Solving a PDE with fixed endpoints

• To: mathgroup at smc.vnet.net
• Subject: [mg70756] Solving a PDE with fixed endpoints
• From: "Charlie Brummitt" <cbrummitt at wisc.edu>
• Date: Thu, 26 Oct 2006 02:38:58 -0400 (EDT)

```Hello,

I am trying to solve PDE's with fixed endpoints. That is, my function
is u[x,t], and I fix u[0,t] = A and u[L,t]=B, where A and B are
constants.

However, when I solve these PDE's out to, say, time t = 8,000, I find
that the endpoints don't always stay fixed. Instead, they sometimes
"drift" away from A and B. An example is shown in the following JPG
images:

3D plot of times 7000 through 8000 and plot of u at the final time.
Note that the endpoints should be fixed at 0.1 on both ends, but the
left end (u(0,t)) drifts downward to -120.
http://i13.tinypic.com/2roqszm.jpg

Here is the initial condition, showing that the endpoints should be
fixed at 0.1 on both ends:
http://i14.tinypic.com/4humgww.jpg

Should I be circumspect of the solution that NDSolve gives me if the
endpoint condition isn't satisfied correctly for all time? Is there
any way I could fix this?

I think this phenomenon of the end points drifting away from A, B
occurs whenever there is a constant term in the equation. That is,
du/dt = const. + (rest of equation). Thus, this constant term
increases the value of u at every spatial point, which makes it hard
(impossible?) to satisfy the endpoint conditions.

Thanks for any ideas/suggestions you may have,
Charlie

```

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