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Homework Help: Laplace's equation on a ring

  1. Jan 16, 2012 #1


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    1. The problem statement, all variables and given/known data
    Solve Laplace's equation on a ring in the plane with r_1<r<r_2. And arbitrary functions on the edges of the ring.

    3. The attempt at a solution

    After separation of variables the solution to the radial factor is an often-seen problem. It's eigenvalues are λ=n² the eigenfunctions Θ(θ)~sin(nθ) + cos(nθ).

    The differential equation for the radial part becomes

    r²R" + rR' - n²R = 0

    I don't know how to solve this thing but my book gives solutions:

    n=0: r → 1 and r → ln(r)
    n>0: r → r^n and r → r^(-n)

    In my book the functions ln(r) and r^(-n) are thrown away because they are singular in r=0. But for this problem r=0 is not part of the domain, so I should keep all these solutions.

    Now I wonder, since ln(r) and r^(-n) can both be written as a Taylor series on the interval [r_1,r_2], can I throw them away anyways?

    If so, the solution will be of the form

    u(r,θ) = Ʃ A_n r^n cos(nθ) + Ʃ B_n r^n sin(nθ)

    Boundary condtions

    u(r_1,θ) = Ʃ A_n r_1^n cos(nθ) + Ʃ B_n r_1^n sin(nθ)
    u(r_2,θ) = Ʃ A_n r_2^n cos(nθ) + Ʃ B_n r_2^n sin(nθ)

    But this seems to be fixing A_n and B_n twice, since both are Fourier series.
  2. jcsd
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