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.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1) |
| DOI | 10.11648/j.ijtam.20170301.17 |
| Page(s) | 43-48 |
| Creative Commons |
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. |
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Copyright © The Author(s), 2017. Published by Science Publishing Group |
Singularly Perturbed, Cauchy Problem, Asymptotic Stability, Limited Equation, Solutions Asymptotic, Analytic Continue, Turning Point
| [1] | Sobolev V. A., Shhepakina E. A. Reduction models and critical phenomena in macrokinetics. Moscov, Fizmatlit. 2010. 320 p. |
| [2] | Semenov M. E., Kolupaeva S. N., Rozhnov A. I. Mathematical modeling of plastic deformation of GCK materials under varying strain rate. Herald of Perm National Research Polytechnic University. Mechanics. 2011. №3. pp. 100–117. |
| [3] | Neishtadt A. I. and Sidorenko V. V., Stability loss delay in a Ziegler system. J. App. MathsMechs. 1997. Vol. 61, No. 1, pp. 15-25. |
| [4] | Golodova E. S., Shhepakina E. A. Modelling of safe combustion with a maximum temperature. Mathematical modeling. 2008. Vol. 20. No. 5. pp. 55–58. |
| [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. |
| [7] | Tihonov A. N. Systems of differential equations containing small parameters in the derivatives. Mat. compilation. Mathematics collection. 1952. 31 (73). no. 3. pp. 575-586. |
| [8] | Shishkova, M. A. Consideration of a system of differential equations-tions with a small parameter in the highest derivatives. Dokl. USSR Academy of Sciences. 1973. vol. 209, no. 3. pp. 576-579. |
| [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. |
| [10] | Tursunov D. A. Uniform asymptotic solutions of the Cauchy problem for a generalized model equation of L. S. Pontryagin in the case of violation of conditions of asymptotic stability. Science Journal of Applied Mathematics and Statistics. New York. 2013. Vol. 1, Nо. 3. pp. 25-29. |
| [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. |
| [12] | Karimov S. Asymptotic behavior of solutions of certain classes of differential equations with a small parameter in the derivatives in the case of change of stability in the plane of the rest point «quick movements». Dr. Diss. Osh, 1983. 260 p. |
| [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. |
| [16] | Tursunov D. A. Asymptotic expansion solution of singular perturbed problems, when complex conjugation eigenvalues has n multiple poles. PhD diss. Osh, 2005. 110 p. |
| [17] | Fedorjuk M. V. Saddle-point method. Moscov, Librokom Publ., 2010. 368 p. |
APA Style
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
ACS 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
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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Download@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://article.sciencepublishinggroup.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}
}
TY - JOUR T1 - A Limiting Transition in a Singularly Perturbed Equation with the Loss of Stability AU - Dilmurat Abdillajanovich Tursunov Y1 - 2017/01/16 PY - 2017 N1 - https://doi.org/10.11648/j.ijtam.20170301.17 DO - 10.11648/j.ijtam.20170301.17 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 43 EP - 48 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20170301.17 AB - 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. VL - 3 IS - 1 ER -
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