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Partial DFQ Dirichlet Problem

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data

    ∇[itex]^{2}[/itex]u=0 on 0<x<∏, 0<y<2∏
    subject to u(0,y)=u(∏,y)=0
    and u(x,0)=0, u(x,2∏)=1


    2. Relevant equations

    --

    3. The attempt at a solution

    I've solved the SLP, and now I am trying to solve the Y-equation that results from separation of variables:

    Y''-λY=0, Y(0)=0
    Y[itex]_{n}[/itex](y)=Acosh(ny)+Bsinh(ny)
    Y(0)=(0)=Acosh(n*0)+Bsinh(n*0)[itex]\Rightarrow[/itex]A=0

    Doesn't this effectively "kill" the problem? Or is this the solution:
    Y[itex]_{n}[/itex](y)=Bsinh(ny)
     
    Last edited: Sep 23, 2012
  2. jcsd
  3. