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Fourier transform

  1. Dec 12, 2008 #1
    Find the Fourier transform [tex]\hat{u}(w,t) = \frac{1}{\sqrt{2 \pi}} \int^{\infty}_{- \infty}u(x,t)e^{(-ixw)}dx[/tex] of the general solution u(x,t) of the PDE [tex]u_{t}= u_{xx} - u[/tex]

    Should I start by solving the PDE, or is there another way to do it?
     
    Last edited: Dec 12, 2008
  2. jcsd
  3. Dec 12, 2008 #2
    Anyone?
     
  4. Dec 12, 2008 #3

    Avodyne

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    The PDE is much easier to solve when Fourier transformed (it becomes an ODE), so first write [itex]u[/itex] in terms of [itex]\hat u[/itex], plug into the PDE, and solve.
     
  5. Dec 14, 2008 #4
    How du I transform u?
     
  6. Dec 14, 2008 #5

    Avodyne

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    [tex]u(x,t) = \frac{1}{\sqrt{2 \pi}} \int^{\infty}_{- \infty}\hat{u}(w,t)e^{(+ixw)}dw[/tex]

    This is the inverse of the Fourier transform.
     
  7. Dec 14, 2008 #6
    How can i find the inverse when I don't know u?

    My book says that the Fourier transform of the PDE is

    [tex]U_t = -w^2 U - U[/tex]

    How is that achieved?
     
  8. Dec 14, 2008 #7

    Avodyne

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    First of all, [itex]U=\hat u[/itex].

    To get your book's equation, first substitute the formula for [itex]u[/itex] in terms of [itex]\hat u[/itex] (that I gave in my last response) into the original PDE. What do you get?
     
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