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Re: NDSolve and vector functions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg87881] Re: NDSolve and vector functions
*From*: Szabolcs Horvát <szhorvat at gmail.com>
*Date*: Sat, 19 Apr 2008 03:35:02 -0400 (EDT)
*Organization*: University of Bergen
*References*: <fu9vl0$ie2$1@smc.vnet.net>
dh wrote:
> Hello all,
>
> does anybody know a way to compute a vector valued function using
>
> NDSolve without explicitely specifying all vector components. Here is a
>
> simple example: Although NDSolve[{p'[t]==p[t],p[0]=={1,0}},p,{t,0,1}] works,
>
> NDSolve[{p'[t]==p[t]+{1,1},p[0]=={1,0}},p,{t,0,1}]
>
> does not work because p[t] in "p[t]+{1,1}" is treated as a scalar and
>
> the expression is evaluated to {1+p[t],1+p[t]} what is clearly not
>
> intended. Even in "IdentityMatrix[2].p[t]+{1,1}"
>
> "IdentityMatrix[2].p[t]" is treated like a scalar and added to the
>
> components of {1,1}.
>
> do I miss something???
Hi Daniel,
Obviously, there are workarounds, but they might not be pretty ... I
think that this is a fundamental limitation of Mathematica ... Most
functions (incuding Plus) assume that symbols represent scalars ...
A workaround is to define a new `plus' function that is only evaluated
when its arguments are vectors:
vecPlus[arg__?VectorQ] := Plus[arg]
Now this works:
f = p /. First@
NDSolve[{p'[t] == vecPlus[p[t], {1, 1}], p[0] == {1, 0}},
p, {t, 0, 1}]
Plot[f[x], {x, 0, 1}]
One disadvantage of this approach is that it prevents NDSolve from
transforming the ODE symbolically. But this is really a limitation
stemming from using vector functions, and not a problem introduced by
vecPlus. A more serious problem is that most probably vecPlus cannot be
compiled, so this might slow NDSolve down.
Szabolcs
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