## 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.

### 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
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### 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).
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