Re: NDSolve and Piecewise

• To: mathgroup at smc.vnet.net
• Subject: [mg92108] Re: NDSolve and Piecewise
• From: "M.G. Bartlett" <marshall.bartlett at gmail.com>
• Date: Sat, 20 Sep 2008 04:59:11 -0400 (EDT)
• References: <gavt1g\$g61\$1@smc.vnet.net>

```On Sep 18, 11:57 pm, "M.G. Bartlett" <marshall.bartl... at gmail.com>
wrote:
> Folks,
>
> I am having some trouble with getting Piecewise and NDSolve to play
> nicely together.  My problem is to find a solution to the heat flow
> equation with an arbitrary time-varying upper boundary condition and a
> Neumann-type lower boundary (steady head flow condition).  My code
> looks like this:
>
> NDSolve[{D[u[t, x], t] == D[u[t, x], x, x],
>   u[t, 0] == Piecewise[{{t/10, 0 <= t < 5}, {(10 - t)/10, 5 <= =
t <
> 10}}],
>   u[0, x] == x/5, (D[u[t, x], x] /. x -> 5) == 1/5}, u, {t, 0, =
10},
> {x, 0, 5}]
>
> This returns and NDSolve::ndum error on my system, which past
> experience tells me is usually me leaving some symbolic value hanging
> around somewhere it ought not to be.  I can't see any such problem
> this time.  I'm pretty sure that I have written similar code in the
> past (using Piecewise and NDSolve, though it was some time ago and I
> can't locate the file now) and had it work, and it works if you
> replace Piecewise with another functional form (like Sin[t]).  Am I
> missing something?
>
> Thanks,
>
> Marshall

Thanks to those of you who replied and indicated (as I discovered
shortly after posting) that the error is associated with the piecewise
function not having a well-defined derivative at the boundaries (t=0
and t=10, in the example above).  A couple of you indicated that this
can be remedied by extending the piecewise function across the model
domain boundaries (i.e., replace the condition 0 <= t < 5 with t < 5
and likewise for the boundary at the other end of the time domain).
This "fix" does the trick.

I guess I just figured that Mathematica would be able to tell that it
should approach the derivative at the boundary from only a single
side.  I get so use to having the program see a problem the same way
that I do that it can be surprising when it behaves differently
(though, as in this case, Mathematica is of course technically correct
to complain).

Thanks again to all those who responded.

Cheers,

Marshall

```

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