= Inner expansion
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An <inner expansion> is an <asymptotic expansion> in a stretched variable that resolves a region of rapid variation. If its width is $\delta(\epsilon)$ near $x_0$, write $X=(x-x_0)/\delta(\epsilon)$ and expand $Y(X;\epsilon)=u(x_0+\delta X;\epsilon)$ with $X$ fixed. The <chain rule> makes $d/dx=\delta^{-1}d/dX$, so even a small coefficient of a high <derivative> can contribute at leading order.
For $\epsilon u''+u'=0$ at a left endpoint, <dominant balance> gives $\delta=\epsilon$. The leading inner equation is $Y_0''+Y_0'=0$, and $Y_0(0)=0$, $Y_0(\infty)=1$ give $Y_0=1-e^{-X}$. Its nonconstant profile supplies the endpoint information lost by the constant <outer solution>. The condition at infinity is an <asymptotic matching condition> in an <overlap region>; it is not a claim that a truncated inner formula is uniformly accurate arbitrarily far from the layer.<Asymptotic Analysis and Singular Perturbation Theory by John K. Hunter>[§4.3.3, inner expansion; §4.3.4, matching]{ref}
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