 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,has the exact solution $u(x)=\tanh (x/\epsilon )$. As $\epsilon \to 0^{+}$, 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 $X=x/\epsilon$ gives $U^{\prime }(X)=1-U^2$ and $U=\tanh X$.

Thus the transition has width proportional to $\epsilon$ and central slope $1/\epsilon$. Matching requires $U\to -1$ as $X\to -\infty$ and $U\to 1$ 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.