Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Regularized Asymptotics of Solution a Singularly Perturbed Cauchy Problem in the Presence of the "weak" Turning Point at the Limit Operator

Author(s):

Alexander Georgievich Eliseev

National Research University "Moscow Power Engineering Institute"
Associate Professor, Department of Higher Mathematics
111250, Moscow, st. Krasnokazarmennaya, d. 14

eliseevag@mpei.ru

Pavel Vladimirovich Kirichenko

National Research University "Moscow Power Engineering Institute"
Senior Lecturer, Department of Higher Mathematics
111250, Moscow, st. Krasnokazarmennaya, d. 14

kirichenkopv@mpei.ru

Abstract:

The article is devoted to the development of the regularization method of S. A. Lomov for singularly perturbed Cauchy problems in the case of violation of the stability conditions for the spectrum of the limit operator. In particular, the problem is considered in the presence of a "weak" turning point,in which the eigenvalues "stick together" at the initial instant of time. Problems with this kind of spectral features are well known to specialists in mathematical and theoretical physics, as well as in the theory of differential equations, but from the point of view of the regularization method they have not been previously considered. This work fills this gap. Based on the ideas of asymptotic integration of problems with spectral features of S. A. Lomov and A. G. Eliseev, it indicates how to introduce regularizing functions, describes in detail the algorithm of the regularization method in the case of a "weak" turning point, justifies this algorithm and an asymptotic solution of any order with respect to a small parameter is constructed.

Keywords

• asymptotic solution
• regularization method
• singularly perturbed Cauchy problem
• turning point

References:

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2. Eliseev, A. G., Lomov, S. A. [Theory of singular perturbations in the case of spectral singularities of the limit operator]. Matematicheskii sbornik, 1986; vol. 131, № 173: 544-557. (in Russian)
3. Bobodzhanov, A. A., Safonov, V. F. [Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernels]. Ufa Mathematical Journal, 2018; vol. 10, № 2: 3-12. (in Russian)
4. Kucherenko, V. V. [Asymptotics of solutions of the system A(x, -ih δ/δx) as h → 0 in the case of characteristics of variable multiplicity]. Mathematics of USSR-Izvestiya, 1974, vol. 38, № 3: 625-662. (in Russian)
5. Eliseev, A. G., Salnikova, T. A. [Construction of a solution to the Cauchy problem in the case of a weak turning point of the limit operator]. Matem. metody i prilojeniya. Trudy 20 matem. chtenii RGSU, 2011: 46-52. (in Russian)
6. Voevodin, V. V. [Computational foundations of linear algebra], Moscow, Nauka Publ., 1977. 304 p. (in Russian)

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