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MathGroup Archive 2006

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Re: Why doesn't Mathematica solve this simple differential equation?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69355] Re: Why doesn't Mathematica solve this simple differential equation?
  • From: Joseph Gwinn <JoeGwinn at comcast.net>
  • Date: Thu, 7 Sep 2006 23:59:01 -0400 (EDT)
  • Organization: Gwinn Instruments
  • References: <eddqq8$3vq$1@smc.vnet.net> djhhr$llh$11smccvnee.neee dm117$cc7$11smccvnee.neee <edom8a$hve$1@smc.vnet.net>

In article <edom8a$hve$1 at smc.vnet.net>, p-valko at tamu.edu wrote:

> If you want Mathematica to be smart, let her do the job symbolically:
> 
> In:
> DSolve[eqns41, {Vc[t], Il[t]}, t] // First
> 
> Out:
> {Il[t] -> Vo/(E^(t/(C1*R1 + C1*R2))*(R1 + R2)), Vc[t] -> Vo/E^(t/(C1*R1
> + C1*R2))}
> 
> In:
> % /. {C1 -> 1.0*10^-6, R1 -> 16, R2 -> 27, Vo -> 4.0}
> 
> Out:
> {Il[t] -> 0.093023 * E^(-23255.81*t), Vc[t] -> 4. * E^(-23255.81*t)}

I did try this.  When I do this with the diode equation, the symbolic 
answer is far from simple, and really isn't very useful.

Thanks,

Joe

> Regards
> P.V.
> 
> Joseph Gwinn wrote:
> > In article <edjh7r$luh$1 at smc.vnet.net>,
> >  Joseph Gwinn <JoeGwinn at comcast.net> wrote:
> >
> > > In article <eddqq8$3vq$1 at smc.vnet.net>,
> > >  Joseph Gwinn <JoeGwinn at comcast.net> wrote:
> > >
> > > > Here is the system I'm trying to solve.  It's an electrical circuit
> > > > consisting of a capacitor C1 (with initial voltage 4.0 volts), a
> > > > resistor R1, and a diode in series.
> > >
> > > I just read the Mathematica 5.2 NDSolve Advanced Documentation section
> > > on  Differential Algebraic Equations.
> > >
> > > It occurs to me that, aside from the typos, what may be happening is
> > > that I'm handing Mathematica something that looks like a Differential
> > > Algebraic Equation (DAE) of index exceeding 1, as Mathematica was able
> > > to solve the system with one implicit equation (for the diode, with a
> > > capacitor but no resistor).
> >
> > I now have a clean example of the problem:
> >
> > eqns41 = {Vc'[t] == -Il[t]/C1, Vc[t] == (R1 + R2)*Il[t], Vc[0] == Vo}
> >
> > eqns42 = eqns41 /. {C1 -> 1.0*10^-6, R1 -> 16, R2 -> 27, Vo -> 4.0}
> >
> > eqns42soln = NDSolve[eqns42, Vc, {t, 0, 1}]
> >
> > The above fails: "NDSolve::overdet: There are fewer dependent variables,
> > {Vc[t]}, than equations, so the system is underdetermined."
> >
> > eqns41a = Eliminate[eqns41, Il[t]]
> >
> > eqns42a = eqns41a /. {C1 -> 1.0*10^-6, R1 -> 16, R2 -> 27, Vo -> 4.0}
> >
> > eqns42asoln = NDSolve[eqns42a, Vc, {t, 0, 1}]
> >
> > The above works.
> >
> > Eliminate[] cannot make a self-consistent system from an inconsistent
> > system, so the original system must also be consistent.  Yet it fails.
> >
> > Does Mathematica think that this system is some kind of complicated DAE?
> > Apparently, given that Eliminate[] solves the problem.  Why couldn't
> > NDSolve do its own algabraic reduction?  The variable Il[t] was not
> > requested as an output.
> >
> > I have a few more such examples, so the issue isn't restricted to this
> > example.
> >
> > The other issue is that *all* combinations of real electronic components
> > (that is, circuits) lead to a numerically solvable system of ODEs,
> > because all circuits will do something real if constructed and tested.
> > So, aside from eliminating silly mistakes, this should not be hard, and
> > I'm trying to figure out the root cause of these random-appearing
> > failures.  Mathematica is probably trying to do something that ordinary
> > circuit simulators (such as SPICE) wouldn't dream of.
> > 
> > Joe Gwinn


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