The advection-diffusion problem, where $0<x<1$,
$$u_t = (-au + du_x)_x$$
with Dirichlet boundary conditions, $ u(0)=1,~u(1)=0 $ , has the steady-state solution,
$$ u(x) = \frac{e^{\lambda} - e^{x\lambda} }{e^{\lambda} - 1} $$
This of course assumes both $a$ and $d$ are constants and $\lambda=a/d$.
Q. Does this problem also have time-dependent analytical solution?
Below I have written some tests for numerical code on various grids and I would like to compare with the analytical solution (the black line) as a function of time.