ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

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

Regularized Solution of a Singularly Perturbed Cauchy Problem in the Presence of Irrational Simple Turning Point


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.



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