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Math Help - Separation of variables in Fourier Series

  1. #1
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    Separation of variables in Fourier Series

    Problem: Find the Fourier Series of f(x) = 0.05sin \pi x, with the initial velocity g(x)=0, c= \frac{1}{n}, L = 1.

    My solution: I use the separation of variables method, then I have:

    u(x,t)= \sum^{\infty}_{n=1}sin \frac{n \pi x}{L}[b_{n}cos( \frac {cn \pi}{L})t+ \hat{b_{n}} sin( \frac {cn \pi}{L})t]

    where

    b_{n}=\frac{2}{L} \int^{L}_{0}f(x)sin( \frac {cn \pi}{L})dx
    \hat{b_{n}} = \frac{2}{cnx} \int^{L}_{0}g(x)sin( \frac {cn \pi}{L})dx

    Now I know that [tex]\hat{b_{n}}=0[tex] since g(x)=0, but when I solve for b_{n}, I have:

    b_{n}=2 \int^{1}_{0}0.05sin( \pi x)sin(n \pi x)dx

    I use the table of integrals, and I get bn = 0, is there something I'm doing wrong here? (Most likely)

    Please check, thanks.

    K
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  2. #2
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    What are you trying to do?
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  3. #3
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    I'm trying to find the Fourier Series representation of this f(x), and I'm using the separation method to do it.
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  4. #4
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    So you are trying to solve the wave equation.

    Use the following facts,
    \int_0^L \sin \frac{\pi n x}{L}\sin \frac{\pi m x}{L} dx= L\delta_{nm}

    \int_0^L \sin \frac{\pi nx}{L}\cos \frac{\pi mx}{L}dx=0

    \int_0^L \cos \frac{\pi nx}{L}\cos \frac{\pi mx}{L} dx = L\delta_{nm}
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