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

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.

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References

  1. 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)

    MathSciNet  Article  Google Scholar 

  2. 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)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    Google Scholar 

  6. 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)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 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. 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)

    MathSciNet  Article  Google Scholar 

  11. 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. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 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

    MathSciNet  Article  MATH  Google Scholar 

  15. 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. 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. 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. 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. 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)

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

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Acknowledgements

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

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Correspondence to S. Kumbinarasaiah.

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

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Keywords

  • Operational matrix
  • Partial differential equations
  • Collocation method
  • Clique polynomials

Mathematics Subject Classification

  • 35-XX
  • 65M70
  • 05Cxx