Nonlinear Dynamics

, Volume 69, Issue 1–2, pp 459–471 | Cite as

Numerical solution of the regularized long wave equation using nonpolynomial splines

  • N. G. Chegini
  • A. Salaripanah
  • R. Mokhtari
  • D. Isvand
Original Paper


In this paper, we employ nonpolynomial spline (NPS) basis functions to obtain approximate solutions of the regularized long wave (RLW) equation. By considering suitable relevant parameters, it is shown that the local truncation error behaves O(k 2+h 2) with respect to the time and space discretization. Numerical stability of the method is investigated by using a linearized stability analysis. To illustrate the applicability and efficiency of the aforementioned basis, we compare obtained numerical results with other existing recent methods. Motion of single solitary wave and double and triple solitary waves, wave undulation, generation of solitary waves using the Maxwellian initial condition and conservation properties of mass, energy, and momentum of numerical solutions of the equation are dealt with.


Nonpolynomial splines Regularized long wave equation Solitary waves 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulloev, K.O., Bogolubsky, I.L., Markhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56, 427–428 (1976) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Islam, S.U., Haq, S., Ali, A.: A meshfree method for the numerical solution of the RLW equation. J. Comput. Appl. Math. 223, 997–1012 (2009) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bona, J.L., Bryant, P.J.: A mathematical model for long waves generated by wave makers in nonlinear dispersive systems. Proc. Camb. Philos. Soc. 73, 391–405 (1973) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Mokhtari, R., Mohammadi, M.: Numerical solution of GRLW equation using Sinc-collocation method. Comput. Phys. Commun. 181, 1266–1274 (2010) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Mohammadi, M., Mokhtari, R.: Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J. Comput. Appl. Math. 235, 4003–4014 (2011) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Mokhtari, R., Torabi Ziaratgahi, S.: Numerical solution of RLW equation using integrated radial basis functions. Int. J. Appl. Comput. Math. 10, 428–448 (2011) MATHGoogle Scholar
  7. 7.
    Griewanka, A., El-Danaf, T.S.: Efficient accurate numerical treatment of the modified Burgers’ equation. Appl. Anal. 88, 75–87 (2009) MathSciNetCrossRefGoogle Scholar
  8. 8.
    El-Danaf, T.S., Ramadan, M.A., Abd-Alaal, F.E.I.: Numerical studies of the cubic non-linear Schrödinger equation. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-0014-6
  9. 9.
    Biswas, A.: Solitary waves for power-law regularized long-wave equation and R(m,n) equation. Nonlinear Dyn. 59, 423–426 (2010) MATHCrossRefGoogle Scholar
  10. 10.
    Biswas, A., Kara, A.H.: Conservation laws for regularized long wave equation and R(m,n) equation. Adv. Sci. Lett. 4, 168–170 (2011) CrossRefGoogle Scholar
  11. 11.
    Mokhtari, R., Mohammadi, M.: New exact solutions to a class of coupled nonlinear PDEs. Int. J. Nonlinear Sci. Numer. Simul. 10, 779–796 (2009) CrossRefGoogle Scholar
  12. 12.
    Raslan, K.R.: The first integral method for solving some important nonlinear partial differential equations. Nonlinear Dyn. 53, 281–286 (2008) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dehghan, M., Shokri, A.: A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn. 50, 111–120 (2007) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mokhtari, R.: Exact solutions of the Harry-Dym equation. Commun. Theor. Phys. 55, 204–208 (2011) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mokhtari, R., Samadi Toodar, A., Chegini, N.G.: Numerical simulation of coupled nonlinear Schrödinger equations using the generalized differential quadrature method. Chin. Phys. Lett. 28, 020202 (2011). doi: 10.1088/0256-307X/28/2/020202 CrossRefGoogle Scholar
  16. 16.
    Mokhtari, R., Samadi Toodar, A., Chegini, N.G.: Application of the generalized differential quadrature method in solving Burgers’ equations. Commun. Theor. Phys. 56, 1009–1015 (2011) CrossRefGoogle Scholar
  17. 17.
    Dag, I., Dogan, A., Saka, B.: B-spline collocation methods for numerical solutions of the RLW equation. Int. J. Comput. Math. 80, 743–757 (2003) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Dag, I., Saka, B., Irk, D.: Galerkin method for the numerical solution of the RLW equation using quintic B-splines. J. Comput. Appl. Math. 190, 532–547 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Esen, A., Kutluay, S.: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174, 833–845 (2006) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Korkmaz, A.: Numerical solutions of some nonlinear partial differential equations using differential quadrature method. Thesis of Master Degree, Eskişehir Osmangazi University, Eskişehir, Turkey (2006) (Unpublished) Google Scholar
  21. 21.
    Korkmaz, A.: Numerical solutions of some one dimensional partial differential equations using B-spline differential quadrature methods. PhD. Dissertation, Eskişehir Osmangazi University, Eskişehir, Turkey (2010) (Unpublished) Google Scholar
  22. 22.
    Korkmaz, A.: Numerical algorithms for solutions of Korteweg-de Vries equation. Numer. Methods Partial Differ. Equ. 26, 1504–1521 (2010) MathSciNetMATHGoogle Scholar
  23. 23.
    Korkmaz, A., Dag, I.: A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dyn. 56, 69–83 (2009) MATHCrossRefGoogle Scholar
  24. 24.
    Korkmaz, A., Dag, I.: A differential quadrature algorithm for simulations of nonlinear Schrödinger equation. Comput. Math. Appl. 56, 2222–2234 (2008) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Korkmaz, A., Dag, I.: Solitary wave simulations of complex modified Korteweg-de Vries equation using differential quadrature method. Comput. Phys. Commun. 180, 1516–1523 (2009) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Korkmaz, A., Dag, I.: Crank-Nicolson differential quadrature algorithms for the Kawahara equation. Chaos Solitons Fractals 42, 65–73 (2009) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Korkmaz, A., Dag, I., Saka, B.: Cosine expansion based differential quadrature (CDQ) algorithms for numerical solution of the RLW equation. Numer. Methods Partial Differ. Equ. 26, 544–560 (2010) MathSciNetMATHGoogle Scholar
  28. 28.
    Saka, B., Dag, I., Dogan, A.: Galerkin method for the numerical solution of the RLW equation using quadratic B-spline. Int. J. Comput. Math. 81, 727–739 (2004) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Saka, B., Dag, I.: A Collocation method for the numerical solution of the RLW equation using cubic B-spline basis. Arab. J. Sci. Eng. 30, 39–50 (2005) MathSciNetGoogle Scholar
  30. 30.
    Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Comput. Phys. Commun. 181, 78–91 (2010) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rashidinia, J., Mohammadi, R.: Numerical methods based on non-polynomial sextic spline for solution of variable coefficient fourth-order wave equations. Int. J. Comput. Methods Eng. Sci. Mech. 10, 266–276 (2009) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181, 1868–1872 (2010) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • N. G. Chegini
    • 1
  • A. Salaripanah
    • 1
  • R. Mokhtari
    • 2
  • D. Isvand
    • 1
  1. 1.Department of MathematicsTafresh UniversityTafreshIran
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

Personalised recommendations