Numerical Algorithms

, Volume 74, Issue 4, pp 1145–1168 | Cite as

A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation

  • Rezvan SalehiEmail author
Original Paper


In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo’s time fractional derivatives are approximated by a scheme of order O(τ 3−α ), 1< α < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method.


Multi-term time fractional diffusion-wave equation Fractional derivatives Caputo’s derivative Moving least squares reproducing kernel method Meshless methods Convergence and stability 

Mathematics Subject Classification (2010)

65M70 65N35 65M15 65N12 35R11 


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  1. 1.
    Abbaszadeh, M., Dehghan, M.: A meshless numerical procedure for solving fractional reaction–subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method. Comput. Math. Appl. 70, 2493–2512 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atanackovic, T.M., Pilipovic, S., Zorica, D.: A diffusion wave equation with two fractional derivatives of different order. J. Phys. A: Math. Theor. 40, 5319–5333 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atluri, S.N., Shen, S.: The meshless local petrov-galerkin (mlpg) method. Technical Science Press, Encino, CA (2002)zbMATHGoogle Scholar
  4. 4.
    Atluri, S.N., Shen, S.: The basis of meshless domain discretization: the meshless local Petrov-Galerkin (MLPG) method. Adv Comput. Math. 23, 73–93 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Belystchko, T., Liu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Meth. Eng. 37, 229–256 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection–dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  7. 7.
    Burrage, K., Hale, N., Kay, D.: An efficient implementation of an implicit FEM scheme for fractional-in-space reaction–diffusion equations. SIAM J. Sci. Comput. 34, 2145–2172 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, W., Pang, G.: A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J. Comput. Phys. 309, 350–367 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine–Gordon and Klein–Gordon equations. Eng. Anal. Bound. Elem. 50, 412–434 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Error estimate for the numerical solution of fractional reaction–subdiffusion process based on a meshless method. J. Comput. Appl. Math. 280, 14–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dehghan, M., Safarpoor, M., Abbaszadeh, M.: Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J. Comput. Appl. Math. 290, 174–195 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ding, X.L., Jiang, Y.L.: Analytical solutions for the multi-term time-space fractional advection–diffusion equations with mixed boundary conditions. Nonlin. Anal. RWA. 14, 1026–1033 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duarte, C.A., Oden, J.T.: H-p clouds—an h-p meshless method. Numer. Meth. Partial Diff. Eq. 12, 673–705 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fu, Z.-J., Chen, W., Yang, H.-T.: Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. phys. 235, 52–66 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fu, Z.-J., Chen, W., Ling, L.: Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Engng. Anal. Bound. Elem. 57, 37–46 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gao, G., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astr. Soc. 181, 375–389 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous–time finance III: The diffusion limit. In: Mathematical Finance, pp 171–80. Mathematics of Birkhauser, Basel (2001)CrossRefGoogle Scholar
  22. 22.
    Gu, Y.T., Zhaung, P., Liu, F.: An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput. Model. Eng. Sci. 56, 303–333 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Han, W., Meng, X.: Error analysis of the reproducing kernel particle method. Comput. Meth. Appl. Mech. Eng. 190, 6157–6181 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hilfer, R.: Applications of fractional calculus in physics. World Scientific, Singapore (2000)CrossRefzbMATHGoogle Scholar
  25. 25.
    Jiang, Y., Ma, J.: High-order finite element methods for time fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi–term time fractional diffusion–wave/diffusion equations in a finite domain. Comput. Math. Appl. 64, 3377–3388 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain. J. Math. Anal. Appl. 389, 1117–1127 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jiang, H., Liu, F., Meerschaert, M.M., McGough, R.J., Liu, Q.: The fundamental solutions for multi-term modified power law wave equations in a finite domain. Electron. J. Math. Anal. Appl. 1, 1–12 (2013)Google Scholar
  29. 29.
    Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. Comput. Math. Appl. 19, 127–145 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Katsikadelis, J.T.: Numerical solution of multi-term fractional differential equations. J. Appl. Math. Mech. 89, 593–608 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kelly, J.F., McGough, R.J., Meerschaert, M.M.: Analytical time-domain Green’s functions for power-law media. J. Acoust. Soc. Am. 124, 2861–2872 (2008)CrossRefGoogle Scholar
  32. 32.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equation. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  33. 33.
    Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 229–307 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kwon, K.C., Park, S.H., Jiang, B.N., Youn, S.K.: The least-squares meshfree method for solving linear elastic problems. Comput. Mech. 30, 196–211 (2003)CrossRefzbMATHGoogle Scholar
  35. 35.
    Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Inter. J. Numer. Meth. Flui. 20, 1081–1106 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liu, W.K., Li, S., Belytschko, T.: Moving least-square reproducing kernel methods (I) Methodology and convergence. Comput. Meth. Appl. Mech. Eng. 143, 113–154 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Li, S., Liu, W.K.: Moving least square reproducing kernel method part II: Fourier analysis. Comput. Meth. Appl. Mech. Eng. 139, 159–194 (1996)CrossRefzbMATHGoogle Scholar
  38. 38.
    Li, S., Liu, W.K.: Meshfree particle methods. Springer, Berlin (2007)zbMATHGoogle Scholar
  39. 39.
    Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35, 4103–4116 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Liu, Q., Gu, Y.T., Zhuang, P., Liu, F., Nie, Y.F.: An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48, 1–12 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Liu, Q., Liu, F., Turner, I., Anh, V., Gu, Y.T.: A RBF meshless approach for modeling a fractal mobile/immobile transport model. Appl. Math. Comput. 226, 336–347 (2014)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Liu, Q., Liu, F., Gu, Y.T., Zhuang, P., Chen, J., Turner, I.: A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation. Appl. Math. Comput. 256, 930–938 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Lin, Y., Xu, C.: Finite difference/spectral approximation for the time–fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Luchko, Y.: Some uniqueness and existence results for the initial–boundary–value problems for the generalized time–fractional diffusion equation. Comput. Math Appl. 59, 1766–1772 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time–fractional diffusion equation. J. Math. Anal. Appl. 374, 538–548 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Melenk, J.M., Babuska, I.: The partition of unity finite element method: basic theory and applications. Comput. Meth. Appl. Mech. Eng. 139, 289–314 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Miller, K.S., Ross, B.: An introduction to fractional calculus and fractional differential equations (1974)Google Scholar
  49. 49.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximation for two-sided space–fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Phys. A 278, 107–125 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 301–318 (1992)CrossRefzbMATHGoogle Scholar
  52. 52.
    Nigmatullin, R.R.: To the theoretical explanation of the universal response. Phys. Status (B): Basic Res. 123, 739–745 (1984)CrossRefGoogle Scholar
  53. 53.
    Nigmatullin, R.R.: Realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status (B): Basic Res. 133, 425–430 (1986)CrossRefGoogle Scholar
  54. 54.
    Oldham, K.B., Spanier, J.: The fractional calculus: theory and application of differentiation and integration of arbitrary order. Academic Press, New York London (1974)zbMATHGoogle Scholar
  55. 55.
    Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L., Sacco, C.: A finite point method for analysis of fluid mechanics problems. Applications to convective transport and fluid flow. Int. J. Numer. Meth. Eng. 39, 3839–3866 (1996)CrossRefzbMATHGoogle Scholar
  56. 56.
    Pedas, A., Tamme, E.: Spline collocation methods for linear multi-term fractional differential equations. J. Comput. Appl. Math. 236, 167–176 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  58. 58.
    Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Phys. A 314, 749–755 (2002)CrossRefzbMATHGoogle Scholar
  59. 59.
    Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62, 1135–1142 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Scher, H., Montroll, E.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 24–55 (1975)CrossRefGoogle Scholar
  61. 61.
    Schiessel, H., Metzler, R., Blumen, A., Nonnenmacher, T.F.: Generalized viscoelastic models: their fractional equations with solutions. J. Phys. A: Math. Gen. 28, 6567–6584 (1995)CrossRefzbMATHGoogle Scholar
  62. 62.
    Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616–624 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behaviour of real materials. J. Appl. Mech. 51, 294–298 (1984)CrossRefzbMATHGoogle Scholar
  65. 65.
    Wang, S., Zhang, H.: Partition of unity-based thermomechanical meshfree method for two-dimensional crack problems. Arch. Appl. Mech. 81, 1351–1363 (2011)CrossRefzbMATHGoogle Scholar
  66. 66.
    Wei, L., He, Y.: Analysis of a fully discrete local discontinuous Galerkin method for time–fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Yang, J.Y., Zhao, Y.M., Liu, N., Bu, W.P., Xu, T.L., Tang, Y.F.: An implicit MLS meshless method for 2-D time dependent fractional diffusion–wave equation. Appl. Math. Model. 39, 1229–1240 (2015)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Ye, H., Liu, F., Anh, V., Turner, I.: Maximum principle and numerical method for the multi–term time–space Riesz–Caputo fractional differential equations. Appl. Math. Model. 227, 531–540 (2014)MathSciNetGoogle Scholar
  69. 69.
    Zeng, F., Li, C., 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)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Zeng, F., Liu, F., Li, C.P., Burrage, K., Turner, I., Anh, V.: A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time–fractional diffusion equations. Appl. Math. Model. 40, 4970–4985 (2016)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Zhuang, P., Gu, Y.T., Liu, F., Turner, I., Yarlagadda, P.K.D.V.: Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method. Int. J. Numer. Meth. Eng. 88, 1346–62 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Zhuang, P., Liu, F., Turner, I., Gu, Y.T.: Finite element methods for solving a one-dimensional space–fractional Boussinesq equation. Appl. Math Model. 38, 3860–3870 (2014)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

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