Interior layer at a simple zero of advection ($z=(x-x_{\ast })/\epsilon$)

In ${\epsilon }^2{y}^{\prime \prime }+a(x)y^{\prime }+\cdots =f$, a simple zero $a(x_{\ast })=0$, $a^{\prime }(x_{\ast })\ne 0$, balances diffusion and advection on width $\epsilon$. With $a(x)\simeq 2(x-x_{\ast })$, the leading homogeneous inner equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, whose bounded solutions are constants and the error function. Thus distinct order-one outer solutions can match across an interior transition. Reversing the diffusion sign makes the nonconstant homogeneous solution grow at both ends, changing the matching structure.

For $-{\epsilon }^2{y}^{\prime \prime }+2(x-x_{\ast })y^{\prime }+\cdots =f$ on an interval straddling $x_{\ast }$, the bounded leading interior solution cannot connect unequal constants. A common bulk value is instead selected by an inner solvability condition, and width-${\epsilon }^2$ endpoint layers enforce the two boundary values. At endpoints $x=0,2$ with $x_{\ast }=1$, their leading decaying factors are $e^{-2x/{\epsilon }^2}$ and $e^{-2(2-x)/{\epsilon }^2}$. The interior layer still corrects derivative mismatches at smaller amplitude.