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MathGroup Archive 2004

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f'[t]== x[t]+u[t] ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51580] f'[t]== x[t]+u[t] ?
  • From: ames_kin at yahoo.com (ames kin)
  • Date: Sat, 23 Oct 2004 00:22:30 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I was thinking about solving that using Mathematica. Clearly I do not have
enough math background. I wanted to get some feedbacks from the group.

let, 

f'[t]== x[t]+u[t] where u[t] is the noise term. or fluctuation term.

When you have a system with noise present, ( as in stochastic ode) is
it Ok or acceptable to obtain the deterministic solution in the form
of Mathematica InterpolatingFunction, and then add the noise when you Evaluate
the InterpolatingFunction.

ie, 

<< Graphics`Colors`
<< Graphics`Graphics`

sol = NDSolve[{Derivative[3][y][x] + Derivative[2][y][x] +
Derivative[1][y][x] == -y[x]^3, y[0] == 1, Derivative[1][y][0] ==
Derivative[2][y][0] == 0}, y, {x, 0, 20}];

which outputs an interpolating function which has to be "evaluated"
before it can be plotted.  If so, If I throw in fluctuation as one
Table the evaluation, it will have an appearance of wide fluctuation.

so, 

Table[Evaluate[y[x] /. sol[[1]]] + Random[Real, {-0.3, 0.3}], {x, 0,
20, 0.2}]

which then can be plotted. 

 ListPlot[Table[Evaluate[y[x] /. sol[[1]]] + Random[Real, {-0.3,
0.3}], {x, 0, 20, 0.2}], PlotStyle -> {GrayLevel[0.8]},  PlotJoined ->
True, PlotRange -> All, Axes -> False, Frame -> True, DisplayFunction
-> Identity];

What is wrong with this method of solution for a system such as 
f'[t] == x[t] + u[t] ? (where u is the noise. this is my, most likely
erroneous, interpretation of stochastic ode)

I mean, what is wrong with this method mathematically.

thanks in advance for any feedback from the group.


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