# Fourier series of interpolating function result of NDSolve

I am having a tough time formulating the right question but here goes.

I know that solving the pde as in here gives me an interpolating function. I understand that the interpolating function object is different from the interpolating polynomial.

So if I wanted to approximate the interpolating function through FourierSinSeries, I don't quite get how I might go about it.

I can't just do:

FourierSinSeries[InterpolatingFunction[{{0.,...},{0.,...},{0.,TMax}},<>],x,4]

I tried that it didn't quite give me a series expansion.

# Edit:

Here's what I tried to do to get fourier coefficients describing my interpolating function.

## Mathematica code:

Off[NDSolve::ibcinc];
k=0.0677;
{xMin,xMax}={-(\[Pi]/k),\[Pi]/k};
TMax=2500;
uSol[t_,x_]=u[t,x]/.NDSolve[{\!$$\*SubscriptBox[\(\[PartialD]$$, $$t$$]$$u[t, x]$$\)==-100 \!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]$$( \*SuperscriptBox[\(u[t, x]$$, $$3$$]\
\*SubscriptBox[$$\[PartialD]$$, $$x, x, x$$]u[t, x])\)\)+1/3 \!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]$$( \*SuperscriptBox[\(u[t, x]$$, $$3$$]\
\*SubscriptBox[$$\[PartialD]$$, $$x$$]u[t, x])\)\)-5 \!$$\*SubscriptBox[\(\[PartialD]$$, $$x$$]$$( \*SuperscriptBox[\(( \*FractionBox[\(u[t, x]$$, $$1 + u[t, x]$$])\), $$2$$]\
\*SubscriptBox[$$\[PartialD]$$, $$x$$]u[t, x])\)\),u[0,x]==1-0.1 Cos[k x],
(*Piecewise Function for INITIAL CONDITION*)
(*u[0,x]== Piecewise[{{-0.1,-\[Pi]/k<= x<-\[Pi]/(10 k)},{ Cos[x],-\[Pi]/(10 k)<= x<= \[Pi]/(10 k)},{-0.1,\[Pi]/(10 k)<x<= \[Pi]/k}}],*)

(u^(0,1))[t,xMin]==0,
(u^(0,1))[t,xMax]==0,
(u^(0,3))[t,xMin]==0,
(u^(0,3))[t,xMax]==0

(*(u^(0,3))[t,xMin]==0,
(u^(0,3))[t,xMax]==0*)},
u,{t,0,TMax},{x,xMin,xMax},MaxStepSize->0.1,MaxSteps->100000,Method->{"BDF", "MaxDifferenceOrder"-> 5}][[1]]


## Fourier series:

Since I now have uSol which, my fourier coeffs should be:

I1 = NIntegrate[uSol[0.1 TMax, x], {x, xMin, xMax}] (*I1 at time=0.1 TMax*)
a0 = (1/(2*xMin))*I1
an = NIntegrate[uSol[0.1 TMax, x] Cos[n \[Pi] x/xMax], {x, xMin, xMax}]
bn = NIntegrate[uSol[0.1 TMax, x] Sin[n \[Pi] x/xMax], {x, xMin, xMax}]


Using NIntegrate errors out with:

NIntegrate::inumr: The integrand Cos[0.0677 n x] <<1>>[250.,x] has evaluated to non-numerical values for all sampling points in the region with boundaries {{-46.4046,-46.3046}}. >>


However, using Integrate instead of NIntegrate gave me just the input as symbols.

Neither is the coefficient.

So what am I missing? There has to be a simpler way of figurijng out the fourier coeffs of an interpolating function. can I export data out of mathematica into .csv or some other format which is not dependent on mathematica to be interpreted?

-
You might have to expand manually, using the definition for the Fourier coefficients (and thus NIntegrate[]). –  Ｊ. Ｍ. Aug 1 '12 at 14:27
Something like 2 NIntegrate[f[t] Sin[n t], {t, 0, Pi}]/Pi to generate your series coefficients, I meant. –  Ｊ. Ｍ. Aug 1 '12 at 14:31
you could also fourier sine transform the functions you're solving, end up with equations for the coefficients and solve those. but that is more work than what JM says. –  acl Aug 1 '12 at 14:47
Instead of (u^(0,1))[t,xMin], maybe it's Derivative[0,1][u][t,xMin] what you want? –  Silvia Aug 2 '12 at 20:14
NIntegrate can't integrate symbolic arguments. You haven't fixed n so the integrand is not numerical; fix n to something and it should work. Unfortunately cut and pasting doesn't work for the reason @Silvia gave; if you fix that maybe we can help more easily. –  acl Aug 2 '12 at 20:40

# Preliminary

First let me change the PDE that is being solve to make things go a bit faster:

k = 0.0677;
{xMin, xMax} = {-(\[Pi]/k), \[Pi]/k};
TMax = 100;
uSol[t_, x_] = u[t, x] /. NDSolve[{\!$$\*SubscriptBox[\(\[PartialD]$$, $$t$$]$$u[t, x]$$\) == (u[t, x] \!$$\*SubscriptBox[\(\[PartialD]$$, $$x, x$$]$$u[t, x]$$\)),
u[0, x] == 1 - 0.1 Cos[k x],
Derivative[0, 1][u][t, xMin] == 0,
Derivative[0, 1][u][t, xMax] == 0},
u, {t, 0, TMax}, {x, xMin, xMax}, MaxStepSize -> 0.1][[1]]


You want to calculate the nth coefficient, and try

an = NIntegrate[uSol[0.1 TMax, x] Cos[n \[Pi] x/xMax], {x, xMin, xMax}]
bn = NIntegrate[uSol[0.1 TMax, x] Sin[n \[Pi] x/xMax], {x, xMin, xMax}]


which fails with

NIntegrate::inumr: The integrand Cos[0.0677 n x] <<1>> [10.,x] has evaluated to
non-numerical values for all sampling points in the region with boundaries
{{-46.4046,-46.3046}}.


This is literal and obviously true: n isn't fixed here. If I fix n it does give a result:

ClearAll[an, bn]
With[
{n = 1},
an = NIntegrate[
uSol[0.1 TMax, x] Cos[n \[Pi] x/xMax], {x, xMin, xMax}];
bn = NIntegrate[
uSol[0.1 TMax, x] Sin[n \[Pi] x/xMax], {x, xMin, xMax}];
]
an
bn

(*
-4.43265
5.73641*10^-12
*)


(it also emits some warnings probably related to the fact that the $b_n$ actually is zero--although I could be wrong).

Anyway, I don't know if this is the best way to go about it, but this seems to work.

-
Yes, this probably works but I don't know yet know the physical significance of holding n constant. I'll have to think deep and hard about this. –  drN Aug 3 '12 at 2:11
What do you mean? You need n numerical to do the integral numerically, but you could do it for n=1, 2 etc. there's nothing deeper going on. –  acl Aug 3 '12 at 8:44
There might be something deeper going on as far as what n represents as far as harmonics being used as an input to simulate a very thin evaporating liquid film. But thank you! –  drN Aug 7 '12 at 12:44
I am not sure what deeper could be going on (or what you are trying to say), but OK :) –  acl Aug 7 '12 at 12:46