# Solution of the 2D stationary diffusion equation

I'm trying to find the solution to the 2D stationary diffusion equation

$$-D\nabla^2P(\vec{\rho_2})=\delta(\vec{\rho_1}-\vec{\rho_2})$$

where $$\vec{\rho}=(x,y)$$ and $$D$$ is the diffusion coefficient.

Help would be very much appreciated. Also, I'd be happy to get some references as well.

I assume that your sign is correct: that there is a sink rather than a source at $${\bf r}_2$$?
Then this is standard result from electrostatics --- the potential of a line charge: $$P({\bf r}_1, {\bf r}_2)= \frac 1{2\pi D} \ln (\mu|{\bf r}_1- {\bf r}_2|),$$ where $$\mu$$ is a constant that it is not determined by your equation. You need to specify the radius $$\mu^{-1}$$ at which $$P$$ becomes zero. This has to be bigger than zero for a solution to exist.
• Hi Mike, and thanks for the answer. Is this the 2D solution? Does $\vec{r}$ have a $z$ component as well? Nov 16 '20 at 13:59
• I was intending ${\bf r}$ to be two dimensional. In three D with a line sink on the $z$ axis, $|r_1-r_2|$ should be the radial distance from the sink, ie the $r$ in cylindrical polar coordinates. Nov 16 '20 at 14:02
• Feels like this solution still holds if, say, I define $W=-P$. Nov 16 '20 at 16:25
• Yes. That's why I asked if there was a sink rather than a source at the origin. Just change the sign of $P$ and you are OK. Nov 16 '20 at 16:41