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

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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.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic press, New York (1975)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Bhrawy, A.H., Alofi, A.: The operational matrix of fractional integration for shifted chebyshev polynomials. Appl. Math. Lett. 26(1), 25–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag (2006)

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  13. Deng, W.: Numerical algorithm for the time fractional fokker–planck equation. J. Comput. Phys. 227(2), 1510–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  22. Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol. 204. Elsevier Science Limited (2006)

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Luchko, Y.: Fractional wave equation and damped waves. J. Math. Phys. 54(3), 031505 (2013)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  39. Podlubny, I.: Fractional differential equations, vol. 198. Academic press (1998)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Saadatmandi, A.: Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model 38(4), 1365–1372 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62(3), 1135–1142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sousa, E.: A second order explicit finite difference method for the fractional advection diffusion equation 64(10), 3141–3152 (2012)

  47. Sun, Z.-z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  49. Szegö, G.: Orthogonal polynomials, volume 23. American Mathematical Society New York (1959)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tomovski, ž., Sandev, T.: Fractional wave equation with a frictional memory kernel of mittag-leffler type. Appl. Math. 218(20), 10022–10031 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yousefi, S., Behroozifar, M.: Operational matrices of bernstein polynomials and their applications. Int. J. Syst. Sci. 41(6), 709–716 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yousefi, S., Razzaghi, M.: Legendre wavelets method for the nonlinear volterra–fredholm integral equations. Mathem. comput. simul. 70(1), 1–8 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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. Zhang, Y., Ding, H.: Improved matrix transform method for the riesz space fractional reaction dispersion equation. J. Comput. Appl. Math. 260, 266–280 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    A. H. Bhrawy

  2. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    A. H. Bhrawy

  3. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    M. A. Zaky

  4. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    R. A. Van Gorder

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  1. A. H. Bhrawy
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  3. R. A. Van Gorder
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Bhrawy, A.H., Zaky, M.A. & Van Gorder, R.A. A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer Algor 71, 151–180 (2016). https://doi.org/10.1007/s11075-015-9990-9

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