##
**Algebraic-differential systems with large and rapidly oscillating coefficients. Asymptotics of solutions.**
*(English.
Russian original)*
Zbl 1405.34013

J. Math. Sci., New York 232, No. 4, 428-436 (2018); translation from Probl. Mat. Anal. 93, 23-29 (2018).

The topic of this paper is to develop asymptotic expansions of the solutions of a real linear DAE
\[
C(t)\frac{dx}{dt}=(\omega A(t,\omega)+\sqrt{\omega}B(t,\omega t,\omega))x
\]
in \(\mathbb R^n\), where the oscillating parameter \(\omega\) is supposed to be a large number. The matrix \(C(t)\) is not full-rank, so we have a DAE.

The main contribution of the authors is the generalization of a result proved by Krutenko and Levenshtam in 2009 : \(r<n\) being the constant rank of \(C(t)\), there is a basis of \(r\) independent solutions \(x_k\) and the authors provide a constructive method to get an asymptotic expansion of this basis as \[ x^k_N(t)=[u_{k0}(t)+\sum_{j=1}^N \omega^{-1/2}(u_{kj}(t)+v_{kj}(t,\omega t))]e^{q_k(\omega,t)}. \] The uniform convergence on an interval \([0, T]\) is proved upon regularity conditions (in particular no small divisor).

This is a technical paper for specialists of this field or physicists that may model systems leading to the initial DAE.

The main contribution of the authors is the generalization of a result proved by Krutenko and Levenshtam in 2009 : \(r<n\) being the constant rank of \(C(t)\), there is a basis of \(r\) independent solutions \(x_k\) and the authors provide a constructive method to get an asymptotic expansion of this basis as \[ x^k_N(t)=[u_{k0}(t)+\sum_{j=1}^N \omega^{-1/2}(u_{kj}(t)+v_{kj}(t,\omega t))]e^{q_k(\omega,t)}. \] The uniform convergence on an interval \([0, T]\) is proved upon regularity conditions (in particular no small divisor).

This is a technical paper for specialists of this field or physicists that may model systems leading to the initial DAE.

Reviewer: Gabriel Thomas (Domene)

### MSC:

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

### Keywords:

linear DAEs; asymptotic expansions; quasi-periodic coefficients; Fourier decomposition; matrix pencil; normal form
PDF
BibTeX
XML
Cite

\textit{V. B. Levenshtam} and \textit{S. P. Prika}, J. Math. Sci., New York 232, No. 4, 428--436 (2018; Zbl 1405.34013); translation from Probl. Mat. Anal. 93, 23--29 (2018)

Full Text:
DOI

### References:

[1] | N. I. Shkil’, I. I. Starun, and V. P. Yakovets, Asymptotic Integration of Linear Systems of Ordinary Differential Equations [in Russian], Vyshcha Shkola, Kiev (1989). · Zbl 0743.34060 |

[2] | Krutenko, EV; Levenshtam, VB, Asymptotic analysis of certain systems of linear differential equations with a large parameter, Comput. Math. Math. Phys., 49, 2047-2058, (2009) |

[3] | V. B. Levenshtam, Differential Equations with Large High-Frequency Terms [in Russian], Southern Federal Univ., Rostov-on-Don (2010). |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.