Advection–diffusion equation

Remarks

Example: in one dimension with constant $u = U$, constant $D$ and $R = 0$, the function $$c(x, t) = \frac{1}{\sqrt{4 \pi D t}} \exp\!\left( -\frac{(x - U t)^2}{4 D t} \right), \quad t > 0,$$ is a solution. Indeed $c(x, t) = G(x - U t, t)$ where $G$ solves the heat equation $G_t = D G_{yy}$, and then $c_t + U c_x = G_t = D c_{xx}$.

Non-example: with $u = U \ne 0$, the heat kernel $G(x, t)$ itself is not a solution, since $G_t + U G_x - D G_{xx} = U G_x \ne 0$.

For $u = 0$ the equation is the heat equation, and for $D = 0$ it is the transport equation. When $c$ is a temperature, $D$ is the thermal diffusivity.