= Boundary layer around a moving endpoint
{title2=$z=(x-\varepsilon)/\varepsilon^2$}
= Moving-endpoint boundary layer
{synonym}
For $\varepsilon xw''+w'+2xw=0$ with left endpoint $x=\varepsilon$, diffusion there is of size $\varepsilon^2$. The distinguished coordinate is $z=(x-\varepsilon)/\varepsilon^2$, giving $(1+\varepsilon z)W''+W'+2\varepsilon^3(1+\varepsilon z)W=0$. Matching to one with zero left value gives $W=1-e^{-z}-\varepsilon z(z+2)e^{-z}/2+O(\varepsilon^2)$. The layer width is different from the endpoint's distance from zero.
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