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Help with Fourier transform of T'(x)/x

  1. Aug 16, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]T(x,t)[/tex]

    What is the fourier transform of
    [tex]
    \frac{1}{x}\frac{\partial T}{\partial x}
    [/tex]

    [tex]
    F(\frac{1}{x}\frac{\partial T}{\partial x}) = \int^{\infty}_{-\infty} \frac{1}{x}\frac{\partial T}{\partial x} e^{i \theta x}dx = ??
    [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Can this be split up using convolution into...

    [tex] F(\frac{1}{x}\frac{\partial T}{\partial x}) = F(\frac{1}{x})F(\frac{\partial T}{\partial x}) =\int^{\infty}_{-\infty} \frac{1}{x} e^{i \theta x}dx \int^{\infty}_{-\infty} \frac{\partial T}{\partial x} e^{i \theta x}dx
    [/tex]
     
  2. jcsd
  3. Aug 16, 2009 #2
    You will need to prove that the convolution of 1/x and dT/dx is (1/x)(dT/dx).
     
  4. Aug 17, 2009 #3
    Okay thanks. I was trying to solve
    [tex]
    \frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}

    [/tex]

    for [tex]
    0 < x < \infty
    [/tex] and initial condition like [tex]
    T(x,0) = g(x)
    [/tex]
    with boundary conditions [tex]
    T(\infty,t) = C_{1}
    [/tex] and [tex]
    T(0,t) = f(t)
    [/tex]

    I got stuck with separation of variables and method of characteristics so I was going to try fourier method
     
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