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Re: DSolve test
*To*: mathgroup at smc.vnet.net
*Subject*: [mg132732] Re: DSolve test
*From*: carlos.felippa%colorado.edu at gtempaccount.com
*Date*: Thu, 15 May 2014 02:25:58 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*Delivered-to*: l-mathgroup@wolfram.com
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*References*: <lkhrc6$e0r$1@smc.vnet.net> <lkvcqh$fff$1@smc.vnet.net>
On Wednesday, May 14, 2014 3:25:37 AM UTC-6, Roland Franzius wrote:
> Am 09.05.2014 08:08, schrieb carlos.felippa%colorado.edu at gtempaccount.com:
>
> > Hi, could somebody try this in Mathematica 9 to see if the bug is fixed?
>
> >
>
> > sol = DSolve[{u'[t] == 2*Sqrt[u[t]], u[0] == 0}, u[t], t]; Pint[sol];
>
>
>
> In all versions until now, Mathematica does not check the Lipshitz
>
> condition |f(u(t))-f(u(0))| < C |u(t)-u(0)| that guaranties uniqueness.
>
>
>
> The results for nonlinear ordinary differential equations are just those
>
> you find in the usual lists like Kamke oder EquationWorld.
>
>
>
> There exist no general algebraic nonlinear solving methods for nonlinear
>
> ODE's except linear substitutions, separation of variables and lookup
>
> tables.
>
>
>
> Generally as a student of ODE one learns to look for points of
>
> discontinuities, here u=0, which generally allow branching:
>
>
>
> f(t):=0 /;t<=t0
>
> f(t):=(t-t0)^2/; t>t0>=0
>
>
>
> is the solution family on (-oo,oo).
>
>
>
> Dsolve has no entry for a domain construct like NDSolve
>
>
>
> NDSolve[{u'[t] == 2*Sqrt[u[t]], u[0] == 0}, u[t],{t,0,10}]
>
>
>
> that will give you the missing constant solution and. But
>
>
>
> In[20]:= v[t_] =
>
> u[t] /. NDSolve[{u'[t] == 2 Sqrt[u[t]], u[1] == 10^-12},
>
> u[t], {t, 0, 7}][[1]]
>
>
>
> There occurs an error in the internal procedure at the critical point 1:
>
>
>
> >>During evaluation of In[20]:= NDSolve::mxst: Maximum number of 10000=
>
> >>steps reached at the point t == 0.9998803430478553`.
>
>
>
> So even with CASystems, it is still the fact that differentiation is an
>
> algebraic functor, numerical integration is a trivialtity for smooth
>
> functions, algebraic integration is a mystery and a historical source of
>
> many branches of mathematics.
>
>
>
> The solution of nonlinear differential equations remains an ingenious
>
> kind of art, Ricatti, Clairaut, d'Alembert are some of the protagonists.
>
>
>
> --
>
>
>
> Roland Franzius
This can be trivially solved by hand using separation of variables. No need for Count Ricatti, or Jean Le Rond d'Alembert, who are long gone.
I gave that ~4 yrs ago as an exercise: integrate with a CAS, integrate by hand, and integrate numerically via classical RK (C++). The numerical solution is of course u=0, the CAS solution (Mathematical 7) was u=t^2, and the hand solution is the branch pair. Students were confused.
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