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Asymptotic expansions of solutions to $n \times n$ systems of ordinary differential equations with a large parameter
A. P. Kosarevab,
A. A. Shkalikovab a Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
Abstract:
The
$n \times n$ system of ordinary differential equations
$$ y' - \sum_{l=0}^{m}\lambda^{-l}B_l(x)y - \lambda^{-m}C(x, \lambda)y=\lambda A(x)y, \qquad x \in [0, 1], \quad m \in \mathbb{N}, $$
is considered, where
$$ A=\operatorname{diag}\{a_1, \dots, a_n\}, \qquad B_l=\{b_{jk}^l\}, \qquad C=\{c_{jk}(\cdot, \lambda)\}, \qquad y=(y_1, \dots, y_n)^\top. $$
It is assumed that, for some integer
$m\ge 1$, the entries of the matrices
$A(x)$ and
$B_l(x)$ are complex-valued functions satisfying the conditions
\begin{gather*} a_i \in W_1^m[0, 1], \quad b^{0}_{ii} \in W_1^{m-1}[0, 1], \quad b^{0}_{jk} \in W_1^{m}[0, 1], \qquad j \ne k, \quad i, j, k=1, \dots, n, \\ b_{jk}^{l} \in W_1^{m-l}[0, 1], \qquad j, k=1, \dots, n, \quad l=1, \dots, m, \end{gather*}
where the
$W^k_1$ are the Sobolev spaces, and the entries of the matrix
$C(\cdot, \lambda)$ are integrable functions on the interval
$[0, 1]$ such that
$\|c_{ij}(\cdot, \lambda)\|_{1} \to 0$ in the metric of the space
$L_1[0, 1]$ uniformly as
$\lambda \to \infty$,
$\lambda \in \mathbb{C}$.
The main results of the paper refine and supplement classical results of
Birkhoff–Tamarkin–Langer theory
concerning asymptotic expansions of the fundamental solutions of the system
under consideration
in sectors of the complex plane.
Special attention is given to minimal
smoothness requirements on
the matrix entries and to explicit expressions for matrices in asymptotic expansions.
Keywords:
asymptotics of solutions of ordinary differential equations and systems,
spectral asymptotics, Birkhoff asymptotics.
UDC:
517 Received: 15.09.2024
DOI:
10.4213/mzm14508
Fulltext:
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