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MathGroup Archive 2005

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Re: Using InterpolatingFunction from NDSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58565] Re: Using InterpolatingFunction from NDSolve
  • From: "Kevin J. McCann" <kjm at KevinMcCann.com>
  • Date: Fri, 8 Jul 2005 00:45:56 -0400 (EDT)
  • References: <daitir$su4$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I added a couple of definitions at the end, but most importantly, I used 
NIntegrate rather than Integrate. The latter fails in its attempt to 
find an exact expression for the integral.

Kevin

\!\(nds = NDSolve[{
     D[u\ [t, x], {t,
       1}] == D[u\ [t, x], {x, 2}] + u[t, x]\^2 -
           u[t, x], u[0, x] == Sin[x], u[t, 0] == 0, u[t, 2\
             Ï?] == 0}, u[t, x], {t, 0, 1}, {x, 0, 2\ Ï?}]\n
   Plot3D[Evaluate[u[t, x] /. nds[\([\)\(1\)\(]\)]], {t, 0, 1}, {x,
          0, 2\ Ï?}]\n
   Plot3D[Evaluate[Integrate[\((u[t, x] /. nds[\([\)\(1\)\(]\)])\), {t,
          0, s}]], {s, 0, 1}, {x, 0, 2  Ï?}]\n
   Plot3D[Evaluate[D[u[t,
        x] /. nds[\([\)\(1\)\(]\)], {x, 2}]], {t, 0, 1}, {x, 0, 2  Ï?}]\n
   Integrate[D[u[t, x] /. nds[\([\)\(1\)\(]\)], {x, 2}], {t, 0, s}]\n
   Plot3D[Evaluate[Integrate[D[u[t, x] /. nds[\([\)\(1\)\(]\)], {
             x, 2}], {t, 0, s}]], {s, 0, 1}, {x, 0, 2
               Ï?}]\n\[IndentingNewLine]
   u[t, x]^2 /. nds[\([1]\)]\n
   Plot3D[Evaluate[u[t, x]^2 /. nds[\([\)\(1\)\(]\)]], {t,
     0, 1}, {x, 0, 2\ Ï?}, AxesLabel -> {t, x, u[t, x]}]\n
   u2[t_, x_] = u[t, x]^2 /. nds[\([1]\)]\n
   f[x_, s_] := NIntegrate[u2[t, x], {t, 0, s}]\n
   Plot3D[f[x, s], {s, 0, 1}, {x, 0, 2  Ï?}]\n
   f[1,  .5]\)

Tamás wrote:
> I solved a PDE with NDSolve in Mathematica 5.1. I could plot,
> differentiate and integrate the obtained InterpolatingFunction object,
> the result being a similar object. I was able to integrate the 2nd
> derivative of it. What I need is to integrate the square of the
> obtained InterpolatingFunction object (the square itself does not
> simplify to such an object).
> You can see the details on my homepage:
> http://www.math.bme.hu/~tladics/nds.nb
> 
> Every suggestions are welcome!
> 
> Thank you,
> Tamás
> 


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