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PDE by Fourier Transform

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data
    (a) Solve [tex]\frac{\partial u}{\partial t}=k\frac{\partial ^{2} u}{\partial x^{2}} - Gu[/tex]

    where -inf < x < inf
    and u(x,0) = f(x)

    (b) Does your solution suggest a simplifying transformation?

    2. Relevant equations

    I used the fourier transform as:
    F[f(x)] = F(w) = [tex] \frac{1}{2*pi} \int_{-inf}^{inf} f(x) e^{iwx} dx [/tex]

    3. The attempt at a solution

    I solved part a using fourier transform. Although I'm not 100% certain, I think my answer is pretty plausible. I'm happy to elaborate on how I solved this, but I didn't want to type it all out for naught, because that's not really my question. Anyway, I got:

    [tex] u(x,t) = \int_{-inf}^{inf} [ \frac{1}{2*pi} \int _{-inf}^{inf} f(x) e^{iwx} dx ] e^{(-w^{2}k-G)t} e^{-iwx} dw [/tex]

    I'm not sure how to answer part b. Any ideas?
     
  2. jcsd
  3. Feb 17, 2009 #2
    So I think maybe the problem statement is asking me to change the order of integration and take the middle integral "offline" by substituting for x (as x - xbar, for example). Or maybe this is expected in part A...
     
  4. Feb 17, 2009 #3
    following my own logic, I find:

    [tex]
    u(x,t) = \frac{1}{2 \pi} \int_{-inf}^{inf} f(X) ( \int_{-inf}^{inf} e^{-iw(x-X)} e^{-(w^{2}k-G)t}dw)dX
    [/tex]

    and I need to find a function g(x-X) such that the Fourier transform is:

    [tex] G(w) = e^{(-w^{2}k-G)t} [/tex]


    So the substitution must simplify G(w) such that i can get an analytical form of the inverse fourier transform... Any idears?
     
  5. Feb 17, 2009 #4
    So it turns out that the substitution is a = t(K-G). Thus you can take the integral offline by evaluating the resultant Gaussian.

    Thanks! At least PF lets me talk to myself better :P
     
  6. Feb 17, 2009 #5
    Well, I dunno if this qualifies as a "transformation" -- again, welcoming comments.
     
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