(Differencialnie Uravnenia i Protsesy Upravlenia)

About
History
Editorial Page
Addresses
Scope
Editorial Staff
Submission Review
For Authors
Publication Ethics
Issues
Русская версия

**Alexander Georgievich Eliseev**

National Research University "Moscow Power Engineering Institute"

Associate Professor, Department of Higher Mathematics

111250, Moscow, st. Krasnokazarmennaya, d. 14

Basing on the regularization method of S. A. Lomov, we construct an asymptotic solution for a singularly perturbed Cauchy problem for the case when the stability conditions for the spectrum of the limit operator are violated. In particular, we consider the problem with a simple turning point, when one eigenvalue at the initial moment of time has zero of arbitrary irrational order (the limit operator is discretely irreversible). This work is a development of the ideas described in the works of S. A. Lomov and A. G. Eliseev. The irrational turning point and the problems that arise in constructing the asymptotic of the solution of the Cauchy problem have not previously been considered from the point of view of the regularization method. In the present work, basing on the theory of normal and unique solvability of iterative problems developed by the author, we design and justify the algorithm for the regularization method and construct an asymptotic solution of any order with respect to a small parameter.

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

- Lomov, S. A.
*Vvedenie v obshyj teorij singuliajrnikh vozmyshenii*[Introduction to the General Theory of Singular Perturbations]. Moscow, Nauka Publ., 1981. 400 p. (in Russian) - Safonov, V. F., Bobodzhanov, A. A.
*Kurs vishej matematiki. Singuliajrno voznushennie zadachi i metod reguliarizacii*[Higher mathematics course. Singularly perturbed problems and regularization method]. Moscow, MPEI Publ., 2012. (in Russian) - 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) - Eliseev A, Ratnikova T. Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational "Simple" Turning Point in Two-Dimensional Case.
*Axioms*. 2019, 8(4):124. (in English) - Eliseev, A. G., Ratnikova, T. A. [Singularly Perturbed Cauchy Problem in the Presence of the Rational " simple" Pivot Point of the Limit Operator].
*Differential Equations and Control Processes*, 2019, № 3: 63-73. (in Russian) - Tursunov, D. A., Kozhobekov, K. G. [The asymptotics of solutions of singularly perturbed differential equations with fractional turning point].
*Izvestiya Irkutskogo Gosudarstvennogo Universiteta*, 2017; vol. 21: 108-121. (in Russian) - Lioville J. [Second memoir on the development of series functions of which various terms are subject to the same equation].
*J. Math. Pure Appl.*, 1837; vol. 2: 16-35. (in French)