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Re: NDSolve and Piecewise
*To*: mathgroup at smc.vnet.net
*Subject*: [mg92126] Re: NDSolve and Piecewise
*From*: Szabolcs Horvát <szhorvat at gmail.com>
*Date*: Sat, 20 Sep 2008 05:02:34 -0400 (EDT)
*Organization*: University of Bergen
*References*: <gavt1g$g61$1@smc.vnet.net>
M.G. Bartlett 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?
>
I think that NDSolve attempts to take the derivative of that Piecewise
expression symbolically, which results in a function whose value is
indeterminate at some points (notably 0 and 5):
In[3]:= D[
Piecewise[{{t/10, 0 <= t < 5}, {(10 - t)/10, 5 <= t < 10}}], t]
Out[3]= Piecewise[{{0, t < 0}, {1/10, 0 < t < 5}, {-(1/10),
5 < t < 10},
{0, t > 10}}, Indeterminate]
The error will go away if you wrap the expression into a function that
only evaluates for numerical arguments:
bval[t_?NumericQ] :=
D[Piecewise[{{t/10, 0 <= t < 5}, {(10 - t)/10, 5 <= t < 10}}], t]
(Use bval[t] as the boundary condition.)
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