Singularly perturbed problems with rapidly oscillating inhomogeneities in the case of discrete irreversibility of the limit operator
Bobodzhanov A.A. Kalimbetov B.T. Safonov V.F. Tuychiev O.D.
30 September 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 397 - 106 pp.
We consider a linear singularly perturbed differential system, one of the points of the spectrum of the limiting operator of which goes to zero on some discrete subset of the segment of the independent variable. The problem belongs to the class of problems with unstable spectrum. Previously, S.A. Lomov’s regularization method was used to construct asymptotic solutions of a similar system. However, it was applied in the case of absence of fast oscillations. The presence of the latter does not allow us to approximate the exact solution by a degenerate one, since the limit transition in the initial system when a small parameter tends to zero in a uniform metric is impossible. Therefore, when constructing the asymptotic solution, it is necessary to take into account the effects introduced into the asymptotics by fast oscillations. In developing the corresponding algorithm, one could use the ideas of the classical Lomov regularization method, but considering that its implementation requires numerous calculations (e.g., to construct the main term of the asymptotics in the simplest case of the second-order zero eigenvalue of the limit operator one has to solve three algebraic systems of order higher than the first), the authors considered it necessary to develop a more economical algorithm based on regularization by means of normal forms.
asymptotic solution , discrete irreversibility of the operator , instability of the spectrum , limit transition , normal form , regularized asymptotics , singularly perturbed problem , solvability of iterative problems
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National Research University, Moscow Power Engineering Institute, 14 Krasnokazarmennaya street, Moscow, 111250, Russian Federation
A. Kuatbekov Peoples’ Friendship University, 32b Tole bi street, Shymkent, 160011, Kazakhstan
B. Gafurov Khudjant State University, 1 Mavlonbekova street, Khudjant, 735700, Tajikistan
National Research University
A. Kuatbekov Peoples’ Friendship University
B. Gafurov Khudjant State University
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