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Approach to this PDE system?

  1. Mar 7, 2013 #1
    [tex]
    (r^2 \nabla^2 - 1) X(r,\theta,z) + 2 \frac{\partial}{\partial \theta} Y(r,\theta,z) = 0
    [/tex]
    [tex]
    (r^2 \nabla^2 - 1) Y(r,\theta,z) - 2 \frac{\partial}{\partial \theta} X(r,\theta,z) = 0
    [/tex]


    any suggestions are greatly appreciated :)
     
  2. jcsd
  3. Mar 7, 2013 #2

    Ben Niehoff

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    Science Advisor
    Gold Member

    I would start with defining

    [tex]Z(r,\theta,z) \equiv X(r,\theta,z) + i Y(r,\theta,z)[/tex]

    Then you have less work to do. It looks like separation of variables might be successful, have you tried that? Or a Fourier transform would certainly work.
     
  4. Mar 7, 2013 #3
    I tried separation of variables, but it didn't work because of the coupling. Using z=x+iy I was able to reduce the system to a single equation (multiply the 2nd equation by i and add it to the first).

    [tex]
    (r^2\nabla^2-1)Z(r,\theta,z) - 2i \frac{\partial Z(r,\theta,z)}{\partial \theta} = 0
    [/tex]

    Is it necessary that the real and complex parts individually equal 0? I was thinking of trying separation of variables for this equation in Z, but I'm unsure how to deal with the complex portion.
     
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