Items tagged with fourier

how to solve  Fourier Series on maple ??

I have computed the eigenfunction expansion for f(x)=x on 0<x<1 in terms of the eigenfunctions exp(-x/2)*sin(n*Pi*x).

I wish to calculate the weighted L2 error in this expansion (the weight function is w(x)=exp(x)).

Specifically, I want to determine how many terms in the eigenfunction expansion are necessary for the error to be less than say 0.3.

Here is the code:

f := x -> x
w := x -> exp(x)
assume('n', integer);
y :=  (n, x) -> exp(-x/2) sin(n Pi x)       
c := n-> (int(f(x)*y(n, x)*w(x), x = 0 .. 1))/(int(y(n, x)^2*w(x), x = 0 .. 1))
Fourierf := (n, x) -> sum(c(j)*y(j, x), j = 1 .. n)

fsolve(Lerror(n) = 0.3, n);

This seems to run forever without giving a value of n.  I know this is a large computation, but it seems that Maple should be able to handle it.  Does anyone have any suggestions?

Heather

 

Hi

 

I want to write the functional Z of J Z = exp(Int(Int(J(x)*Delta(x-y)*J(y), x), y))with Delta(x) = Int(I*exp(-I*k*x)*(1/(k^2-m^2)), k) in terms of the fourier transform of J: J(x) = Int(J(p)*exp(-I*p*x), p).

Actually I'm in Minkowski space and all the integrals should be over 4 dimensions, x,y,k,p should all be four-vectors, but I wanted to keep things short. (The only way I have found to express a 4D integral is using Physics-Intc with the singleparameters of the four vector. Is there a more convenient way to get d^4x?) But still in 1D I cannot solve it.

To find the solution, an exponential of only one integral, is actually pretty easy, since there are integrals over e. g. exp(-I*x*(p-k)) deliver a delta distribution, but I cannot reproduce this in Maple since he doesn't perform the integral over x.

I have found that I can/have to use the command inttrans-fourier to gain the delta distribution, but when I try to use it for the problem mentioned above I run into all kinds of problems. Not to mention that I cannot manage to perform a fourier transformation in 4D.

Does anybody know how to solve this problem? Thanks!

hi,i am studying the maple most recent.But when calculating function integral,I ran into trouble.I hope to get your help.Here is the code I wrote, but it runs a very long time. How to effectively reduce the integration time?

restart;
with(student);
assume(n::integer);
Fourierf := proc (sigma, a, b, N) local A, A0, B, T, S, Ff; T := b-a; A0 := int(sigma, t = a .. b); A := int(sigma*sin(n*Pi*t/T), t = a .. b); B := int(sigma*cos(n*Pi*t/T), t = a .. b); S := sum(A*sin(n*Pi*t/T)+B*cos(n*Pi*t/T), n = 1 .. N)+(1/2)*A0; Ff := unapply(S, t) end proc;

f := proc (t) options operator, arrow; piecewise(t < .13*2.6 and 0 <= t, 100*t/(.13*2.6), .13*2.6 <= t and t < 2.6, 100, 2.6 <= t and t < 2.6*1.1, 0) end proc;

sigma := f(t);
a := 0;
b := 1.1*2.6;
s1 := unapply((Fourierf(sigma, a, b, 500))(t)/uw0, t);

s2 := unapply((Fourierf(sigma, a, b, 500))(t)/ua0, t);
A1 := (2*n+1)^2*Pi^2*(C3+1+sqrt(4*C1*C2*C3+C3^2-2*C3+1))/(8*C1*C2-8);
A2 := (2*n+1)^2*Pi^2*(C3+1-sqrt(4*C1*C2*C3+C3^2-2*C3+1))/(8*C1*C2-8);
g := -C2*Cww*(diff(s1(x), `$`(x, 2)))+Caa*(diff(s2(x), `$`(x, 2))+(n+1/2)^2*Pi^2*(diff(s2(x), x)));
f1 := -(1/2)*(n+1/2)^2*Pi^2*sqrt(4*C1*C2*C3+C3^2-2*C3+1)+C2*Cww*((D@@1)(s1))(0)-Caa*((D@@1)(s2))(0)+(n+1/2)^2*Pi^2*(C2-(1/2)*C3+1/2);

