Discrete Fourier Transform of NDsolve in Mathematica?

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Discussion Overview

The discussion revolves around performing a discrete Fourier transform on solutions obtained from NDSolve in Mathematica, specifically addressing the challenges posed by the use of Interpolating functions instead of numerical data.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses difficulty in applying a discrete Fourier transform to the output of NDSolve due to the presence of Interpolating functions.
  • Another participant suggests using the InterpolatingFunction to generate a table of equally spaced points before applying the Fourier transform.
  • A subsequent reply indicates that the suggested approach resulted in a table of interpolating functions rather than numerical values.
  • A participant provides a detailed example of using NDSolve, including the equations and the resulting output, to illustrate the issue further.
  • One participant acknowledges the help received from the suggestions provided in the thread.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus on how to effectively convert the output of NDSolve into a format suitable for Fourier analysis, with multiple approaches being proposed but no clear resolution on their effectiveness.

Contextual Notes

The discussion does not clarify the specific limitations of the InterpolatingFunction or the assumptions underlying the proposed methods for generating numerical data.

Catrin
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I want to do a discrete Fourier transform of the solution I have found using NDSolve, however, because the NDSolve creates Interpolating functions rather than numbers I can't do this. Any help is appreciated. I've attatched the file I'm working with.

Catrin
 

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Use the InterpolatingFunction to create a Table of equally spaced points and then do Fourier on the table?
 
That's what I tried, but I just get a table of interpolating functions.
 
It is sometimes "less than obvious" how to correctly use an InterpolatingFunction.

In[1]:=eq1 = {D[a[t], {t, 2}] + 0.22 (a[t])^2 + a[t]==0, a[0]==2, a'[0]==0}

Out[1]={a[t]+0.22a[t]^2+a''[t]==0,a[0]==2,a'[0]==0}

In[2]:= timesol01=NDSolve[eq1,a, {t,0,200}];

In[3]:=Table[First[a[t]/.timesol01],{t,0,200}]

Out[3]=
{2.,0.759429,-1.40946,-2.76245,-3.0405,-2.28904,-0.466852,1.58763,1.71456,-0.\
252706,-2.15986,-3.01434,-2.83345,-1.58573,0.552881,1.98694,0.957919,-1.22436,\
-2.6816,-3.05752,-2.40789,-0.677208,1.44126,1.82006,-0.0364096,-2.02039,-2.\
97895,-2.89475,-1.75264,0.340689,1.94796,1.14587,-1.03115,-2.59077,-3.06543,-\
2.51646,-0.882344,1.27765,1.90243,0.180197,-1.87073,-2.93423,-2.9465,-1.90977,\
0.125161,1.88374,1.32079,-0.830714,-2.48981,-3.06428,-2.61483,-1.08103,1.\
09918,1.96031,0.395086,-1.71108,-2.88007,-2.98884,-2.05685,-0.0915471,1.79535,\
1.48022,-0.62413,-2.37863,-3.05405,-2.70311,-1.27225,0.908322,1.99273,0.\
606058,-1.54176,-2.81631,-3.02187,-2.1937,-0.307455,1.68423,1.62185,-0.412673,\
-2.25715,-3.03472,-2.78143,-1.45516,0.707557,1.99915,0.81078,-1.36319,-2.\
74282,-3.04569,-2.32024,-0.520773,1.55216,1.7435,-0.197828,-2.12536,-3.00622,-\
2.84995,-1.62909,0.499359,1.97946,1.00682,-1.17596,-2.65945,-3.06039,-2.43647,\
-0.729919,1.40119,1.84326,0.0187196,-1.98329,-2.96847,-2.90881,-1.79353,0.\
286117,1.93399,1.19168,-0.980831,-2.56606,-3.066,-2.54244,-0.933512,1.23359,1.\
91951,0.235077,-1.83108,-2.92136,-2.95815,-1.94812,0.0701023,1.86351,1.36288,-\
0.778733,-2.46252,-3.06254,-2.63822,-1.13038,1.05175,1.97099,0.449173,-1.\
66895,-2.86477,-2.99811,-2.0926,-0.146573,1.76919,1.518,-0.570791,-2.34873,-3.\
05,-2.72395,-1.31955,0.858138,1.99684,0.658774,-1.49723,-2.79856,-3.02879,-2.\
22683,-0.361975,1.65257,1.65473,-0.358335,-2.22464,-3.02835,-2.79977,-1.50022,\
0.655263,1.99662,0.861512,-1.3164,-2.72258,-3.05029,-2.35075,-0.574364,1.\
51551,1.77098,-0.142899,-2.09023,-2.99751,-2.86581,-1.67178,0.445571,1.97033,\
1.05494,-1.1271,-2.63668,-3.06267,-2.46436,-0.782212,1.3601,1.86491,0.0737764,\
-1.94558,-2.95739,-2.92224,-1.83374,0.231426,1.91842,1.23655,-0.930117,-2.\
54072,-3.06598}

In[4]:= Plot[Evaluate[First[a[t]/.timesol01]],{t,0, 4*Pi}]

Out[4]=<<<plot snipped>>>
 
Thank you, that really helped.
 

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