This question seems so simple, yet the answer eludes me..
I'm trying to solve the advection-diffusion equation on the half-line x > 0:
$$ u_t = u_x + u_{xx} $$
The advection is in the direction of the origin. There is no flux across the origin, i.e., $u(0, t) + u_x(0, t) = 0$.
The equation represents the transient concentration of buoyant particles in the ocean. The first term is the buoyancy and the second one is the turbulent diffusion. $x$ is the depth below sea level and $x = 0$ is the sea surface.
Solutions are welcome of course, but any insight into the problem is helpful as well.