Regularized Solution of a Singularly Perturbed Cauchy Problem in the Presence of Irrational Simple Turning Point
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
Abstract:
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.
Keywords
- asymptotic solution
- regularization method
- singularly perturbed Cauchy problem
- turning point
References:
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- 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)
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