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 $x+ϵ$ changes near $x=O(ϵ)$ even when the fast inner expansion has width $O({ϵ}^2)$. The intermediate solution must be matched at both ends rather than assigned an independent boundary value.