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Two dimensional Poisson's equation, Green's function technique

  1. Aug 4, 2010 #1
    Hi,
    While considering perturbed gravitational potential of incompressible fluid in rectangular configuration, I encountered two dimensional Poisson's equation including the step function.
    I want to solve this equation

    [tex] \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} \right) \psi(x, z) = [ \theta( z - ( a + \epsilon \cos(kx) ) } ) - \theta( z - a ) ] + [ \theta( z - ( - a - \epsilon \cos(kx) ) ) - \theta( z - (- a) ) ] [/tex]

    [tex]a[/tex] is the height from [tex]z=0[/tex] plane and [tex]\epsilon [/tex] is a small number much smaller than [tex]a[/tex].
    The source term is periodic in x direction with wavenumber [tex] k [/tex] and has a reflection symmetry.
    Hence I expect [tex]\psi[/tex] would be also periodic in x-direction and be an even function about z=0 plane.

    Do I have to use green's technique here to solve Poisson's equation involving periodic load?
    Can it be reduced to Helmholtz equation in one dimension like [tex] \psi^{\prime \prime} - k^2 \psi = ... [/tex] ?

    Any help would be greatly appreciated.

    Thank you~
     
    Last edited: Aug 4, 2010
  2. jcsd
  3. Aug 4, 2010 #2
    Ok..
    What if I simplify the problem?

    [tex] \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} \right) \psi(x, z) = \theta(z - ( a + \epsilon \cos(kx) ) } [/tex]

    If I can solve the above one then the superposed solution can be obtained.

    help me. T.T
     
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