Skip to main content
SpringerLink
Log in

A new approach for the numerical solution for nonlinear Klein–Gordon equation

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

In this article, we generated a new operational matrix of integration using Clique polynomials of complete graphs and also introducing a new numerical technique to solve nonlinear Klein–Gordon equation. These equations describe a variety of physical phenomena such as ferroelectric and ferromagnetic domain walls, and DNA dynamics. We obtain an approximate solution for the nonlinear Klein–Gordon equation using the present method by transforming a system of nonlinear algebraic equations. The proposed scheme is applied to some examples and compared with another method in the literature that demonstrates the effectiveness of this method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Availability of data and materials

All data generated or analyzed during this study are included.

References

  1. Chowdhury, M.S.H., Hashim, I.: Application of Homotopy-perturbation method to Klein–Gordon and sine-Gordon equations. Chaos Solitons Fract. 39, 1928–1935 (2009)

    Article  MathSciNet  Google Scholar 

  2. Dehghan, M., Shokri, A.: Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions. J. Comput. Appl. Math. 230, 400–410 (2009)

    Article  MathSciNet  Google Scholar 

  3. Diudea, M.V., Gutman, I., Lorentz, J.: Molecular Topology. Nova Science, Hauppauge (1999)

    Google Scholar 

  4. El-Sayed, S.M.: The decomposition method for studying the Klein–Gordon equation. Chaos Solitons Fract. 18(5), 1025–1030 (2003)

    Article  MathSciNet  Google Scholar 

  5. Harary, F.: Graph Theory. Addison-Wesley, Oxford (1969)

    Book  Google Scholar 

  6. Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., Fereidouni, F.: Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions. Eng. Anal. Bound. Elem. 37, 1331–1338 (2013)

    Article  MathSciNet  Google Scholar 

  7. Hoede, C., Li, X.: Clique polynomials and independent set polynomials of graphs. Discrete Math. 125, 219–228 (1994)

    Article  MathSciNet  Google Scholar 

  8. Kanth, A.R., Aruna, K.: Differential transform method for solving the linear nonlinear Klein–Gordon equation. Comput. Phys. Commun. 180, 708–711 (2009)

    Article  MathSciNet  Google Scholar 

  9. Khuri, S.A., Sayfy, A.: A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation. Appl. Math. Comput. 216, 1047–1056 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Mohammadi, A., Aghazadeh, M., Rezapour, S.: Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent Emden–Fowler equations with initial and boundary conditions. Math. Sci. 13, 255–265 (2019)

    Article  MathSciNet  Google Scholar 

  11. Rashidinia, J., Jokar, M.: Numerical solution of nonlinear Klein–Gordon equation using polynomial wavelets. In: Advances in Intelligent Systems and Computing, pp. 199–214 (2016)

  12. Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein–Gordon equation. Comput. Phys. Commun. 181, 78–91 (2010)

    Article  MathSciNet  Google Scholar 

  13. Rashidinia, J., Ghasemia, M., Jalilian, R.: Numerical solution of the nonlinear Klein/Gordon equation. J. Comput. Appl. Math. 233, 1866–1878 (2010)

    Article  MathSciNet  Google Scholar 

  14. Raza, N., Rashid Butt, A., Javid, A.: Approximate solution of nonlinear Klein–Gordon equation using Sobolev gradients. Hindawi Publ. Corpor. J. Funct. Sp. 2016, 1391594 (2016). https://doi.org/10.1155/2016/1391594

    Article  MathSciNet  MATH  Google Scholar 

  15. Shiralashetti, S.C., Kumbinarasaiah, S.: Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane-Emden type equations. Appl. Math. Comput. 315, 591–602 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Shiralashetti, S.C., Kumbinarasaiah, S.: Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alexandria Eng. J. 57(4), 2591–2600 (2018)

    Article  Google Scholar 

  17. Shiralashetti, S.C., Kumbinarasaiah, S.: Laguerre wavelets collocation method for the numerical solution of the Benjamina Bona Mohany equations. J. Taibah Univ. Sci. 13(1), 9–15 (2019)

    Article  Google Scholar 

  18. Shiralashetti, S.C., Kumbinarasaiah, S.: CAS wavelets analytic solution and Genocchi polynomials numerical solutions for the integral and integrodifferential equations. J. Interdiscip. Math. 22(3), 201–218 (2019)

    Article  Google Scholar 

  19. Yin, F., Tian, T., Song, J., Zhu, M.: Spectral methods using Legendre wavelets for nonlinear Klein/Sine-Gordon equations. J. Comput. Appl. Math. 275, 321–334 (2015)

    Article  MathSciNet  Google Scholar 

  20. Yusufoglu, E.: The variational iteration method for studying the Klein–Gordon equation. Appl. Math. Lett. 21, 669–674 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author wish to express his thanks to Karnatak University and Bangalore University for the necessary support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, India

    S. Kumbinarasaiah

Authors
  1. S. Kumbinarasaiah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Kumbinarasaiah.

Ethics declarations

Conflict of interest

The author declare that he has no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumbinarasaiah, S. A new approach for the numerical solution for nonlinear Klein–Gordon equation. SeMA 77, 435–456 (2020). https://doi.org/10.1007/s40324-020-00225-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-020-00225-y

Keywords

Mathematics Subject Classification

Search

Navigation