Asymptotics of the solution of a singularly perturbed system of equations with a multizone internal layer
type: Item
instance of: scholarly article
title: Asymptotics of the solution of a singularly perturbed system of equations with a multizone internal layer
zbMATH Open document ID: 1469.34077
DOI: 10.1134/S0012266121040017
author: V. F. Butuzov
author: R. E. Simakov
published in: Differential Equations
publication date: 2021-05-26T00:00:00Z
review text: The authors consider the asymptotic behavior of the solutions for a singularly perturbed boundary value problem \[ \varepsilon^2 \frac{\mathrm{d}^2u}{\mathrm{d}x^2}=F(u,v,x,\varepsilon), \ \varepsilon\frac{\mathrm{d}^2v}{\mathrm{d}x^2}=f(u,v,x,\varepsilon),\ 0 < x < 1, \] \[ \frac{\mathrm{d}u}{\mathrm{d}x}(0,\varepsilon)=\frac{\mathrm{d}u}{\mathrm{d}x}(1,\varepsilon)=0, \ \frac{\mathrm{d}v}{\mathrm{d}x}(0,\varepsilon)=\frac{\mathrm{d}v}{\mathrm{d}x}(1,\varepsilon)=0, \] where \(\varepsilon > 0\) is a small parameter and \(u(x,\varepsilon)\) and \(v(x,\varepsilon)\) are scalar functions. The function \(F\) has the form \[ F(u,v,x,\varepsilon)=\left(u-\varphi_1(v,x)\right)^2\left(u-\varphi_2(v,x)\right)\left(u-\varphi_3(v,x)\right)-\varepsilon F_1(u,v,x,\varepsilon) \] with \[ \varphi_1(v,x) < \varphi_2(v,x) < \varphi_3(v,x), \ (v,x)\in I_v\times [0,1]. \] The regular part and internal layer part of the asymptotics are constructed in the form of series in integer powers of \(\sqrt{\varepsilon}\) in the Subsection 3.1 and Subsection 2.3, respectively. By combining the asymptotic expansion techniques and the method of lower and upper solutions, the authors prove that for sufficiently small parameter values the problem has a solution with a rapid transition from the double root of the degenerate equation to a simple root in a neighborhood of some interior point of the interval.
reviewed by: Robert Vrabel
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 7351174
zbMATH Keywords: boundary value problem
zbMATH Keywords: singular perturbation
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: multizone internal layer
MaRDI profile type: Publication
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full work available at URL: s0012266121040017
OpenAlex ID: W3164303687
description: scientific article; zbMATH DE number 7351174