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PDE Cylindrical and Spherical Symmetry

  1. Mar 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the solution u(r,theta) of Laplace's equation (nabla^2)*u=0 in the semi-circular region r<a, 0<theta<pi, which vanishes on theta=0 and takes the constant value A on theta=pi and on the curved boundary r=a, is
    u(r,theta)=(A/pi)[theta + 2*summation ((r/a)^n*((sin n*theta)/n)])



    2. Relevant equations



    3. The attempt at a solutionI didn't now where to even begin. I am struggling and am requesting help in explaining what I do not understand. Thanks!
     
  2. jcsd
  3. Mar 15, 2009 #2

    gabbagabbahey

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    Start with the general soltion to Laplace's equation for cylindrical symmetry...what's that?

    Now, what are your boundary conditions for this problem?
     
  4. Mar 15, 2009 #3
    This was all the data I was given. Our book is very brief and assumes you are very proficient with ODE's. I am struggling and need to understand these concepts. Thanks!
     
  5. Mar 15, 2009 #4

    gabbagabbahey

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    Well, I'm not here to teach an entire course on PDE's...you must at least know the method of Separation of variables right?
     
  6. Mar 15, 2009 #5
    Yes I do understand and know the method of Separation of variables.
     
  7. Mar 15, 2009 #6

    gabbagabbahey

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    Well then, try applying it....show an attempt
     
  8. Mar 15, 2009 #7
    okay I have been studying my book all weekend and this is what I had.
    Laplace Equation (1/r) d/dr(rdu/dr) + (1/r^2)d^2u/ds = 0 (s represents theta)
    u(r,s)=R(r)S(s)
    (1/Rr)d/dr(rdR/dr) + (1/(r^2S))(d^2S/ds^2) = 0
    (1/S)d^2S/ds^2 = -w^2 and (r/R)d/dr(rdR/dr)= w^2
    Therefore S = A cos ws + B sin ws
    r(d/dr(rdR/dr)) - w^2R = 0
    w = n n=integer
    r^2(d^2R/dr^2) + r(dR/dr) - n^2R = 0
    R(r) = (C/r^n) + Dr^n
    u(r,s) = ((C/r^n) + Dr^n)(A cos ns + B sin ns)

    r=a therefore un(r,s) = r^n(An cos ns + Bn sin ns)

    This is a interior Dirichlet problem therefore the solution is
    u(r,s) = A0/2 + Summation (r/a)^n(An cos ns + Bn sin ns)

    This is where I get lost, I think I need to do the following:
    Bn = 1/(2*pi) integral 0 to pi m sin nm dm.
     
  9. Mar 15, 2009 #8

    gabbagabbahey

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    So far so good, but you will need to analyze 3 different cases:

    (1)w^2>0
    (2)w^2=0
    (3)w^2<0

    You may find non-trivial solutions in more than one of these cases for this problem:wink:....

    This is correct for the first case , but don't forget to analyze the other two cases as-well!

    How are you deducing that w is an integer? This can only come from applying your boundary conditions....
     
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