Comput. Math. 94, 405–426 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hu, J.W., Tang, H.M.: Numerical Methods for Differential Equations. Science Press, Beijing (1999)Google Scholar
  14. 14.
    Ji, C.C., Sun, Z.Z.: A high-order compact finite difference schemes for the fractional sub-diffusion equation. J. Sci. Comput. 64, 959–985 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Laub, A.J.: Matrix Analysis for Scientists and Engineers. SIAM (2005)Google Scholar
  16. 16.
    Liu, F., Meerschaert, M., McGough, R., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9–25 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lubich, Ch: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mainardi, F. In: Carpinteri, A, Mainardi, F (eds.) : Fractal calculus: some basic problems in continuum and statistical mechanics. Springer, Berlin (1997)Google Scholar
  19. 19.
    Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)MathSciNetMATHGoogle Scholar
  20. 20.
    McLean, W., Thomée, V.: Numerical solution of an evolution equation with a positive-type memory term. J. Austral. Math. Soc. Ser. B 35, 23–70 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Mustapha, K., Furati, K., Knio, O.M., Le Maître, O.P.: A finite difference method for space fractional differential equations with variable diffusivity coefficient. arXiv:1706.00971
  23. 23.
    Mustapha, K., Mclean, W.: Superconvergence of a discontinous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 1–12 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATHGoogle Scholar
  26. 26.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (2008)MATHGoogle Scholar
  27. 27.
    Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville drivative. Appl. Numer. Math. 90, 22–37 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sun, Z.Z.: Numerical Methods for Partial Differential Equations. Science Press, Beijing (2005)Google Scholar
  29. 29.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Higher Education Press, Beijing and Springer, Berlin (2010)Google Scholar
  31. 31.
    Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wang, P.D., Huang, C.M.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    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. Modell. 38, 3652–3661 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zeng, F.H.: Second-order stable finite difference schemes for the time-fractional diffusion-wave equation. J. Sci. Comput. 65, 411–430 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    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)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhouPeople’s Republic of China

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