Afrika Matematika

, Volume 29, Issue 7–8, pp 1203–1214 | Cite as

An efficient meshless method based on RBFs for the time fractional diffusion-wave equation

  • Mohammad AslefallahEmail author
  • Elyas Shivanian


This paper proposes a numerical method to deal with time-fractional diffusion-wave equation (one-dimensional and two-dimensional). The time-fractional term of the problem is scheduled in Caputo sense which is popular in analyzing time-fractional dependent problems. The proposed technique is based on radial basis functions and more, it is a kind of meshless method, therefore it is not difficult applying the method to handle two or three dimensional time-fractional diffusion-wave problems especially when the domain are more general and not regular forms. The generalized thin plate splines (GTPS) radial basis functions are employed. Numerical examples are given to test the accuracy. Three numerical experiments reveal that proposed method is very convenient for solving such problems.


Radial basis function Fractional PDE Diffusion-wave equation Kansa’s method Finite differences \(\theta \)-method 

Mathematics Subject Classifcation

65M06 65N12 26A33 


  1. 1.
    Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finance III: the diffusion limit. In: Kohlmann, M., Tang, S. (eds.) Mathematical Finance. Trends in Mathematics, pp. 171–180. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  2. 2.
    Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B 27, 273–275 (2002)MathSciNetGoogle Scholar
  3. 3.
    Aslefallah, M., Rostamy, D.: A numerical scheme for solving space-fractional equation by finite differences theta-method. Int. J. Adv. Aply. Math. Mech. 1(4), 1–9 (2014)zbMATHGoogle Scholar
  4. 4.
    Aslefallah, M., Rostamy, D.: Numerical solution for Poisson fractional equation via finite differences theta-method. J. Math. Com. Sci. TJMCS 12(2), 132–142 (2014)CrossRefGoogle Scholar
  5. 5.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  6. 6.
    Aslefallah, M., Shivanian, E.: Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions. Eur. Phys. J. Plus 130(3), 1–9 (2015)CrossRefGoogle Scholar
  7. 7.
    Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains. J. Comput. Appl. Math. 193(1), 243–268 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Meerschaert, M.M., Tadjeran, C.: Finite Difference Approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zeng, S., Baleanu, D., Bai, Y., Wua, G.: Fractional differential equations of Caputo-Katugampola type and numerical solutions. Appl. Math. Comput. 315, 549–554 (2017). MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ahmadian, A., Ismail, F., Salahshour, S., Baleanu, D., Ghaemi, F.: Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution. Commun. Nonlinear Sci. Numer. Simul. 53, 44–64 (2017). MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kumar, D., Singh, J., Baleanu, D.: A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math. Meth. Appl. Sci. 40, 5642–5653 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo-Cheng, Wu, Baleanu, Dumitru, Luo, Wei-Hua: Lyapunov functions for RiemannLiouville-like fractional difference equations. Appl. Math. Comput. 314, 228–236 (2017). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ciment, M., Leventhal, S.H.: Higher order compact implicit schemes for the wave equation. Math. Comp. 29, 985–994 (1975)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ciment, M., Leventhal, S.H.: A note on the operator compact implicit method for the wave equation. Math. Comp. 32, 143–147 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dahlquist, G.: On accuracy and unconditional stability of linear multi-step methods for second order differential equations. BIT 18, 133–136 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mohanty, R.K., Jain, M.K., Arora, U.: An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions. Int. J. Comput. Math. 79, 133–142 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, G., Gu, Y.: An introduction to meshfree methods and their programing. Springer, New York (2005)Google Scholar
  18. 18.
    Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Duarte, C., Oden, J.: An h-p adaptative method using clouds. Comput. Methods Appl. Mech. Eng. 139, 237–262 (1996)CrossRefGoogle Scholar
  20. 20.
    Atluri, S.N., Zhu, T.L.: A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22(2), 117–127 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Abbasbandy, S., Sladek, V., Shirzadi, A., Sladek, J.: Numerical simulations for coupled pair of diffusion equations by MLPG method. CMES Compt. Model. Eng. Sci. 71(1), 15–37 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shirzadi, A., Ling, L., Abbasbandy, S.: Meshless simulations of the two-dimensional fractional-time convectiondiffusion- reaction equations. Eng. Anal. Bound. Elem. 36, 1522–1527 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhu, T., Zhang, J.D., Atluri, S.N.: A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach. Comput. Mech. 21, 223–235 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Melenk, J.M., Babǔska, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289–314 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kansa, E.J.: Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II. J. Comput. Math. Appl. 19, 147–161 (1990)CrossRefGoogle Scholar
  26. 26.
    Aslefallah, M., Shivanian, E.: A nonlinear partial integro-differential equation arising in population dynamic via radial basis functions and theta-method. J. Math. Com. Sci. TJMCS 13(1), 14–25 (2014)CrossRefGoogle Scholar
  27. 27.
    Hosseini, V.R., Chen, W., Avazzadeh, Z.: Numerical solution of fractional telegraph equation by using radial basis functions. Eng. Anal. Boundary Elem. 38, 31–39 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Abbasbandy, S., Ghehsareh, H.R., Hashim, I.: Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function. Eng. Anal. Boundary Elem. 36(12), 1811–1818 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Abbasbandy, S., Ghehsareh, H.R., Hashim, I.: A meshfree method for the solution of two-dimensional cubic nonlinear schrödinger equation. Eng. Anal. Boundary Elem. 37(6), 885–898 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kansa, E.: Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates. Comput. Math. Appl. 19(8–9), 127–145 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Dehghan, M., Shokri, A.: A numerical method for solution of the two dimensional sine-Gordon equation using the radial basis functions. Mathematics and Computers inSimulation 79, 700–715 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lucy, L.B.: A numerical approach to the testing of fusion process. Astron. J. 88, 1013–1024 (1977)CrossRefGoogle Scholar
  33. 33.
    Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 21, 1081–1106 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Liu, G.R.: Mesh Free Methods: Moving beyond the Finite Element Method. CRC Press, Boca Raton (2003)zbMATHGoogle Scholar
  35. 35.
    Shivanian, E.: Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Eng. Anal. Boundary Elem. 37, 1693–1702 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Dehghan, M., Ghesmati, A.: Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput. Phys. Commun. 181, 772–786 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Shivanian, E., Aslefallah, M.: Stability and convergence of spectral radial point interpolation method locally applied on two-dimensional pseudoparabolic equation. Numer. Methods Partial Differ. Eq. 33, 724–741 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Heydari, M.H., Hooshmandasl, M.R., Maleki Ghaini, F.M., Cattani, C.: Wavelets method for the time fractional diffusion-wave equations. Phys. Lett. A 379, 71–76 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hu, X., Zhang, L.: On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, 5019–5034 (2012)MathSciNetzbMATHGoogle Scholar
  40. 40.
    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. Comput. 219, 1737–1748 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Aslefallah, M., Rostamy, D., Hosseinkhani, K.: Solving time-fractional differential diffusion equation by theta-method. Int. J. Adv. Aply. Math. Mech. 2(1), 1–8 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  43. 43.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsImam Khomeini International UniversityQazvinIran

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