Exponential-reaction terminal boundary layer
= Exponential-reaction terminal boundary layer
{title2=$y_c=-\log(e^{-A}-x)+\log(e^{-A}-1)e^{(x-1)/\epsilon}$}
For $\epsilon y''-y'+e^y=0$ with $y(0)=A<0$ and $y(1)=0$, the <outer solution> is $-\log(e^{-A}-x)$. The right-end <boundary layer> has width $\epsilon$ and leading correction $\log(e^{-A}-1)e^{(x-1)/\epsilon}$. This follows by setting $X=(x-1)/\epsilon$, solving $Y''-Y'=0$, and matching its constant part. The leading correction changes sign at $A=-\log2$.