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Solve the PDE by Fourier Transforms

  1. Nov 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve:

    ∂u/∂t = k ∂2u/∂x2 - ζu

    with the initial condition

    u(x,0) = f(x)

    where k and ζ are constants.

    x is on an infinite domain.


    2. Relevant equations

    Define Fourier transforms:

    f(x) = ∫[-∞,∞]F(w)e-iwxdw

    F(w) = 1/2∏ ∫[-∞,∞]f(x)eiwxdx


    From tables of Fourier Transforms:

    2f/∂x2 = (-iw)2F(w)



    3. The attempt at a solution

    I have little experience with transforms so please don't berate me if this is completely wrong.

    Began by taking transform of entire eqn:

    F(∂u/∂t) = k F(∂2u/∂x2) - ζF(u)

    I will call F(u) = U*

    ∂U*/∂t = w2k U* - ζU*

    ∂U*/∂t = (w2k - ζ) U*

    So the general solution is:

    U* = C(w) e(w2k-ζ)t

    Where C(w) = 1/2∏ ∫[-∞,∞]f(x)eiwxdx

    Is this even remotely correct? It seems too easy, but then again, I suppose that is the point of transforms, to put something into a simpler form.
     
    Last edited: Nov 28, 2012
  2. jcsd
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