CN := ((2*(int(exp(-A1*x)*g, x = 0 .. t)-f1))*exp(A1*t)-(2*(int(exp(-A2*x)*g, x = 0 .. t)-f1))*exp(A2*t))/((n+1/2)^3*Pi^3*sqrt(4*C1*C2*C3+C3^2-2*C3+1));
ua := sum(CN*sin((n+1/2)*Pi*z), n = 0 .. 100);

 

 

Hi all,

 

I have recentry constructed Fourier amplitude and Fouier Power graphs which are plotted against the frequency of the oscillations, like the picture below.

 

I would like to change the frequency scale along the x axis into a period scale in order to Fourier analyse future ODE systems in  terms of period rather than frequency.

But I unsure how to manipulate my code to do so.

 

Any help would be much apreciated, and the Maple file that I am using is attached.

 

Thanks in advance!

 

Fourier_with_Period.mw

 

 

Hi All,

 

Where can I download Fourier Series Package? I found other posts related to this but the link is not working.

 

Thanks.

 

Hi,

 

I was wondering how to go about plotting a Fourier Tranform in Maple.

My assignment is to plot a simple harmonic equation as a Fourier transform, depicting amplitude against fequency.

I've been given: x'' + w^2 x = 0

And want to obtain both the f(x) = a0 sin(wt) + b0cos(wt) form, and a graph of the the amplitude (c^2 =a0^2 + b0^2) against frequency.

I know how to do this on paper but not in Maple, so any help with line commands and layout would be very much appreciated.

 

Thanks

If I do even expansion for the function 2*x-2, x from [0,1], to the real line,how do I make Maple to produce

the Fourier cosine expansion in the form 2*x-2=a_0/2+Sum(a_n cos n*Pi*x)

 

?

 

hi

how can I download Fourier series package?

Does Maple have any tool or package that computes the Fourier & Fourier-Bessel series expansions of a given funtion "f(x)" over a specified interval "[a,b]"?

I need to know if the Software Maple solve, step-by-step series of Fourier and Laplace transforms? The Maple command has to solve step by step series of Fourier and Laplace transforms? or commands show only the direct solution?

Below is the function that I have.

 

f := (t-1)^(1/3)

p:=2;

b[n] := 2/p*(Int(f*sin(2*Pi*n*t/p), t = 0 .. p))

 

I also included a picture below to show what it is doing. Some help would be greatly appreciated. All I need to know is why maple doesn't want to evaluate bn?

 

Maple Code

 

I've been instructed to create an animation showing the changing plots of a single square waveform using 5,10,20,40,80,160,320, and 640 terms in my Fourier series. This is my code right now: 

 

with (plots):
L := [seq(2^i, i = 0 .. 6)];


[1, 2, 4, 8, 16, 32, 64]


animate( plot, [2/((2*n-1)*Pi))*sin((2*n-1)*Pi*x], n=L);
Error, `)` unexpected

 

It doesn't work. Can anyone explain what I'm doing wrong, or how to solve my question?

I want to do a step by step computation for obtaining the coefficents of the sine fourier series expansion of f(x)=x over the interval [-L,L]. The steps are as follows:

1-write the fourier expansion as: Sum(A[n]*sin(n*pi*x/L),n=1..N)
2-multiply the series by: sin(m*pi*x/L)
3-integrate the series over the interval [-L,L]
3-using the orthogonality properties of the set {sin(n*pi*x/L} compute the A[n].

I can't do these steps since I have problem with the series manipulations in maple!
Can any one suggest a way from begining to the end?

Thanks. :)
Below shows what I did in Maple 17.

Using the Fourier convolution theorem to solve f(t) =sin (t)

f(t)=R dJ(t)/dt+J(t)/C

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