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Finite difference/Finite element simulation of the two-dimensional linear and nonlinear Higgs boson equation in the de Sitter space-time

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Abstract

In this work, finite element simulation of the two-dimensional linear and nonlinear form of the Higgs boson equation in de Sitter space-time is presented. The mathematical model of the problem is linear and power type nonlinear Klein–Gordon-like partial differential equations. Therefore, we discretize the temporal variable using the finite difference method and we also discretize the spatial variable using the finite element method. We use the Newton linearization technique which is one of the most useful linearization techniques for the linearization of the nonlinear partial differential equations. In the Newton method, we consider the Jacobian matrix numerically. Applying the considered numerical scheme we obtain the Bubble-like solutions in good agreement with the numerical results and theory available in the literature.

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References

  1. Yagdjian K (2012) On the global solution of the Higgs boson equation. Comm Part Diff Equ 37(3):447–478

    Article  MathSciNet  Google Scholar 

  2. Yagdjian K, Balogh A (2018) The maximum principle and sign changing solutions of the hyperbolic equationwith the Higgs potential. J Math Anal Appl 465:403–422

    Article  MathSciNet  Google Scholar 

  3. Balogh A, Banda J, Yagdjian K (2019) High-performance implementation of a Runge–Kutta finite-difference scheme for the Higgs boson equation in the de sitter spacetime. Commun Nonlinear Sci 68:15–30

    Article  MathSciNet  Google Scholar 

  4. Selvitopi H, Yazici M (2019) Numerical results for the Klein–Gordon equation in de sitter spacetime. Math Methods Appl Sci 42:5446–5454

    Article  MathSciNet  Google Scholar 

  5. Tsuchiya T, Nakamura M (2019) On the numerical experiments of the cauchy problem for semi-linear Klein–Gordon equation in the de sitter spacetime. J Comput Appl Math 361:378–412

    Article  MathSciNet  Google Scholar 

  6. Zaky MA, Handy AS (2020) An efficient dissipation-preserving Legendre–Galerkin spectral method for the Higgs boson equation in the de Sitter spacetime universe. Appl Nume Mathe 160:281–295

    Article  MathSciNet  Google Scholar 

  7. Handy AS, Zaky MA (2020) Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations. Appl Numer Math 156:276–302

    Article  MathSciNet  Google Scholar 

  8. Handy AS, Zaky MA (2021) Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation. Nonlinear Dyn 103(3):1–15

  9. Zaky MA, Handy AS (2020) Convergence analysis of an L 1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations. Int J Comput Math 1–18

  10. Zaky MA, Handy AS, De Staelen RH (2021) Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system. Mathematics 9(2):183

    Article  Google Scholar 

  11. Muũoz-Pérez LF, Macías-Díaz JE, Guerrero JA (2020) A dissipation-preserving finite-difference scheme for a generalized Higgs boson equation in the de Sitter space-time. Appl Math Lett 107:106425

    Article  MathSciNet  Google Scholar 

  12. Muũoz-Pérez LF, Macías-Díaz JE (2020) On the solution of a generalized Higgs boson equation in the de Sitter space-time through an efficient and Hamiltonian scheme. J Comput Phys 417:109568

    Article  MathSciNet  Google Scholar 

  13. Macías-Díaz JE (2020) Design and analysis of a dissipative scheme to solve a generalized multi-dimensional Higgs boson equation in the de Sitter space–time. J Comput Appl Math:113120

  14. Hafez RM, Zaky MA (2019) High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems. Eng Comput 36:1813–1829

    Article  Google Scholar 

  15. Handy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput: 1–13

  16. Reddy JN (1993) An introduction to finite element method. McGraw-Hill, Newyork

    Google Scholar 

  17. Pekmen B, Tezer-Sezgin M (2012) Differential quadrature solution of nonlinear Klein–Gordon and sine-Gordon equations. Comput Phys Commun 183(8):1702–1713

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the reviewers for their valuable contributions and suggestions to improve the paper. In the revised version, all the reviewers’ comments were taken into consideration, resulting in a substantial improvement with respect to the original submission.

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  1. Department of Mathematics, Faculty of Sciences, Erzurum Technical University, 25050, Erzurum, Turkey

    Harun Selvitopi

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  1. Harun Selvitopi
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Correspondence to Harun Selvitopi.

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Selvitopi, H. Finite difference/Finite element simulation of the two-dimensional linear and nonlinear Higgs boson equation in the de Sitter space-time. Engineering with Computers 38, 891–900 (2022). https://doi.org/10.1007/s00366-021-01415-6

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  • DOI: https://doi.org/10.1007/s00366-021-01415-6

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