RUS  ENG
Full version
JOURNALS // Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory // Archive

Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 2022 Volume 207, Pages 37–47 (Mi into977)

Asymptotic estimates for the solution of the Cauchy problem for a differential equation with linear degeneration

D. P. Emel'yanov, I. S. Lomov

Lomonosov Moscow State University

Abstract: Application of the method of separation of variables to problems for the linearly degenerate equation $u''_{xx}+yu''_{yy}+c(y)u'_y-a(x)u=f(x,y)$ in a rectangle leads to problems for the singularly perturbed ordinary differential equation with degeneration $yY''+c(y)Y'-(\pi^2k^2+a(y))Y=f_k(y)$, $k\in\mathbb{N}$. In this paper, we examine the asymptotic behavior of solutions of this equation with given initial data at $0$ and zero right-hand side as $k\to+\infty$ and obtain the leading term of the asymptotics in the explicit form.

Keywords: degenerate differential equation, singularly perturbed differential equation.

UDC: 517.928.2

MSC: 34E15

DOI: 10.36535/0233-6723-2022-207-37-47



© Steklov Math. Inst. of RAS, 2025