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The Stationary Case

Part of the International Series of Numerical Mathematics book series (ISNM, volume 156)

Abstract

In this chapter we consider the following coupled boundary value problems of the elliptic-elliptic type
$$ {\begin{array}{*{20}c} {\left( { - \varepsilon u\prime \left( x \right) + \alpha _1 \left( x \right)u\left( x \right)} \right)\prime + \beta _1 \left( x \right)u\left( x \right) = f\left( x \right),x \in \left( {a,b} \right)} \\ {\left( {\left( { - \mu } \right)\left( x \right)v\prime \left( x \right) + \alpha _2 \left( x \right)v\left( x \right)} \right)\prime + \beta _2 \left( x \right)v\left( x \right) = g\left( x \right),x \in \left( {b,c} \right)} \\ \end{array} } $$
(1)
with the following natural transmission conditions at x = b
$$ u\left( b \right) = v\left( b \right),{\mathbf{ }} - \varepsilon u\prime \left( b \right) + \alpha _1 \left( b \right)u\left( b \right) = - \mu \left( b \right)v\prime \left( b \right) + \alpha _2 \left( b \right)v\left( b \right), $$
(2)
and one of the following types of boundary conditions
$$ \begin{gathered} u\left( a \right) = v\left( c \right) = 0, \hfill \\ u\prime \left( a \right) = v\left( c \right) = 0, \hfill \\ u\left( a \right) = 0,{\mathbf{ }} - v\left( c \right) = \gamma _0 \left( {v\left( c \right)} \right), \hfill \\ \end{gathered} $$
(3)
where a, b, c ∈ ℝ, a < b < c and ε is a small parameter, 0 < ε ≪ 1. In this chapter, we denote again by (P.k)ε the problem which comprises (S), (TC), and (BC.k), k = 1, 2, 3, just formulated above. We will make use of the following assumptions on the data:
$$ \begin{gathered} \left( {h_1 } \right)\alpha _1 \in H^1 \left( {a,b} \right),\beta _1 \in L^2 \left( {a,b} \right),\left( {1/2} \right)\alpha _1^\prime + \beta _1 \geqslant 0,{\mathbf{ }}a.e. in{\mathbf{ }}\left( {a,b} \right); \hfill \\ \left( {h_2 } \right)\alpha _2 \in H^1 \left( {b,c} \right),\beta _2 \in L^2 \left( {b,c} \right),\left( {1/2} \right)\alpha _2^\prime + \beta _2 \geqslant 0,{\mathbf{ }}a.e. in{\mathbf{ }}\left( {b,c} \right); \hfill \\ \left( {h_3 } \right)\mu \in H^1 \left( {b,c} \right),\mu \left( x \right) > 0{\mathbf{ }}for all{\mathbf{ }}x{\mathbf{ }} \in \left[ {b,c} \right](equivalently, there is a constant \hfill \\ \mu _0 > 0{\mathbf{ }}such that{\mathbf{ }}\mu \left( x \right) \geqslant \mu _0 {\mathbf{ }}for all{\mathbf{ }}x{\mathbf{ }} \in \left[ {b,c} \right]); \hfill \\ \left( {h_4 } \right)f \in L^2 \left( {a,b} \right),g \in L^2 \left( {b,c} \right); \hfill \\ \left( {h^5 } \right)\alpha _1 > 0{\mathbf{ }}in{\mathbf{ }}\left[ {a,b} \right]{\mathbf{ }}or \hfill \\ \left( {h^5 } \right)\prime \alpha _1 < 0{\mathbf{ }}in{\mathbf{ }}\left[ {a,b} \right]; \hfill \\ \left( {h^6 } \right)\gamma _0 :D\left( {\gamma _0 } \right) = \mathbb{R} \to \mathbb{R}{\mathbf{ }}is a continuous nondecreasing function. \hfill \\ \end{gathered} $$

Keywords

Boundary Layer Unique Solution Asymptotic Expansion Asymptotic Analysis Transmission Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2007

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