 Interior layer at a simple zero of advection

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.