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Re: Solving systems of differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105248] Re: Solving systems of differential equations
  • From: dh <dh at metrohm.com>
  • Date: Wed, 25 Nov 2009 02:35:08 -0500 (EST)
  • References: <hegdl4$kk$1@smc.vnet.net>


Yun Zhao wrote:

> I have a question as to an apparent inconsistency I noticed while using

> Mathematica to solve differential equations, I have 2 ODE like this, and I

> solve them symbolically using Mathematica 7.0:

> 

> system = {p'[t] == -(rho + d)*p[t],

> n'[t] == rho*p[t] + (rho - d)*n[t]};

> sol = DSolve[system, {p, n}, t][[1]]

> 

> I get the following solution:

> 

> {p -> Function[{t}, E^((-d - rho) t) C[1]],

> n -> Function[{t},

> 1/2 (-E^((-d - rho) t) + E^((-d + rho) t)) C[1] +

> E^((-d + rho) t) C[2]]}

> 

> It looks to be correct.

> 

> Then, manually, I know p(0)=C1, n(0)=0, so C1=p0, C2=0.

> 

> I know the values for rho and d. So for t from 0 to 120, I can calculate and

> plot the values for p(t) and n(t).

> 

> Up to now, this is all good.

> 

> Now I want to solve the system of ODEs numerically (even though I have

> already solved it, I have good reasons to want to learn to solve it

> numerically, as I am going to add more complexity to this model), so I did

> the following:

> 

> sol4 = NDSolve[{p'[t] == -(0.0258 + 0.0123)*p[t],

> n'[t] == 0.0258*p[t] + (0.0258 - 0.0123)*n[t], p[0] == 30000,

> n[0] == 0}, {p, n}, {t, 0, 120}]

> 

> Now I get the following solution:

> 

> {{p-> InterpolatingFunction[{{0.`,120.`}},"<>"],

> n->InterpolatingFunction[{{0.`,120.`}},"<>"]}}

> 

> It may be correct, but I then plotted it using

> 

> Plot[Evaluate[{n[t]} /. First[sol4]], {t, 0, 120},

> PlotRange -> All]

> 

> Now my question is:

> 

> I generated plot #1 of n(t) by solving the system of ODEs first

> symbolically, then manually inputting in values (C1, C2, rho, d).

> I generated plot #2 of n(t) by solving the system of ODEs numerically, then

> had mathematica plot the solutions of and n(t) for me.

> 

> Why does plot #1 and #2 look so different? By that I mean, n(t) in plot 2 is

> much lower than n(t) in plot 1.

> 

> Could you please explain what I did wrong? Thank you.

> 

> I know mathgroup doesn't like attachment, but is there anyway I can show

> these plots?

> 

> 

Hi,

I can not see a difference:



rho0 = 0.0258; d0 = 0.0123; p0 = 30000; n0 = 0;

system = {p'[t] == -(rho + d)*p[t],

    n'[t] == rho*p[t] + (rho - d)*n[t]};

sol = DSolve[system, {p, n}, t][[1]] /. {rho -> rho0, d -> d0,

    C[1] -> p0, C[2] -> n0}

Plot[Evaluate[{p[t], n[t]} /. sol], {t, 0, 120}]



sol4 = NDSolve[{p'[t] == -(rho + d)*p[t],

     n'[t] == rho*p[t] + (rho - d)*n[t], p[0] == p0,

     n[0] == n0} /. {rho -> rho0, d -> d0}, {p, n}, {t, 0, 120}]

Plot[Evaluate[{p[t], n[t]} /. sol4], {t, 0, 120}]



Daniel




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