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.

We have 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)? We have been suggested the method of images, but not sure how it works with differential equations.

NOTE: I know this is more a mathematical issue (solving a differential EQ.) but thought that as it is a mechanics problem it could be useful asking here.

https://mathoverflow.net/questions/289748/how-to-define-and-solve-the-diffusion-equation-with-a-sticky-boundary-at-the-o

https://home.icts.res.in/~abhi/notes/lect3.pdf