= Intermediate asymptotic region
An intermediate asymptotic region has a scale between two other scales and provides their connecting <matched asymptotic expansion>. It can be needed because a coefficient changes its dominant form before the narrowest region's highest-derivative balance is reached. For example, a coefficient $x+\epsilon$ changes near $x=O(\epsilon)$ even when the fast <inner expansion> has width $O(\epsilon^2)$. The intermediate solution must be matched at both ends rather than assigned an independent boundary value.
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