Matched asymptotic expansion

A matched asymptotic expansion constructs approximations in regions with different distinguished scalings and determines their free constants by requiring agreement in an overlap region.

An inner expansion resolves a region in which a stretched independent variable makes otherwise small effects enter the leading balance.

An outer expansion describes the solution away from a thin or distant inner region and is matched to the inner expansion in their common limit.

A switchback term such as $ϵ\log ϵ$ is forced into a matched expansion when logarithms of a stretched coordinate split into parameter-dependent and local pieces in the overlap region.