1D drift-diffusion equation with single absorbing boundary 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?
 A: 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. 
A: Here's one way to solve it analytically:
The diffusion operator with a drift term is diagonalized by Hermite polynomials. Scaling the drift-diffusion equation gives
\begin{equation}
{\partial \over \partial t} f(t,v) = {\partial \over \partial v} \left(v+{\partial \over \partial v} \right) f(t,v),
\end{equation}
Expanding $f(t,v)$ in terms of (probabilist's) Hermite polynomials
\begin{equation}
f(t,v) = \sum_{n=0}^\infty f_n(t) e^{-v^2/2} \mathit{He}_n(v),
\end{equation}
and applying the operator ${1 \over \sqrt{2 \pi} n!} \int_{-\infty}^\infty dv\; \mathit{He}_n(v)$, to both sides of the first equation, gives
$$
{\partial \over \partial t} f_n(t) = -n f_n(t),
~~~~~~~
\Rightarrow~~~~~~~
f_n(t) = f_n(0) e^{-nt}.
$$
Using
$$
f_n(t) = {1 \over \sqrt{2 \pi} n!} \int_{-\infty}^\infty dv\; f(t,v) \mathit{He}_n(v),
$$
allows one to write
\begin{equation}
f(t,v) = {e^{-v^2/2} \over  \sqrt{2 \pi}} \int_{-\infty}^\infty dv' \;  \left[ \sum_{n=0}^\infty  {e^{-nt} \over n!}  \mathit{He}_n(v) \mathit{He}_n(v') \right]  f(0,v').
\end{equation}
Now, the Kernel in the above equation can be identified as the Mehler kernel
\begin{equation}
 \sum_{n=0}^\infty \frac{\rho^n}{n!} ~ \mathit{He}_n(x)\mathit{He}_n(y) = 
 \frac 1{\sqrt{1-\rho^2}}\exp\left(-\frac{\rho^2 (x^2+y^2)- 2\rho xy}{2(1-\rho^2)}\right),
\end{equation}
which allows the propagator solution to the drift-diffusion equation to be written explicitly as
$$
f(t,v) = \int_{-\infty}^\infty dv' \; \left\lbrace {1 \over \sqrt{2 \pi(1-e^{-2t})}} \exp \left[ - {(v-v'e^{-t})^2 \over 2(1-e^{-2t})} \right] \right\rbrace f(0,v').
$$
Now, we can define the free-space Green's function
$$
G_{0}(t-t',v,v') = {\Theta(t-t') \over \sqrt{2 \pi(1-e^{-2(t-t')})}} \exp \left[ - {(v-v'e^{-(t-t')})^2 \over 2(1-e^{-2(t-t')})}\right].
$$
To enforce an absorbing boundary condition at $v = 0$, define the new Green's function for $v>0$,
$$
G_{absorbing}(t-t',v,v') = G_0(t-t',v,v')-G_0(t-t',-v,v').
$$
