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NIntegrate within NDSolve-optimization?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg128169] NIntegrate within NDSolve-optimization?
  • From: maves <vesel.mateja at gmail.com>
  • Date: Thu, 20 Sep 2012 00:25:26 -0400 (EDT)
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Hello,
I am trying to numerically solve a set of differential equations, and the problem is that the expression also contains numerical integrals, I believe this is why I am getting large numerical errors.

Firstly, I define three integrals (for instance for b=6, c=0.5):

int1[x_?NumericQ, z_?NumericQ] :=
  NIntegrate[-(R (x - R Cos[\[Phi]]))/
   ((x - R Cos[\[Phi]])^2 + (R Sin[\[Phi]])^2 + (z -
      Z)^2)^(3/2), {\[Phi], 0, 2 Pi}, {R, b, b + c}, {Z,
    R - (b + c), -R + (b + c)}];

int2[x_?NumericQ, z_?NumericQ] :=
  NIntegrate[-(R (z-Z))/
   ((x - R Cos[\[Phi]])^2 + (R Sin[\[Phi]])^2 + (z -
      Z)^2)^(3/2), {\[Phi], 0, 2 Pi}, {R, b, b + c}, {Z,
    R - (b + c), -R + (b + c)}];

int3[x_?NumericQ, z_?NumericQ] :=
  NIntegrate[-R/((x - R Cos[\[Phi]])^2 + (R Sin[\[Phi]])^2 + (z -
      Z)^2)^(1/2), {\[Phi], 0, 2 Pi}, {R, b, b + c}, {Z,
    R - (b + c), -R + (b + c)}];

And then I use them in NDSolve:

s = NDSolve[{
    x''[t] == int1[x[t], z[t]],
    z''[t] == int2[x[t], z[t]],
    x[0] == 7.1, x'[0] == 0, z'[0] == 0, z'[0] == 1}, {x, x=
', z,
    z'}, {t, 0, 5}][[1]]

And plot the solution:
ParametricPlot[Evaluate[{x[t], z[t]} /. s], {t, 0, 5},
 PlotRange -> All]

Plot[Evaluate[0.5*((x'[t])^2+(z'[t])^2+int3[x[t],z[t]]) /. s], {t, 0, 5}]

I change the initial conditions and try to find something "useful", but this second plot shows me, that something just isn't right, as it should be constant, but it oscillates with a huge amplitude. Could this be due to numerical errors and how you think I should optimize my code to get a good result? Is this combination of NIntegrate and NDSolve even acceptable?

Thank you.

But the solution just doesn't seem right



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