[Physics] 1D drift-diffusion equation with single absorbing boundary

Question

If we have just the simple diffusion equation (in 1D):
$$
\frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{\partial x^2}
$$
with an absorbing boundary at x=0 and initial condition $P(x,0) = \delta(x-x_0)$, we can use the method of images to get the solution
$$
P(x,t) = \frac{1}{\sqrt{4 \pi D t}}e^{\frac{-(x-x_0)^2}{4 D t}} – \frac{1}{\sqrt{4 \pi D t}}e^{\frac{-(x+x_0)^2}{4 D t}}.
$$
However I am interested in solving this in the case where there is also a drift (ultimately one that is not constant in time, but to start with just a solution with a constant drift velocity would be great). I haven't been able to find anything about this problem, does anyone have any ideas?

Answer 1

An alternative could be to use separable solutions, that will give solutions which are exponentially decaying in time and trigonometric functions in space. The decay time is identified with the eigenvalues of the spatial part, in accordance with the absorbing boundary at x = 0.