= Mixed-boundary convection-diffusion solution in parabolic coordinates
{title2=$u(x,y)=\operatorname{erfc}\sqrt{\frac{\sqrt{x^2+y^2}-x}{2}}$}
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 $u=\operatorname{erfc}b$. Then $\Delta u=u_{bb}/[4(a^2+b^2)]$ and $u_x=-bu_b/[2(a^2+b^2)]$; the derivative identity $u_{bb}=-2bu_b$ verifies the equation. Its boundary traces are $u(x,0)=\operatorname{erfc}\sqrt{-x}$ 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.
Back to article page
