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Kyrgyzstan Osh
Section Mathematics
Title Asymptotics of the solution to the boundary-value problem when the limit equation has an irregular singular point
Author(-s) Kozhobekov K.G.a, Tursunov D.A.ab
Affiliations Osh State Universitya, Osh Branch of the Russian State Social Universityb
Abstract This article studies the asymptotic behavior of the solutions of singularly perturbed two-point boundary value-problems on an interval. The object of the study is a linear inhomogeneous ordinary differential second-order equation with a small parameter with the highest derivative of the unknown function. The special feature of the problem is that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has an irregular singular point at the left end of the segment. At the ends of the segment, boundary conditions are imposed. Two problems are considered: in one of them the function in front of the first derivative of the unknown function is nonpositive on the segment considered, and in the second it is nonnegative. Asymptotic expansions of the problems are constructed by the classical method of Vishik-Lyusternik-Vasilyeva-Imanaliev boundary functions. However, this method cannot be applied directly, since the external solution has a singularity. We first remove this singularity from the external solution, and then apply the method of boundary functions. The constructed asymptotic expansions are substantiated using the maximum principle, i.e., estimates for the residual functions are obtained.
Keywords irregular singular point, singular perturbation, asymptotic behavior, methods of boundary layer functions, Dirichlet problem, boundary function, small parameter
UDC 517.928.2
MSC 34E05, 34E10, 34E20, 34B05
DOI 10.20537/vm190304
Received 11 May 2019
Language Russian
Citation Kozhobekov K.G., Tursunov D.A. Asymptotics of the solution to the boundary-value problem when the limit equation has an irregular singular point, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 332-340.
  1. Il'in A.M., Danilin A.R. Asimptoticheskie metody v analize (Asymptotic methods in analysis), Moscow: Fizmatlit, 2009, 248 p.
  2. Lomov S.A. Introduction to the general theory of singular perturbations, Providence, Rhode Island: AMS, 1992.
  3. Fruchard A., Schäfke R. Composite asymptotic expansions, Berlin-Heidelberg: Springer-Verlag, 2013.
  4. Nayfeh A.H. Introduction to perturbation techniques, New York, 1993.
  5. Bobochko V.N. An unstable differential turning point in the theory of singular perturbations, Russian Mathematics, 2005, vol. 49, no. 4, pp. 6-14.
  6. Alymkulov K., Khalmatov A.A. A boundary function method for solving the model lighthill equation with a regular singular point, Mathematical Notes, 2012, vol. 92, issue 5-6, pp. 751-755.
  7. Alymkulov K., Asylbekov T.D., Dolbeeva S.F. Generalization of the boundary function method for solving boundary-value problems for bisingularly perturbed second-order differential equations, Mathematical Notes, 2013, vol. 94, issue 3-4, pp. 451-454.
  8. Sibuya Y. Global theory of a second order linear ordinary differential equation with a polynomial coefficient, Elsevier, 1975.
  9. Yakovets V.P., Chornenka (Golovchenko) O.V. The construction of asymptotic solutions of a singularly perturbed linear system of differential equations with an irregular singular point in the case where the main matrix has multiple spectrum, Neliniini kolyvannya, 2008, vol. 11, no. 1, pp. 128-144 (in Ukrainian).
  10. Vyugin I.V. Riemann-Hilbert problem for scalar Fuchsian equations and related problems, Russian Mathematical Surveys, 2011, vol. 66, no. 1, pp. 35-62.
  11. Korikov D.V., Plamenevskii B.A. Asymptotics of solutions of the stationary and nonstationary Maxwell systems in a domain with small cavities, St. Petersburg Math. J., 2017, vol. 28, no. 4, pp. 507-554.
  12. Kuznetsov E.B., Leonov S.S. Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points, Computational Mathematics and Mathematical Physics, 2017, vol. 57, issue 6, pp. 931-952.
  13. Alymkulov K., Tursunov D.A. A method for constructing asymptotic expansions of bisingularly perturbed problems, Russian Mathematics, 2016, vol. 60, issue 12, pp. 1-8.
  14. Tursunov D.A. The asymptotic solution of the three-band bisingularly problem, Lobachevskii Journal of Mathematics, 2017, vol. 38, issue 3, pp. 542-546.
  15. Tursunov D.A. Asymptotic solution of linear bisingular problems with additional boundary layer, Russian Mathematics, 2018, vol. 62, issue 3, pp. 60-67.
  16. Tursunov D.A., Erkebaev U.Z., Tursunov E.A. Asymptotics of the Dirichlet problem solution for a ring with quadratic growths on the boundaries, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2016, issue 2 (48), pp. 73-81 (in Russian).
  17. Tursunov D.A., Erkebaev U.Z. Asymptotics of the Dirichlet problem solution for a bisingular perturbed equation in the ring, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 517-525 (in Russian).
  18. Tursunov D.A. Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of “rapid motions”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2018, no. 54, pp. 46-57 (in Russian).
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