Geodesic
Asymptotics of the solution of one Valley–Ooussin problem with an unstable spectrum
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 2, pp. 72-77 Cet article a éte moissonné depuis la source Math-Net.Ru

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A differential equation describes the relationship between an unknown function and its derivatives. Such connections are sought in various fields of knowledge: mechanics, physics, chemistry, biology, economics, sociology, oceanology, etc. Systems of ordinary differential equations with a small parameter are used in modeling processes of various natures. Typically, when modeling, small factors are discarded in order to obtain a simpler model from which the necessary information can be extracted. Practice has proven that small factors should be included not in equations, but in solutions. Equations containing small factors are called perturbed. Perturbation theory has been widely used in modern applied mathematics. With its help, researchers answer questions about the influence of various factors on the course of the process, about the stability of the obtained solutions, the proximity of the processes described by the obtained solutions to the real objects under study. The article studies the Vall?e-Poussin problem for a system of inhomogeneous linear singularly perturbed ordinary differential equations of the first order. The peculiarity of the problem under consideration is that the spectrum of the matrix, which is the coefficient of the linear part of the system, is unstable at three points of the segment under consideration. It is required to construct a uniform asymptotic expansion of the solution to the problem, modifying the classical method of boundary functions.
Keywords: small parameter, singularly perturbed Vall?e-Poussin problem, unstable spectrum, bisingular problem, smooth external solution, boundary function, boundary layer. 
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D. A. Tursunov; A. S. Sadieva. Asymptotics of the solution of one Valley–Ooussin problem with an unstable spectrum. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 2, pp. 72-77. http://geodesic.mathdoc.fr/item/VYURM_2024_16_2_a6/

[1] W.R. Wasow, Linear Turning Point Theory, Springer-Verlag, 1985, 246 pp. | DOI | MR | Zbl

[2] A.H. Nayfeh, Introduction to Perturbation Techniques, Toronto, New York, 1981, 519 pp. | MR

[3] M. Van Dyke, Perturbation Methods in Fluid Mechanics, California, Stanford, 1975, 271 pp. | MR | Zbl

[4] Ch. J. de la Vallee-Poussin, “Sur l'equation Differentielle du Second Ordre. Determination d'une Integrale par Deux Valeurs Assignees. Extension aux equations d'ordre n”, J. Math. pures et appl., 8:2 (1929), 125–144

[5] I. Kiguradze, B. Puza, “On the Valle-Poussin Problem for Singular Differential Equations with Deviating Arguments”, Archivum Mathematicum, 33:1 (1997), 127–138 | MR | Zbl

[6] Bobochko V.N., “A Singularly Perturbed de la Valle-Poussin Problem with Two Vanishing Points of the Spectrum”, Ukr. Mat. Zh., 35:5 (1983), 545–551 | MR | Zbl

[7] Bobochko V.N., “The Valle-Poussin Problem for a System of Singularly Perturbed Differential Equations with Nonstable Spectrum”, Sov. Math., 32:6 (1988), 16–28 | MR | Zbl

[8] Il'in A.M., Danilin A.R., Asymptotic Methods in Analysis, Fizmatlit Publ, M., 2009, 248 pp. (in Russ.)

[9] Tursunov D.A., Tursunov E.A., “The Asymptotic Solution of a Bisingular Cauchy Problem for Systems of Ordinary Differential Equations”, Vestnik VolGU. Seriya 1. Matematika. Fizika (Mathematical Physics and Computer Simulation), 2017, no. 1(38), 33–41 (in Russ.)