 Mixed-boundary convection-diffusion solution in parabolic coordinates

For $y>0$, the displayed function, expressed using the complementary error function, satisfies $\Delta u=u_x$, with $u=1$ on the positive boundary ray and $u_y=0$ on the negative boundary ray. Put $x=a^2-b^2$, $y=2ab$, so $u=\mathrm{erfc}b$. Then $\Delta u=u_{bb}/[4(a^2+b^2)]$ and $u_x=-b{u}_b/[2(a^2+b^2)]$; the derivative identity $u_{bb}=-2b{u}_b$ verifies the equation. Its boundary traces are $u(x,0)=\mathrm{erfc}\sqrt{-x}$ for $x<0$ and $u_y(x,0)=-1/\sqrt{\pi x}$ for $x>0$. At fixed nonzero polar angle it decays at infinity, but along fixed $y>0$ and $x\to +\infty$ it approaches one. The downstream thermal layer has width $O(\sqrt{x})$, so uniform radial decay would be an incompatible far-field condition.