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

Levenshtam, V. B.; Prika, S. P.

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.

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

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).