Numerical Algorithms

, Volume 73, Issue 2, pp 445–476 | Cite as

Analysis of a meshless method for the time fractional diffusion-wave equation

  • Mehdi Dehghan
  • Mostafa Abbaszadeh
  • Akbar Mohebbi
Original Paper


In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.


Time fractional diffusion-wave equation Fractional derivative Convergence analysis Error estimate Caputo derivative Meshless Galerkin method Radial basis functions 

Mathematics Subject Classification (2010)

65M70 34A34 


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  1. 1.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional calculus models and numerical methods. Series on complexity, nonlinearity and chaos. World Scientific, Boston (2012)CrossRefMATHGoogle Scholar
  2. 2.
    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. 293, 142–156 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezis, H.: Functional analysis Sobolev spaces and partial differential equations springer new york dordrecht heidelberg london (2011)Google Scholar
  4. 4.
    Bagley, R., Torvik, P.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)CrossRefMATHGoogle Scholar
  5. 5.
    Cai, Z.: Convergence and error estimates for meshless Galerkin methods. Appl. Math. Comput. 184, 908–916 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, C.M., Liu, F., Burrage, K.: Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dacorogna, B.: Introduction to the calculus of variations. Imperial College Press, London (2004)CrossRefMATHGoogle Scholar
  9. 9.
    Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16–30 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dehghan, M., Mirzaei, D.: The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrodinger equation. Eng. Anal. Bound. Elem. 32, 747–756 (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Dehghan, M., Salehi, R.: The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math. 236, 2367–2377 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dehghan, M., Manafian, J., Saadatmandi, A.: The solution of the linear fractional partial differential equations using the homotopy analysis method. Zeitschrift fur Naturforschung - Section A 65, 935–949 (2010)MATHGoogle Scholar
  13. 13.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Duan, Y., Tan, Y.J.: Meshless Galerkin method based on regions partitioned into subdomains. Appl. Math. Comput. 162, 317–327 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Applic. 59(3), 1326–1336 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Esmaeili, S., Shamsi, M., Luchko, Y.: Numerical solution of fractional differential equations with a collocation method based on Mntz polynomials. Comput. Math. Appl. 62, 918–929 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fasshauer, G. E.: Meshfree approximation methods with MATLAB, USA World Scientific (2007)Google Scholar
  19. 19.
    Gu, Y.T., Zhuang, P., Liu, Q.: An advanced meshless method for time fractional diffusion equation. Int. J. Comput. Methods (CMES) 8, 653–665 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gu, Y.T., Zhuang, P., Liu, F.: An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput. Model. Eng. Sci. CMES 56, 303–334 (2010)MathSciNetMATHGoogle Scholar
  21. 21.
    Franke, C., Schaback, R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 381–399 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jiang, H., Liu, F., Turner, I., Burrag, K.: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64, 3377–3388 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kansa, E.J.: Multiquadrics A scattered data approximation scheme with applications to computational fluid-dynamics I. Comput. Math. Appl. 19, 127–145 (1990)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kansa, E.J.: Multiquadrics A scattered data approximation scheme with applications to computational fluid dynamics - II. Comput. Math. Appl. 19, 147–161 (1990)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kansa, E.J., Aldredge, R.C., Ling, L.: Numerical simulation of two–dimensional combustion using mesh-free methods. Eng. Anal. Bound. Elem. 33, 940–950 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Li, J., Chen, Y.: Computational partial differential equations using MATLAB. CRC Press, Boca Raton (2008)Google Scholar
  27. 27.
    Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Liu, Q., Gu, Y., Zhuang, P., Liu, F., Nie, Y.: An implicit, RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48, 1–12 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Li, X.L., Zhu, J.L.: Galerkin boundary node method and its convergence analysis. J. Comput. Appl. Math. 230, 314–328 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Li, X.L.: Meshless Galerkin algorithms for boundary integral equations with moving least square approximations. Appl. Numer. Math. 61, 1237–1256 (2011)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37, R161–208 (2004)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Miller, K.S., Ross, B.: An introductional the fractional calculus and fractional differential equations. Academic Press, New York and London (1974)Google Scholar
  34. 34.
    Mirzaei, D., Dehghan, M.: A meshless based method for solution of integral equations. Appl. Numer. Math. 60, 245–262 (2010)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Mohebbi, A., Dehghan, M.: The use of compact boundary value method for the solution of two-dimensional Schrödinger equation. J. Comput. Appl. Math. 225, 124–134 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mohebbi, A., Abbaszadeh, M., Dehghan, M.: A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. J Comput. Phy. 240, 36–48 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Murillo, J.Q., Yuste, S.B.: An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form. J. Comput. Nonl. Dynam. 6, 021–014 (2011)Google Scholar
  38. 38.
    Momani, S., Odibat, Z.M.: Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics. J. Appl. Math. Comput. 24, 167–178 (2007)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Odibat, Z.M.: Computational algorithms for computing the fractional derivatives of functions. Math. Comput. Simul. 79, 2013–2020 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Oldham, K.B., Spanier, J.: The fractional calculus: Theory and application of differentiation and integration to arbitrary order academic press (1974)Google Scholar
  41. 41.
    Oldham, K.B., Spanier, J.: The fractional calculus. Academic Press, New York and London (1974)MATHGoogle Scholar
  42. 42.
    Podulbny, I.: Fractional differential equations. Academic Press, New York (1999)Google Scholar
  43. 43.
    Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer-Verlag, New York (1997)MATHGoogle Scholar
  44. 44.
    Shokri, A., Dehghan, M.: Meshless method using radial basis functions for the numerical solution of two–dimensional complex Ginzburg-Landau equation. Comput. Model. Eng. Sci. CMES 34, 333–358 (2012)MathSciNetGoogle Scholar
  45. 45.
    Shokri, A., Dehghan, M.: A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation. Comput. Phys. Commun. 181, 1990–2000 (2010)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    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
  47. 47.
    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
  48. 48.
    Wendland, H.: Scattered Data Approximation. In: Cambridge Monograph on Applied and Computational Mathematics, Cambridge University Press (2005)Google Scholar
  49. 49.
    Wendland, H.: Meshless Galerkin methods using radial basis functions. Math. Comput. 68, 1521–1531 (1999)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Wendland, H.: Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93, 258–272 (1998)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Wess, W.: The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1996)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Wu, Z., Shaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Yang, J.Y., Zhao, Y.M., Liu, N., Bu, W.P., Xu, T.L., Tang, Y.F., An implicit, M L S: Meshless method for 2D time dependent fractional diffusion-wave equation. Appl. Math. Model. 39, 1229–1240 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Yuste, S.B., Acedo, L.: An, explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Zhao, X., Sun, Z.Z.: Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 1–25 (2014)MathSciNetGoogle Scholar
  57. 57.
    Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: Algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mehdi Dehghan
    • 1
  • Mostafa Abbaszadeh
    • 2
  • Akbar Mohebbi
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical ScienceUniversity of KashanKashanIran

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