# 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