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Poissons Equation Problem

  1. Jul 31, 2011 #1
    1. The problem statement, all variables and given/known data

    First part of the question is to obtain a fourier series of the function f(theta)= 1 0=<theta<pi
    -1 pi=<theta<2pi
    Then to find the solution of Poissons equation inside the unit disk r=<1
    ∇^2(P) = (1/r)d/dr(r(dP/dr))+ (1/r^2)d^2P/d^theta = f(theta) with Boundary Condition that when r=1, P=0. Given HINT that general solution of (r^2)R'' + rR' -(n^2)R = r^2 is R= a(r^n) + b(r^-n) - (r^2)/(n^2-4)

    Then required to show that the integral of f(theta)*P dA =
    (-4/pi) sum over n odd (1/(n^2)((n+2)^2))

    2. Relevant equations



    3. The attempt at a solution

    I think I have got the Fourier series correct as f = sum over n odd ((4/n*pi)sin(n*theta))

    I have then tried seperation of variables letting P=R(r)THETA(theta), and got
    R''(THETA) + (1/r)R'(THETA) + (1/r^2)R(THETA)'' = f(theta)

    and then said as r=1 when P=0, R(r)=0 at r=1, so THETA''=f(theta) and so by integrating, THETA = (-1/n^2)THETA'', which can be subbed back in to get the form for the HINT. This means I can construct P =R(r)THETA(theta) but then when I go ahead and integrate this, I a) cannot get rid of the a_n, b_n that occur from summing the HINT, as only one BC
    b) cannot get t the right form for the final integral, but I do have a lot of powers of n, n-2, n+2 so feel I am on the right sort of lines.

    Sorry this is so messy, any help would be much appreciated.
     
  2. jcsd
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