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2-D Poisson Equation Boundary Value Prob

  1. Dec 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Solve the equation:
    2F/∂x2 + ∂2F/∂y2 = f(x,y)

    Boundary Conditions:
    F=Fo for x=0
    F=0 for x=a
    ∂F/∂y=0 for y=0 and y=b


    2. Relevant equations
    How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X?


    3. The attempt at a solution
    I can't imagine what the Fourier transform of the generic f(x,y) looks like. Once this is done I'm supposed to be left with an ODE which is solvable.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 11, 2012 #2
    Ok, so based on my boundary conditions, I think I have to expand f(x,y) as a double Fourier cosine expansion, so:

    f(x,y)= 1/2Ʃ(C2cos(n∏x/a) + C3cos(n∏y/b)) + ƩƩC4cos(m∏x/a)cos(n∏y/b)

    Also, I can let F(x,y)=u(x,y)=X(x)Y(y)

    Plugging that into the original Poisson equation I end up with:
    1/X(x)∂2X/∂x2 + 1/Y(y)∂2Y/∂y2 = f(x,y)/X(x)Y(y)

    I think I'm doing something wrong by having the right-hand side of the Poisson equation equal to f(x,y)/F(x,y). It's also not clear to me if I can solve for any of the Fourier expansion coefficients or reduce the equation...
     
  4. Dec 12, 2012 #3
    Ok, so since ∂F/∂y = 0, that means it has a homogeneous solution of the form
    Y=Ʃcos(n∏y/(b/2)) correct?

    So that would cancel out the Y terms in the Fourier Expansion of f(x,y)?
     
  5. Dec 12, 2012 #4

    pasmith

    User Avatar
    Homework Helper

    As with inhomogeneous linear ODEs, Poisson's equation can be solved by taking a particular solution [itex]F_p[/itex] with
    [tex]
    \nabla^2 F_p = f(x,y)
    [/tex]
    subject to [itex]F_p = 0[/itex] on the boundary, and adding a complimentary function [itex]F_c[/itex] where
    [tex]
    \nabla^2 F_c = 0
    [/tex]
    subject to the given boundary conditions for [itex]F[/itex].

    To find [itex]F_p[/itex], you will want to find a family of eigenfunctions [itex]\phi_{nm}(x,y)[/itex] such that
    [tex]
    \nabla^2 \phi_{nm} = k_{nm}\phi_{nm}
    [/tex]
    and such that [itex]\phi_{nm}[/itex] vanishes on the boundary. You can then take a linear combination of these so that
    [tex]
    F_p(x,y) = \sum_{n}\sum_m A_{nm} \phi_{nm}(x,y)
    [/tex]
    with the [itex]A_{nm}[/itex] chosen so that [itex]\nabla^2 F_p = f(x,y)[/itex].
     
    Last edited: Dec 12, 2012
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