The solution of diffusion equation $$ \partial_t\rho=D\nabla^2\rho$$ with a point source $$ \rho(0,z)=\delta(z)$$
is in 1 dimension $$ \rho(t,z)=\frac{1}{\sqrt{4\pi Dt}}e^{-\frac{z^2}{4Dt}}$$
My question is : is there any analytical solution of the same problem with a boundary solution $$v(z_0)=0=-D\partial_z\rho(z_0)$$
meaning if you put a wall somewhere ?
My feeling is that the problem would be overconstrained. Is it right ?
If it's possible, what would be the way to achieve it ?
Thanks in advance
Edit 1 :
And if it's possible is it also possible to add a force towards the wall ? I mean for example if you have diffusion + gravity : $$v=-v_g-\frac{D\nabla\rho}{\rho} $$ which associate with : $$\partial_t \rho +\nabla.(\rho v)=0 $$ to
$$\partial_t \rho=D\nabla^2\rho + v_g \nabla\rho $$
and in 1d : $$\partial_t \rho= D\frac{d^2\rho}{dz^2} + v_g \frac{d\rho}{dz} $$
This can be simplified thanks to a change of variable : $z\rightarrow\tilde{z}=z+v_g t$ into : $$\partial_t \rho= D\frac{d^2\rho}{d\tilde{z}^2} $$
and the solution is $$ \rho(t,z)=\frac{1}{\sqrt{4\pi Dt}}e^{-\frac{(z-vt)^2}{4Dt}}$$
So I was wondering how this solution was changed if there is a wall.
Edit 2
I found this link in wikipedia about the Mason-Weaver equation where a solution for my second equation in the edit seems to be derived.