Numerical Algorithms

, Volume 71, Issue 1, pp 151–180 | Cite as

A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation

Original Paper

Abstract

The space-time fractional diffusion-wave equation (FDWE) is a generalization of classical diffusion and wave equations which is used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. This paper reports an accurate spectral tau method for solving the two-sided space and time Caputo FDWE with various types of nonhomogeneous boundary conditions. The proposed method is based on shifted Legendre tau (SLT) procedure in conjunction with the shifted Legendre operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided fractional derivatives. We focus primarily on implementing this algorithm in both temporal and spatial discretizations. In addition, convergence analysis is provided theoretically for the Dirichlet boundary conditions, along with graphical analysis for several special cases using other conditions. These suggest that the Legendre Tau method converges exponentially provided that the data in the given FDWE are smooth. Finally, several numerical examples are given to demonstrate the high accuracy of the proposed method.

Keywords

Fractional diffusion-wave equation Tau method Shifted legendre polynomials Operational matrix Convergence analysis Riesz fractional derivative 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic press, New York (1975)MATHGoogle Scholar
  2. 2.
    Atabakzadeh, M., Akrami, M., Erjaee, G.: Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl. Math. Model. 37(20), 8903–8911 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Behiry, S.: Solution of nonlinear fredholm integro-differential equations using a hybrid of block pulse functions and normalized bernstein polynomials. J. Comput. Appl. Math. 260, 258–265 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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). doi: 10.1016/j.jcp.2014.03.039
  5. 5.
    Bhrawy, A.H., Alofi, A.: The operational matrix of fractional integration for shifted chebyshev polynomials. Appl. Math. Lett. 26(1), 25–31 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bhrawy, A.H., Baleanu, D.: A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72(2), 219–233 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bhrawy, A.H., Baleanu, D., Assas, L.: Efficient generalized laguerre-spectral methods for solving multi-term fractional differential equations on the half line. J. Vib. Control 20, 973–985 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag (2006)Google Scholar
  9. 9.
    Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. newblock J. Comput. Phys. (2014). doi: 10.1016/j.jcp.2014.10.016
  10. 10.
    Chen, C.-M., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81(277), 345–366 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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. 219(4), 1737–1748 (2012)MathSciNetMATHGoogle Scholar
  12. 12.
    Danfu, H., Xufeng, S.: Numerical solution of integro-differential equations by using cas wavelet operational matrix of integration. Appl. Math 194(2), 460–466 (2007)MathSciNetMATHGoogle Scholar
  13. 13.
    Deng, W.: Numerical algorithm for the time fractional fokker–planck equation. J. Comput. Phys. 227(2), 1510–1522 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ding, Z., Xiao, A., Li, M.: Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients. J. Comput. Appl. Math. 233(8), 1905–1914 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Doha, E.H., Bhrawy, A.H., Abd-Elhameed, W.M.: Jacobi spectral Galerkin method for elliptic Neumann problems. Numer. Algor. 50(1), 67–91 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Numerical approximations for fractional diffusion equations via a chebyshev spectral-tau method. Cent. Eur. J. Phys. 11(10), 1494–1503 (2013)Google Scholar
  17. 17.
    Du, R., Cao, W., Sun, Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34(10), 2998–3007 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Par. Diff. Eqs. 22(3), 558–576 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Garg, M., Manohar, P.: Matrix method for numerical solution of space-time fractional diffusion-wave equations with three space variables. Afrika Matematika 25 (1), 161–181 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gorenflo, R., Mainardi, F.: Signalling Problem and Dirichlet-Neumann Map for Time Fractional Diffusion Wave Equations. Freie Universität Berlin, Fachbereich Mathematik und Informatik: Ser. A, Mathematik. Freie Univ., Fachbereich Mathematik und Informatik (1998)Google Scholar
  21. 21.
    Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algor. 64(4), 707–720 (2013)CrossRefMATHGoogle Scholar
  22. 22.
    Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol. 204. Elsevier Science Limited (2006)Google Scholar
  23. 23.
