# 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

Because the advection-diffusion equation is linear, there are many exact solutions. One reference with many exact solutions (including source terms) is

M. Th. van Genuchten and W. J. Alves Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation U. S. Department of Agriculture, Agricultural Research Service Technical Bulletin 1661, 1982

You can get a pdf of this report here.

If you search for papers that cite this report, you will turn up many additional problems with exact solutions.

• That's a very nice report! – boyfarrell Jul 6 '13 at 1:29

Add a right-hand-side/forcing function, and use the Method of Manufactured Solutions. Then you can have any solution you want. Also, Separation of Variables works on this PDE just fine, so it also has an analytical solution.