Inner expansion

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

The singular perturbation equation $ϵ\sqrt{x}{u}^{\prime \prime }+u^{\prime }=0$ balances its derivatives on the width $x={ϵ}^{2}X$. The inner equation is $\sqrt{X}{U}^{\prime \prime }+U^{\prime }=0$, so $U^{\prime }=C{e}^{-2\sqrt{X}}$. Its solution with $U(0)=0$ and $U(\infty )=1$ is the displayed $H$, since ${\int }_0^{\infty }{e}^{-2\sqrt{q}}dq=1/2$. Near zero $H=2X-8X^{3/2}/3+O(X^2)$, so the first derivative is finite while the second is singular. Treat the differential equation on the open interval and impose the endpoint value by continuous extension.

An inner variable is a stretched coordinate such as $\xi =(x-x_0)/\delta (ϵ)$ that remains order one inside a thin region of width $\delta (ϵ)$. Its scale is chosen by a dominant balance, rather than by the size of a coefficient alone.

For the singular perturbation problem $ϵy^{\prime \prime }+y^{\prime }=0$ with $y(0)=0$, $y(1)=1$, set $X=x/ϵ$ and $Y_{ϵ}(X)=y_{ϵ}(ϵX)$. The chain rule givesThus both derivative terms survive in the inner differential equation $Y_{ϵ}^{\prime \prime }+Y_{ϵ}^{\prime }=0$. The exact domain is $0\le X\le 1/ϵ$, withFor bounded $X$, $Y_{ϵ}\to Y_0$ as $ϵ\to 0^{+}$: the inner expansion has a nonconstant limiting profile even though the outer solution is constant. The finite-$ϵ$ profiles nearly collapse onto $Y_0$ after rescaling, with their remaining difference coming from the normalization at the distant right endpoint.