Skip to main content
Springer Nature Link
Log in

Lie Symmetry Reductions and Solitary Wave Solutions of Modified Equal Width Wave Equation

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we obtained the exact and solitary wave solutions of modified equal width wave equation by using Lie symmetry method. With the help of MAPLE software we obtained infinitesimal generators and commutation table. Lie symmetry transformation has been used for converting nonlinear partial differential equation into nonlinear ordinary differential equation. Then, we used tanh method and power series method for solving reduced nonlinear ordinary differential equations. Convergence of power series solution has also been shown.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Karakoc, S.B.G., Geyikli, T.: Numerical solution of the modified equal width wave equation. Int. J. Differ. Equ. 2012, 1–15 (2012)

    Article  MathSciNet  Google Scholar 

  2. Essa, Y.M.A.: Multigrid method for the numerical soluton of the modified equal width wave equation. Appl. Math. 7, 1140–1147 (2016)

    Article  Google Scholar 

  3. Gardner, L.R.T., Gardner, G.A.: Solitary waves of the regularised long-wave equation. J. Comput. Phys. 91(2), 441–459 (1990)

    Article  MathSciNet  Google Scholar 

  4. Gardner, L.R.T., Gardner, G.A.: Solitary waves of the equal width wave equation. J. Comput. Phys. 101(1), 218–223 (1992)

    Article  MathSciNet  Google Scholar 

  5. Morrison, P.J., Meiss, J.D., Cary, J.R.: Scattering of regularized-long-wave solitary waves. Physica D. Nonlinear Phenomena 11(3), 324–336 (1984)

    Article  MathSciNet  Google Scholar 

  6. Gardner, L.R.T., Gardner, G.A., Geyikli, T.: The boundary forced MKdV equation. J. Comput. Phys. 113(1), 5–12 (1994)

    Article  MathSciNet  Google Scholar 

  7. Abdulloev, Kh O., Bogolubsky, I.L., Makhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56(6), 427–428 (1976)

    Article  MathSciNet  Google Scholar 

  8. Geyikli, T., Karakoc, S.B.G.: Septic B-spline collocation method for numerical solution of modified equal width wave equation. Appl. Math. 2(6), 739–749 (2011)

    Article  MathSciNet  Google Scholar 

  9. Geyikli, T., Karakoc, S.B.G.: Petro-Galerkin method with cubic bsplines for solving the MEW equation. Bull. Belgian Math. Soc. Simon Stevin 19, 215–227 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Arora, R., Siddiqui, Md J., Singh, V.P.: Solution of modified equal width wave equation, its variant and non-homogeneous Burgers’ equation by RDT method. Am. J. Comput. Appl. Math. 1(2), 53–56 (2011)

    Article  Google Scholar 

  11. Saka, B.: Algorithms for numerical solution of the modified equal width wave equation using collocation method. Math. Comput. Model. 45(9–10), 1117–2007 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Wazwaz, A.M.: The tanh and sine–cosine methods for a reliable treatment of the modified equal width wave equation and its variants. Commun. Nonlinear Sci. Numer. Simul. 11(2), 148–160 (2006)

    Article  MathSciNet  Google Scholar 

  13. Din, S.T.M., Yildirim, A., Berberler, M.E., Hosseini, M.M.: Numerical solution of the modified equal width wave equation. World Appl. Sci. J. 8(7), 792–798 (2010)

    Google Scholar 

  14. Hassan, H.N.: An accurate numerical solution for the modified equal width wave equation using the Fourier pseudo-spectral method. J. Appl. Math. Phys. Lett. A 4, 1054–1067 (2016)

    Article  Google Scholar 

  15. Esen, A., Kutluay, S.: Solitary wave solutions of the modified equal width wave equation. Commun. Nonlinear Sci. Numer. Simul. 13(8), 1538–1546 (2008)

    Article  MathSciNet  Google Scholar 

  16. Wang, H., Chen, L., Wang, H.: Exact travelling wave solutions of the modified equal width wave equation via the dynamical system method. Nonlinear Anal. Differ. Equ. 4(1), 9–15 (2016)

    Article  MathSciNet  Google Scholar 

  17. Sahoo, S., Ray, S.S.: Lie symmetry analysis and exact solutions of (3+1) dimensional Yu–Toda–Sasa–Fukuyama equation in mathematical physics. Comput. Math. Appl. 73, 253–260 (2017)

    Article  MathSciNet  Google Scholar 

  18. Yang, S., Hua, C.: Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation. Appl. Math. Comput. 234, 579–583 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Wang, G., Kara, A.H., Fakhar, K., Guzman, J.V.: Group analysis. Exact solutions and conservation laws of a generalized fifth order KdV equation. Chaos Solitons Fractals 86, 8–15 (2016)

    Article  MathSciNet  Google Scholar 

  20. Kumar, M., Kumar, R., Kumar, A.: Some more similarity solutions of the (2+1)-dimensional BLP system. Comput. Math. Appl. 70, 212–221 (2015)

    Article  MathSciNet  Google Scholar 

  21. Kumar, M., Kumar, R., Kumar, A.: On similarity solutions of Zabolotskaya–Khokhlov equation. Comput. Math. Appl. 68, 454–463 (2014)

    Article  MathSciNet  Google Scholar 

  22. Olver, P.J.: Applications of Lie Groups to Differential Equations, pp. 30–130. Springer, New York (1993)

    Book  Google Scholar 

  23. Bluman, G., Cheviakov, A.: Applications of Symmetry Methods to Partial Differential Equations. Appl. Math. Sci., vol. 168, p. 417. Springer, New York (2010)

    MATH  Google Scholar 

  24. Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations. Springer, New York (1974)

    Book  Google Scholar 

  25. Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  26. Goyal, N., Wazwaz, A.M., Gupta, R.K.: Applications of maple software to derive exact solutions of generalized fifth-order Korteweg–De Vries equation with time-dependent coefficients. Rom. Rep. Phys. 68(1), 99–111 (2016)

    Google Scholar 

  27. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)

    Article  MathSciNet  Google Scholar 

  28. Wazwaz, A.M.: Partial Differential Equation and Solitary Wave Theory. Nonlinear Physical Science. Springer, New York (2009)

    Book  Google Scholar 

  29. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. China Machine Press, Beijing (2004)

    MATH  Google Scholar 

Download references

Acknowledgements

The second author is thankful to the “University Grants Commission (UGC)” India for financial support to carry out her research work.

Author information

Authors and Affiliations

  1. Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur, U.P., 247001, India

    Rajan Arora & Antim Chauhan

Authors
  1. Rajan Arora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antim Chauhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antim Chauhan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arora, R., Chauhan, A. Lie Symmetry Reductions and Solitary Wave Solutions of Modified Equal Width Wave Equation. Int. J. Appl. Comput. Math 4, 122 (2018). https://doi.org/10.1007/s40819-018-0557-z

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-018-0557-z

Keywords

Search

Navigation