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Regularization of the Singularly Perturbed Cauchy Problem for a Hyperbolic System

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Abstract

In this paper we construct the asymptotics of the solution of the Cauchy problem for a singularly perturbed hyperbolic system by using the regularization method for singularly perturbed problems of S.A. Lomov. The regularization method for singularly perturbed problems of S.A. Lomov is used for the first time to construct the asymptotic solution of a hyperbolic system.

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References

  1. Yu-cheng Su, “Asymptotics of solutions of some degenerate quasilinear hyperbolic equations of the second order,” Reports of the USSR Academy of Sciences, 138(1), 63–66 (1961).

    Google Scholar 

  2. V. A. Trenogii, “On the asymptotic behavior of solutions of quasilinear hyperbolic equations with a hyperbolic boundary layer,” Proceedings of the Moscow Institute of Physics and Technology, 9, 112–127 (1962).

    Google Scholar 

  3. V. F. Butuzov, “Angular boundary layer in mixed singularly perturbed problems for hyperbolic equations,” Mathematical Collection, 104(146), 460–485 (1977).

    MathSciNet  Google Scholar 

  4. V. F. Butuzov and A. V. Nesterov, “On some singularly perturbed problems of hyperbolic type with transition layers,” Differential Equations, 22 (10), 1739–1744 (1986).

    MathSciNet  Google Scholar 

  5. M. A. Valiev, “Asymptotics of the solution of the Cauchy problem for a hyperbolic equation with a parameter,” Proceedings of the Moscow Energy Institute, 146, 2–12 (1972).

    MathSciNet  Google Scholar 

  6. A. O. Abduvaliev, “Asymptotic expansions of solutions of the Darboux problem for singularly perturbed hyperbolic equations,” Fundamental Mathematics and Applied Informatics, 1 (4), 863–869 (1995).

    MathSciNet  MATH  Google Scholar 

  7. A. B. Vasileva, “On the inner transition layer in the solution of a system of partial differential equations of the first order,” Differential Equations, 21 (9), 1537–1544 (1985).

    Google Scholar 

  8. V. F. Butuzov and A. F. Karashchuk, “On a singularly perturbed system of partial differential equations of the first order,” Mathematical Notes, 57 (3), 338–349 (1995).

    Article  MathSciNet  Google Scholar 

  9. V. F. Butuzov and A. F. Karashchuk, “Asymptotics of the solution of a system of partial differential equations of the first order with a small parameter,” Fundamental and Applied Mathematics, 6 (3), 723–738 (2000).

    MATH  Google Scholar 

  10. A. V. Nesterov and O. V. Shuliko, “Asymptotics of the solution of a singularly perturbed system of first order partial differential equations with small nonlinearity in the critical case,” Journal of Computational Mathematics and Mathematical Physics, 47 (3), 438–444 (2007).

    MathSciNet  MATH  Google Scholar 

  11. A. V. Nesterov, “Asymptotics of the solution of the Cauchy problem for a singularly perturbed system of hyperbolic equations,” Chebyshev collection, 12 (3), 93–105 (2011).

    MathSciNet  MATH  Google Scholar 

  12. A. V. Nesterov, “On the asymptotics of the solution of a singularly perturbed system of partial differential equations of the first order with small nonlinearity in the critical case,” Journal of Computational Mathematics and Mathematical Physics, 52 (7), 1267–1276 (2012).

    MATH  Google Scholar 

  13. A. V. Nesterov and T. V. Pavlyuk, “On the asymptotics of the solution of a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case,” Journal of Computational Mathematics and Mathematical Physics, 54 (3), 450–462 (2014).

    MathSciNet  MATH  Google Scholar 

  14. A. V. Nesterov, “On the structure of the solution of one class of hyperbolic systems with several spatial variables in the far field,” Journal of Computational Mathematics and Mathematical Physics, 56 (4), 639–649 (2016).

    MathSciNet  Google Scholar 

  15. S. A. Lomov, Introduction to the general theory of singular perturbations. Nauka, M. (1981).

  16. S. A. Lomov and I. S. Lomov, Basics of mathematical theory of the boundary layer. MSU, M. (2011).

  17. L. Collatz, Eigenwertaufgaben mit technischen Anwendungen. Nauka, M. (1968).

  18. J. H. Wilkinson, The Algebraic eigenvalue problem. Nauka, M. (1970).

  19. S. Mizohata, Theory of partial differential equations. Mir, M. (1977).

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Authors and Affiliations

  1. Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan

    Asan Omuraliev & Ella Abylaeva

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  1. Asan Omuraliev
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  2. Ella Abylaeva
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Correspondence to Asan Omuraliev.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 19, No. 2, pp. 202–212, April–June, 2022.

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Omuraliev, A., Abylaeva, E. Regularization of the Singularly Perturbed Cauchy Problem for a Hyperbolic System. J Math Sci 264, 415–422 (2022). https://doi.org/10.1007/s10958-022-06008-2

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  • DOI: https://doi.org/10.1007/s10958-022-06008-2

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