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Determine values of $a$ for which the problem:

$\epsilon y^{''} + y^{'}+ae^y=0,$ $ y(0)=y(1)=0$

has a solution with a boundary layer structure.

I am familiar with the procedure for tackling this problem having already known there is a boundary layer at either of the endpoints. My thought process is that I can assume a boundary layer at some point within $(0,1)$ and then apply the matching conditions to the inner and outer solutions. Then hopefully that would yield some constraints for the paramater $a$.

Is this the right approach?

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$e^y$ never goes close enough to infinity to influence the balancing, so you always get $e^{-(x-x_0)/ϵ}$ as inner solution. It is only finite to the right, $x>x_0$, which pushes the boundary layer to the left boundary of the interval.

So you only have to care that the outer solution $y=-\ln(1-a+ax)$ exists on the whole interval and does not have the value $y(0)=0$.

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  • $\begingroup$ why can't the outer solution take on the left boundary condition? $\endgroup$ – John Simpleton Oct 24 '18 at 15:06
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    $\begingroup$ It can, for $a=0$ with $y=0$. But then you do not have a boundary layer, as there is no jump from the outer solution to the boundary condition. The more serious case is "no outer solution at all" for $a>1$. $\endgroup$ – LutzL Oct 24 '18 at 15:10
  • $\begingroup$ does $x_0$ here represent the arbitrary location of the boundary layer? $\endgroup$ – John Simpleton Oct 24 '18 at 15:25
  • $\begingroup$ Yes, it is the point that is tested for the viability of a boundary layer. Set $x=x_0+ϵX$ into the equation, $Y(X)=y(x)$, to get $ϵY''+ϵY'+ϵ^2e^Y=0$ and solve to get $Y=Ce^{-X}+D+O(ϵ)$. $\endgroup$ – LutzL Oct 24 '18 at 15:30

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