= Interior layer
An interior layer is a narrow region of rapid variation lying inside a <singular perturbation> domain, separated from its boundary. Away from it, a slowly varying outer approximation is valid; a stretched <inner variable> resolves the transition near the layer location.
For example,
$$
\varepsilon u'(x)=1-u(x)^2,\qquad u(0)=0,
$$
has the exact solution $u(x)=\tanh(x/\varepsilon)$. As $\varepsilon\to0^+$, it approaches $-1$ for fixed negative $x$ and $+1$ for fixed positive $x$. Neither constant outer branch describes the transition at zero. The inner coordinate $X=x/\varepsilon$ gives $U'(X)=1-U^2$ and $U=\tanh X$.
Thus the transition has width proportional to $\varepsilon$ and central slope $1/\varepsilon$. Matching requires $U\to-1$ as $X\to-\infty$ and $U\to1$ as $X\to+\infty$. The same principle can describe thin internal fronts in differential equations; boundary layers instead lie near boundary data or a domain edge.
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