Constant-concentration inlet solution ($C=\frac{1}{2}\mathrm{erfc}\frac{x-Ut}{2\sqrt{Dt}}+\frac{1}{2}{e}^{Ux/D}\mathrm{erfc}\frac{x+Ut}{2\sqrt{Dt}}$)

The half-line solution of $C_t+U{C}_x=D{C}_{xx}$ with initially zero concentration, unit inlet concentration for positive time, and decay at infinity at fixed time. Both complementary error functions are needed to satisfy the inlet exactly. A Laplace transform gives $\tilde{C}=p^{-1}{e}^{(U-\sqrt{U^2+4Dp})x/(2D)}$. A finite layer requires an additional downstream boundary condition.