I am working on a project on the semiconductor equations and I have arrived at the following PDE for electron transport with no electric field and initial carrier density $n_0$: $$\frac{\partial n}{\partial t} = - \frac{n-n_0}{\tau_n} + D_n\frac{\partial^2 n}{\partial x^2}$$
I have come across some literature saying this has solution $$n(x,t) = \frac{N} {(4\pi D_nt) ^{1/2}}\exp\left(-\frac{x^2}{4D_nt}-\frac{t}{\tau_n}\right)+n_0$$
Where $N$ is the number of electrons/holes initially generated per unit area, $D_n$ is the diffusive constant for electrons and $\tau_n$ is the carrier lifetime. However working through I can't arrive at this myself and ideally need to understand the mathematics behind it.
If anyone could go through how this solution is reached it would be much appreciated.