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Re: DSolve test
*To*: mathgroup at smc.vnet.net
*Subject*: [mg132728] Re: DSolve test
*From*: Roland Franzius <roland.franzius at uos.de>
*Date*: Wed, 14 May 2014 05:25:18 -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>
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]; Print[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
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