On one singularly perturbed system of ordinary differential equations with multiple root of the degenerate equation
type: Item
instance of: scholarly article
title: On one singularly perturbed system of ordinary differential equations with multiple root of the degenerate equation
zbMATH Open document ID: 1440.34059
author: V. F. Butuzov
published in: Journal of Mathematical Sciences (New York)
publication date: 2019-12-03T00:00:00Z
review text: Consider the two-dimensional system of singularly perturbed differential equations \[\begin{aligned}\varepsilon^2\frac{d^2u}{dx^2} & =h(x)(u-\varphi(v,x))^2-\varepsilon F_1(u,v,x,\varepsilon), \\ \varepsilon\frac{d^2v}{dx^2} & =f(u,v,x,\varepsilon)\end{aligned} \tag{1}\] on the interval \([0,1]\) with the boundary conditions \[u(0,\varepsilon)=u^0,\ u(1,\varepsilon)=u^1,\ \frac{dv}{dx} (0,\varepsilon)=q^0,\ \frac{dv}{dx}(1,\varepsilon)=q^1.\tag{2}\] It is assumed that the degenerate system \[h(x)(u-\varphi(x,v))^2=0,\ f(u,v,x,0)=0\] has the double root \(u=\varphi(x,v)\) and the simple root \(v=\overline v_0(x)\) on \([0,1]\). The author proves the existence of a solution \((u(x,\varepsilon),v(x,\varepsilon))\) of (1), (2) for sufficiently small \(\varepsilon\) by means of the method of lower and upper solutions. To construct lower and upper solutions, the author makes the ansatz for the solution \((u(x,\varepsilon),v(x,\varepsilon))\) \[u(x,\varepsilon)= \overline u(x,\varepsilon)+\Pi u(\xi,\varepsilon)+\widetilde\Pi u (\widetilde\xi,\varepsilon)+Pu(\zeta,\varepsilon)+\widetilde Pu (\widetilde\zeta,\varepsilon),\] \[v(x,\varepsilon)=\overline v(x,\varepsilon)+\Pi v(\xi,\varepsilon)+\widetilde\Pi v (\widetilde\xi,\varepsilon)+Pv(\zeta,\varepsilon)+\widetilde Pv (\widetilde\zeta,\varepsilon),\] with \[\xi=\frac{x}{\sqrt{\varepsilon}},\ \zeta=\frac{x}{\varepsilon}, \ \widetilde\xi=\frac{1-x} {\sqrt{\varepsilon}}, \ \widetilde\zeta=\frac{1-x}{\varepsilon}.\] \(\Pi,\widetilde\Pi,P,\widetilde P\) are the boundary layer functions. The author presents a procedure to determine these functions which differs essentially from the case of simple roots. It is shown that the boundary layer functions exhibit a multizone structure.
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 7139492
Wikidata QID: Q115382220
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: s10958-019-04350-6
OpenAlex ID: W2954449992
cites work: Q4002771
cites work: Asymptotics of the solution of a system of singularly perturbed equations in the case of a multiple root of the degenerate equation
cites work: On the special properties of the boundary layer in singularly perturbed problems with multiple root of the degenerate equation
cites work: Q3135823
cites work: Asymptotics of the solution to an initial boundary value problem for a singularly perturbed parabolic equation in the case of a triple root of the degenerate equation
cites work: Asymptotic behaviour and stability of solutions of a singularly perturbed elliptic problem with a triple root of the degenerate equation
DOI: 10.1007/S10958-019-04350-6
description: scientific article; zbMATH DE number 7139492