    Labecca, W., Guimarães, O., Piqueira, J.R.C.: Dirac’s formalism combined with complex fourier operational matrices to solve initial and boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 19(8), 2614–2623 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Langlands, T., Henry, B.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Li, C., Zhao, Z., Chen, Y.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62(3), 855–875 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Liang, J., Chen, Y.: Hybrid symbolic and numerical simulation studies of time-fractional order wave-diffusion systems. Int. J. Control 79(11), 1462–1470 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection–dispersion models. Comput. Math. Appl. 64(10), 2990–3007 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62(3), 1055–1067 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Luchko, Y.: Fractional wave equation and damped waves. J. Math. Phys. 54(3), 031505 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mainardi, F., Paradisi, P.: A model of diffusive waves in viscoelasticity based on fractional calculus. In: Proceedings of the 36th IEEE Conference on Decision and Control, 1997., vol. 5, pp. 4961–4966. IEEE (1997)Google Scholar
  32. 32.
    Meerschaert, M.M., Scheffler, H.-P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211(1), 249–261 (2006)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56(1), 80–90 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Momani, S., Odibat, Z., Erturk, V.S.: Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation. Phys. Lett. A 370(5), 379–387 (2007)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mokhtary, P., Ghoreishi, F.: The L 2-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro-differential equations. Numer. Algor. 58(4), 475–496 (2011)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Murillo, J.Q., Yuste, S.B.: On three explicit difference schemes for fractional diffusion and diffusion-wave equations. Phys. Scripta. 2009(T136), 014025 (2009)CrossRefGoogle Scholar
  38. 38.
    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(2), 021014 (2011)CrossRefGoogle Scholar
  39. 39.
    Podlubny, I.: Fractional differential equations, vol. 198. Academic press (1998)Google Scholar
  40. 40.
    Ren, J., Sun, Z.-z.: Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with neumann boundary conditions. J. Sci. Comput. 56(2), 381–408 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Saadatmandi, A.: Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model 38(4), 1365–1372 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Saadatmandi, A., Dehghan, M.: Numerical solution of the one-dimensional wave equation with integral condition. Numerical Methods for Partial Differential Equations 23(2), 282–292 (2007)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62(3), 1135–1142 (2011)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Saadatmandi, A., Dehghan, M., Azizi, M.-R.: The sinc–legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4125–4136 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Sapora, A., Cornetti, P., Carpinteri, A.: Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach. Commun. Nonlinear Sci. Numer. Simul. 18(1), 63–74 (2013)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Sousa, E.: A second order explicit finite difference method for the fractional advection diffusion equation 64(10), 3141–3152 (2012)Google Scholar
  47. 47.
    Sun, Z.-z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Sweilam, N.H., Khader, M.M., Nagy, A.: Numerical solution of two-sided space-fractional wave equation using finite difference method. J. Comput. Appl. Math. 235(8), 2832–2841 (2011)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Szegö, G.: Orthogonal polynomials, volume 23. American Mathematical Society New York (1959)Google Scholar
  50. 50.
    Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.-P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213(1), 205–213 (2006)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Tomovski, ž., Sandev, T.: Fractional wave equation with a frictional memory kernel of mittag-leffler type. Appl. Math. 218(20), 10022–10031 (2012)MathSciNetMATHGoogle Scholar
  53. 53.
    Wang, H., Wang, K., Sircar, T.: A direct O(N log 2 N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095–8104 (2010)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Yousefi, S., Behroozifar, M.: Operational matrices of bernstein polynomials and their applications. Int. J. Syst. Sci. 41(6), 709–716 (2010)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Yousefi, S., Razzaghi, M.: Legendre wavelets method for the nonlinear volterra–fredholm integral equations. Mathem. comput. simul. 70(1), 1–8 (2005)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Zeng, F.: Second-order stable finite difference schemes for the time-fractional diffusion-wave equation. J. Sci. Comput. (2015). doi: 10.1007/s10915-014-9966-2
  57. 57.
    Zhang, Y., Ding, H.: Improved matrix transform method for the riesz space fractional reaction dispersion equation. J. Comput. Appl. Math. 260, 266–280 (2014)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Zhao, X., Sun, Z.-z.: A box-type scheme for fractional sub-diffusion equation with neumann boundary conditions. J. Comput. Phys. 230(15), 6061–6074 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. H. Bhrawy
    • 1
    • 2
  • M. A. Zaky
    • 3
  • R. A. Van Gorder
    • 4
  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  3. 3.Department of Applied MathematicsNational Research CentreGizaEgypt
  4. 4.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations