# Does the time-dependent advection-diffusion equation have an analytical solution?

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.

http://vimeo.com/69527955

• Your description isn't quite correct: the solution for the boundary values given is $u=1$. I think you have the b.c. wrong. – Wolfgang Bangerth Jul 3 '13 at 15:52
• Thanks for pointing that out. The right had side value should have been $u=0$. Typo. fixed above. – boyfarrell Jul 3 '13 at 23:09