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The first of the Gaussian drift primitives, $E_0$, is an even function; the second, $E_1$, is an odd function. Their derivatives follow by the fundamental theorem of calculus. In particular $E_0^{\prime }$ is the Dawson function, which behaves as $1/(2z)+O(z^{-3})$. Since the error function tends to $\pm 1$, the derivative of $E_1$ behaves as $1/(2|z|)+O(|z{|}^{-3})$ at either end. Integration therefore givesAlso $F\sim z+\sqrt{\pi }/4$ on the left and $F\sim \sqrt{\pi }/4$ on the right. The two inner overlaps are consequentlyExpressing the outer expansions in $r=\epsilon z$ requires the left constant $1+A\log \epsilon$ and the right constant $B\log \epsilon$. Equating those constants gives the stated $\alpha ,\beta$. This is exactly how a logarithmic overlap creates a switchback term: $\log |r|=\log \epsilon +\log |z|$. Adding the outer solutions and removing these overlaps leaves the displayed additive composite expansion.

An integrating factor explains the second hint and the first-order inner calculation. Formultiplication by $e^{z^2}$ gives $(e^{z^2}{Y}^{\prime }{)}^{\prime }=e^{z^2}(2a_{1}z+a_2+a_3\mathrm{erf}z)$. The term $a_{1}z$ already produces $2a_{1}z$, and integrating the remaining terms twice gives $a_2{E}_0+a_3{E}_1$, in terms of the Gaussian drift primitives. The homogeneous solutions are a constant and the error function. HenceFor the piecewise forcing in the inner problem, $F$ replaces $a_{1}z$ and provides the continuously matched particular solution for $2zH(-z)$. The coefficients $2c$ and $2d$ then account for the $2Y_0$ forcing.

The solution and its first derivative are continuous at $x=1$: a jump would create a Dirac delta distribution or its derivative, absent from the forcing. Put $r=x-1$, $A=a+1$ and $B=b$. The outer expansions, fixed by the respective endpoint boundary conditions, areBalancing diffusion and advection near $r=0$ gives the interior layer at a simple zero of advection with inner variable $z=r/\epsilon$. The leading inner expansion solves $Y_0^{\prime \prime }+2z{Y}_0^{\prime }=0$, so matching to $A$ and $B$ givesFor the next term define the Gaussian drift primitivesWriting $H$ for the Heaviside step function, let $F(z)=zH(-z)+(\sqrt{\pi }/4)\mathrm{erf}|z|$. Its value and first derivative match at zero and $(D^2+2zD)F=2zH(-z)$. Thuswhere the logarithmic overlaps fixHere $C_1,C_2$ are the constants in the supplied large-positive-$z$ limits of $E_0,E_1$. The terms $\epsilon \log \epsilon$ are essential switchback terms; discarding them would not give accuracy through $O(\epsilon )$.

Subtracting the common overlap from the inner expansion and outer expansions gives the additive composite expansionThe standard overlap subtraction leaves $\epsilon rH(-r)$. Replacing $H(-r)$ by $(1-\mathrm{erf}z)/2$ changes the value only by $O({\epsilon }^2)$ uniformly and gives a composite with continuous first derivative. The last term restores the endpoint values through the retained order. When $b=a+1$, $d=0$ and the order-one error-function jump disappears. An $O(\epsilon )$ interior adjustment remains to accommodate the leading outer derivative mismatch, together with logarithmic matching when $A\ne 0$.

Reversing the diffusion sign changes the leading inner equation to $-Y_0^{\prime \prime }+2z{Y}_0^{\prime }=0$. Its nonconstant solution grows like the integral of $e^{z^2}$, so bounded matching forces the same leading value $C$ on both sides. The bulk is $C+r$ on the left and $C$ on the right; both endpoint conditions are instead supplied by decaying endpoint layers for reversed diffusion, of width ${\epsilon }^2$. A weaker width-$\epsilon$ interior adjustment matches derivatives and selects $C$. Indeed, its first-order equation has a non-growing solution only ifThe leading structure is consequentlywith the smaller interior correction understood. Unlike the positive-diffusion case, the endpoint values are not transported into two distinct order-one inner limits.