The method of boundary integral equations for determining boundary layer asymptotics of the solution to the nonlinear singularly perturbed heat conduction boundary value problem
type: Item
instance of: scholarly article
title: The method of boundary integral equations for determining boundary layer asymptotics of the solution to the nonlinear singularly perturbed heat conduction boundary value problem
zbMATH Open document ID: 0963.35010
author: V. F. Kravchenko
author: G. A. Nesenenko
published in: Doklady Mathematics
publication date: 2001-01-28T00:00:00Z
review text: Consider the following singularly perturbed heat conduction equation \[ \frac {\partial T} {\partial t}= \varepsilon \frac{\partial^2 T} {\partial x^2}, \quad (x,t)\in M_t \equiv \{(x,t)\mid kt< x< N_2(t), 0<t\leq 1\}\tag{*} \] with nonlinear Stefan-Boltzmann boundary conditions: \(T(x,t)=T_0\) in \(t=+0\), \(0\leq x\leq N_2(0)\); \(\partial T/\partial x=\gamma T^4 (x,t)\) in \(x=kt\), \(0\leq t\leq 1\); \(\partial T/\partial x=0\) in \(x=N_2(t)\), \(0\leq t\leq 1\). Such a problem arises, for example, in plasma-coating processes in a vacuum. An approach based on the method of boundary integral equations and a modification of the Laplace method, obtaining a boundary-layer asymptotic expansion in Poincaré sense for the solution of the problem (*) is described. It is established that \[ T(x,t){\underset{\varepsilon\to 0}\sim}T_0-\frac\gamma{k}T_0^4 \exp\left\{-\frac{(x-kt)k}\varepsilon\right\}\sum_{i=0}^\infty a_i \varepsilon^{i+1}, \] where \(a_0=1\), \(a_1=4\gamma T_0^3/k\), \(a_2=22\gamma^2 T_0^6/k^2\).
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1557574
zbMATH Keywords: nonlinear Stefan-Boltzmann boundary conditions
zbMATH Keywords: plasma-coating processes in a vacuum
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1557574