International Journal of Theoretical and Applied Mathematics

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A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability

Received: 24 November 2016    Accepted: 03 January 2017    Published: 16 January 2017
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Abstract

A limiting transition is performed in some systems of singularly perturbed differential equations in the case of change of stability. This phenomenon is found in laser physics, chemical kinetics, plastic deformation, biophysics, in the modified Zieglers system, and in the simulation of upland forest fires, safe combustion with maximum temperature, etc. Cases when such equations have explicit solutions are extremely rare. For sufficiently small values of the parameter to determine the behavior of the solution a daunting task even for super computers, but it is possible with the asymptotic series. Therefore, studies of singularly perturbed problems when the condition of asymptotic stability is relevant.

DOI 10.11648/j.ijtam.20170301.17
Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1, February 2017)
Page(s) 43-48
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Singularly Perturbed, Cauchy Problem, Asymptotic Stability, Limited Equation, Solutions Asymptotic, Analytic Continue, Turning Point

References
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[5] Golodova E. S., Shchepakina E. A. Maximal combustion temperature estimation. Journal of Physics: Conf. Series. 2006. V. 55. P. 94-104.
[6] Shhepakina E. A. Singular perturbations in the problem of safe combustion regimes. Mathematical modeling. 2003. Vol. 15. No. 8. pp. 113–117.
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[9] Nefedov N. N., Schneider K. R. On immediate-delayed exchange of stabilities and periodic forced canards. Journal of Computational Mathematics and Mathematical Physics. 2008. Vol. 48. No. 1. pp. 46-61.
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[11] Golodova E. S., Shhepakina E. A. Evaluation of delayed loss of stability in differential systems with trajectories-ducks. Bulletin of the Samara State University. Natural science series. 2013. No. (104). pp. 12–24.
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[13] Alybaev K. S. Method of level lines of research singularly perturbed equations in violation of conditions of stability. Dr. Diss. Zhalalabad, 2001. 203 p.
[14] Pankov P. S., Alybaev K. S., Tampagarov K. B., Narbaev K. B. The phenomenon of boundary-layer lines and the asymptotic behavior of solutions of singularly perturbed linear ordinary differential equations with analytic functions. Bulletin of the Osh state university. 2013. No. 1. pp. 227-231.
[15] Tursunov D. A. Asymptotics of solution of singularly perturbed problem with periodic turning points in complex plane. Bulletin of the Tomsk Polytechnic University. 2014. vol. 324. no. 2. pp. 40–46.
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Author Information
  • Department of Informatics, Osh State University, Osh, Kyrgyzstan

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    Dilmurat Abdillajanovich Tursunov. (2017). A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability. International Journal of Theoretical and Applied Mathematics, 3(1), 43-48. https://doi.org/10.11648/j.ijtam.20170301.17

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    Dilmurat Abdillajanovich Tursunov. A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability. Int. J. Theor. Appl. Math. 2017, 3(1), 43-48. doi: 10.11648/j.ijtam.20170301.17

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    AMA Style

    Dilmurat Abdillajanovich Tursunov. A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability. Int J Theor Appl Math. 2017;3(1):43-48. doi: 10.11648/j.ijtam.20170301.17

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  • @article{10.11648/j.ijtam.20170301.17,
      author = {Dilmurat Abdillajanovich Tursunov},
      title = {A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {3},
      number = {1},
      pages = {43-48},
      doi = {10.11648/j.ijtam.20170301.17},
      url = {https://doi.org/10.11648/j.ijtam.20170301.17},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijtam.20170301.17},
      abstract = {A limiting transition is performed in some systems of singularly perturbed differential equations in the case of change of stability. This phenomenon is found in laser physics, chemical kinetics, plastic deformation, biophysics, in the modified Zieglers system, and in the simulation of upland forest fires, safe combustion with maximum temperature, etc. Cases when such equations have explicit solutions are extremely rare. For sufficiently small values of the parameter to determine the behavior of the solution a daunting task even for super computers, but it is possible with the asymptotic series. Therefore, studies of singularly perturbed problems when the condition of asymptotic stability is relevant.},
     year = {2017}
    }
    

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