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Re: Solving Ordinary differential equations by NDSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104984] Re: Solving Ordinary differential equations by NDSolve
  • From: pratip <pratip.chakraborty at gmail.com>
  • Date: Sun, 15 Nov 2009 20:49:10 -0500 (EST)
  • References: <hdomqf$93a$1@smc.vnet.net>

On Nov 15, 11:57 am, Allamarein <matteo.diplom... at gmail.com> wrote:
> I'd solve this ODE:
>
> V'[t] == A Sqrt[2 (Po (V0/V[t])^gamma - Pa)/rho
> IC: V[0] == V0
>
> I wrote this code:
>
> d = 2*10^-2 ;
> A = d^2  3.14/4;
> Po = 5 *101325;
> Pa = 1*101325 ;
> rho = 1000 ;
> V0 = 5*10^-3 ;
> gamma = 1.14;
> sol = NDSolve[{
>     V'[t] == A Sqrt[2 (Po (V0/V[t])^gamma - Pa)/ rho],
>     V[0] == V0},
>    {V}, {t, 0, 9},
>    MaxSteps -> 1000000, AccuracyGoal -> 10, PrecisionGoal -> 10];
> v[t_] := V[t] /. sol[[1]];
> Plot[Evaluate[V[t] /. sol]*1000, {t, 0, 9}, PlotRange -> All]
> Grid[Table[{t, v[t]}, {t, 0, 9, 1}]]
>
> If it can be useful, I can underline units of these variables:
> d [m]
> P0, Pa [Pa]
> rho [kg/m^3]
> V [m^3]
> gamma [--]
> t [sec]
>
> Running this code, V has got comlex part. This is impossible, because
> it's a volume.
> I should re-write my ODE in order NDSolve can digest better or I can
> set an option in my code where I suggest " V must be positive and
> real"?

Hi,
There is nothing to be panicked about the complex values of V.
Just use Chop to get rid of the complex tail which is nothing but a
artifact generated during numerical computation.

You can see it yourself
Plot[Evaluate[V[t]/.sol//Chop]*1000,{t,0,9},PlotRange->All]
Grid[Table[{t,v[t]//Chop},{t,0,9,1}]]
And also to see the residual error of the solution generated by
NDSolve evaluate the following
Er=(V'[t]==A Sqrt[2 (Po (V0/V[t])^gamma-Pa)/rho])/.a_==b_-> (a-b)=
^2/.V
[t]-> v[t]/.V'[t]-> v'[t];
Grid[Table[{t,Er//Chop},{t,0,9,1}]]
Now you will realize that there is nothing wrong with the solution.
Regards,

Pratip


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