= Matched asymptotic expansion
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A <matched asymptotic expansion> constructs approximations in regions with different <distinguished limits> and fixes their free constants by requiring agreement in an <overlap region>. The <inner expansion> resolves rapid variation; the <outer expansion> describes the surrounding region. Matching compares their common limiting behavior, rather than imposing distant <boundary conditions> directly on a local approximation.<Asymptotic Analysis and Singular Perturbation Theory by John K. Hunter>[§§4.3.3–4.3.4, pp. 57–58: inner expansion and matching]{ref}
For the <singular perturbation> problem $\epsilon y''+y'=0$, $y(0)=0$, $y(1)=1$, the leading <outer solution> is $y_0=1$. The <inner variable> $X=x/\epsilon$ gives $Y_0''+Y_0'=0$ and $Y_0(0)=0$, hence $Y_0=A(1-e^{-X})$. In the <overlap region> $\epsilon\ll x\ll1$, the <inner variable> tends to infinity while the outer coordinate tends to zero. The <asymptotic matching condition> is
$$
\lim_{X\to\infty}Y_0(X)=\lim_{x\to0^+}y_0(x)=1,
$$
so $A=1$. One concrete path through the <overlap region> is $x=\sqrt\epsilon$: then $x\to0$ and $X=\epsilon^{-1/2}\to\infty$. Matching selects the free constant that the left <boundary condition> alone could not determine.
Why choose the right <boundary condition> for the <outer solution>? If instead one selects $y_0=0$ from the left, a proposed right <boundary layer> with $Z=(1-x)/\epsilon$ satisfies $U_0''-U_0'=0$. Its <general solution> is $A+B e^Z$. Matching to zero as $Z\to\infty$ forces $A=B=0$, contradicting $U_0(0)=1$. The direction in which the <exponential function> decays determines which endpoint can carry the <boundary layer>; endpoint selection is part of the calculation.
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