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Math Help - Fourier Series Wave Equation

  1. #1
    Member Haven's Avatar
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    Fourier Series Wave Equation

    It's been forever since I've done a course in PDE's, so I'd really appreciate a step by step process for solving this

    u_{tt}-\alpha^2 u_{xx}  0<x<L, t>0
    u(0,t)=u(L,t)=0      t \geq 0
    u(x,0)=f(x) 0 \leq x \leq L
    u_t(x,0)=g(x)  0 \leq x \leq L

    leads to a solution of the form
    u(x,t) = \sum_{n=1}^\infty \left[a_n \cos\left(\frac{n\pi \alpha}{L}t\right)+b_n \sin\left(\frac{n\pi \alpha}{L}t\right)\right]\sin\left(\frac{n\pi x}{L}\right)

    If \alpha=3,  L=\pi, f(x)=6\sin(2x)+2\sin(6x) and g(x)=11\sin(9x)-14\sin(15x).

    I want to say the a_n=0 due to the boundary condition, but I'm not sure.

    I'm also not sure what the integral's should look like when I'm trying to find the fourier coefficents of u,f and g.

    Thanks in advance.
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  2. #2
    MHF Contributor
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    Re: Fourier Series Wave Equation

    u(x,0) = sum(n,1,Infinity, (a_n cos(n pi alpha 0/L) + b_n sin(n pi alpha 0/L))sin(n pi x/L)

    u(x,0) = sum(n,1,Infinity, a_n sin(n pi x/L)) = f(x)

    so it looks to me like your a_n are a sort of Fourier coefficients of f(x)
    Last edited by romsek; November 25th 2013 at 05:50 PM.
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  3. #3
    Member Haven's Avatar
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    Re: Fourier Series Wave Equation

    Nevermind, just figured it out. Thanks anyways.
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