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Some Difference Algorithms for Nonlinear Klein-Gordon Equations

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Numerical Methods and Applications (NMA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11189))

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Abstract

In this study, sixth and eighth-order finite difference schemes combined with a third-order strong stability preserving Runge-Kutta (SSP-RK3) method are employed to cope with the nonlinear Klein-Gordon equation, which is one of the important mathematical models in quantum mechanics, without any linearization or transformation. Various numerical experiments are examined to verify the applicability and efficiency of the proposed schemes. The results indicate that the corresponding schemes are seen to be reliable and effectively applicable. Another salient feature of these algorithms is that they achieve high-order accuracy with relatively less number of grid points. Therefore, these schemes are realized to be a good option in dealing with similar processes represented by partial differential equations.

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References

  1. Arodz, H., Hadasz, L.: Lectures on Classical and Quantum Theory of Fields. Springer, London (2010). https://doi.org/10.1007/978-3-642-15624-3

    Book  MATH  Google Scholar 

  2. Dodd, R.K., Eilbeck, I.C., Gibbon, J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic, London (1982)

    MATH  Google Scholar 

  3. Cao, W.M., Guo, B.Y.: Fourier collocation method for solving nonlinear Klein-Gordon equation. J. Comput. Phys. 108, 296–305 (1993)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Inc, M., Ergut, M., Evans, D.J.: An efficient approach to the Klein-Gordon equation: an application of the decomposition method. Int. J. Simul. Process Model. 2, 20–24 (2006)

    Article  Google Scholar 

  6. 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 

  7. Bratsos, A.G.: On the numerical solution of the Klein-Gordon equation. Numer. Methods Partial Differ. Equ. 25, 939–951 (2009)

    Article  MathSciNet  Google Scholar 

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

    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. Li, Q., Ji, Z., Zheng, Z., Liu, H.: Numerical solution of nonlinear Klein-Gordon equation using Lattice Boltzmann method. Appl. Math. 2, 1479–1485 (2011)

    Article  Google Scholar 

  11. Mittal, R.C., Bhatia, R.: Numerical solution of nonlinear system of Klein-Gordon equations by cubic B-spline collocation method. Int. J. Comput. Math. 92, 2139–2159 (2015)

    Article  MathSciNet  Google Scholar 

  12. Sarboland, M., Aminataei, A.: Numerical solution of the nonlinear Klein-Gordon equation using multiquadratic quasi-interpolation scheme. Univ. J. Appl. Math. 3, 40–49 (2015)

    Google Scholar 

  13. Guo, P.F., Liew, K.M., Zhu, P.: Numerical solution of nonlinear Klein-Gordon equation using the element-free kp-Ritz method. Appl. Math. Model. 39, 2917–2928 (2015)

    Article  MathSciNet  Google Scholar 

  14. Chang, C.W., Liu, C.S.: An implicit Lie-group iterative scheme for solving the nonlinear Klein-Gordon and sine-Gordon equations. Appl. Math. Model. 40, 1157–1167 (2016)

    Article  MathSciNet  Google Scholar 

  15. Sankaranarayanan, S., Shankar, N.J., Cheong, H.F.: Three-dimensional finite difference model for transport of conservative pollutants. Ocean Eng. 25, 425–442 (1998)

    Article  Google Scholar 

  16. Sari, M., Gurarslan, G., Zeytinoglu, A.: High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation. Commun. Numer. Methods Eng. 27, 1296–1308 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Suleyman Demirel University, Isparta, Turkey

    Asuman Zeytinoglu

  2. Yildiz Technical University, Istanbul, Turkey

    Murat Sari

Authors
  1. Asuman Zeytinoglu
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  2. Murat Sari
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Corresponding author

Correspondence to Murat Sari .

Editor information

Editors and Affiliations

  1. Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria

    Geno Nikolov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Natalia Kolkovska

  3. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir Georgiev

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Zeytinoglu, A., Sari, M. (2019). Some Difference Algorithms for Nonlinear Klein-Gordon Equations. In: Nikolov, G., Kolkovska, N., Georgiev, K. (eds) Numerical Methods and Applications. NMA 2018. Lecture Notes in Computer Science(), vol 11189. Springer, Cham. https://doi.org/10.1007/978-3-030-10692-8_56

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  • DOI: https://doi.org/10.1007/978-3-030-10692-8_56

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10691-1

  • Online ISBN: 978-3-030-10692-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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