There is a one-dimensional diffusion process in which particles start running at $t = 0$ and from $x_o > 0$.
When particles reach x = 0 they are removed from the system, thus the total concentration is not conserved anymore.
I have to solve the diffusion equation, which is the following partial differential equation:
$$\frac{\partial P (R, t)}{\partial t} = D\triangledown^2P(R,t) $$
Where $P(R, t)$ is the probability that the particles arrive at R at time t.
And I am given the initial conditions:
$$c(x,0) = N\delta(x - x_o)$$
$$c(0, t) = 0$$
I have been doing some research in how to do so and I came across with a method which is based on a particular Gaussian function:
$$G (R, t) = (\frac{1}{4\pi Dt})^{\frac{d}{2}} e^{\frac{(R-R_0)^2}{4Dt}}$$
Where d is the dimensionality of the system.
But the issue here is that we are working with an 'absorbing boundary' that makes the condition $c(0, t) = 0$ useless because we work from $x_o > 0$.
Then how could I solve the probability (i.e this differential equation)? I have been suggested the method of images, but not sure about it.