= Solution
{c}
Set $x=\delta X$ and write $y(x)=Y(X)$. The equation becomes
$$
\frac{\epsilon}{\delta^2}Y''+\delta^{\alpha-1}X^\alpha Y'+Y=0.
$$
A <dominant balance> must compare all three coefficients, rather than just the two derivative terms.
For $-1<\alpha<1$, balancing diffusion and drift gives $\delta=\epsilon^{1/(\alpha+1)}$. Their common coefficient tends to infinity, so the reaction term is smaller and the leading inner equation is
$$
Y''+X^\alpha Y'=0,\qquad
Y'=C\exp[-X^{\alpha+1}/(\alpha+1)].
$$
This derivative is integrable at infinity, allowing an ordinary localized endpoint adjustment. Thus \b[the thickness is $\epsilon^{1/(\alpha+1)}$] in this range.
For $\alpha=1$, $\delta=\sqrt\epsilon$ makes all three terms comparable. Keep all of them:
$$
Y''+XY'+Y=0.
$$
One decaying solution is $e^{-X^2/2}$, and reduction of order gives the other. This is a \b[square-root-thickness layer with all terms retained].
For $\alpha>1$, the first variation scale is $\delta=\sqrt\epsilon$. Keep diffusion and reaction, and drop drift at this scale:
$$
Y''+Y=0.
$$
This leading motion is oscillatory, not a monotone decaying boundary-layer profile. To describe localization one must also resolve the slower damping envelope. Removing the first derivative in the original equation by
$$
y=\exp\left[-\frac{x^{\alpha+1}}{2\epsilon(\alpha+1)}\right]v
$$
gives
$$
v''+\left[\frac1\epsilon-\frac{\alpha x^{\alpha-1}}{2\epsilon}
-\frac{x^{2\alpha}}{4\epsilon^2}\right]v=0.
$$
Near the endpoint the wavelength is $O(\sqrt\epsilon)$, while the envelope changes on $x=O(\epsilon^{1/(\alpha+1)})$. Hence an oscillatory endpoint region is possible, but \b[it has distinct wavelength and envelope scales]; calling the diffusion-drift balance alone its leading equation would incorrectly discard the reaction term. This generalizes the <oscillatory endpoint with a quadratically vanishing drift>.
For $\alpha\leq-1$, the diffusion-drift balance would require $\delta^{\alpha+1}=\epsilon$, which gives no shrinking positive power scale. At every shrinking scale the singular drift dominates diffusion, so no regular finite-amplitude diffusion boundary layer of the preceding kind occurs. The coefficient itself is singular at zero; singular endpoint solutions are a different issue from a bounded boundary-layer correction.
Back to article page
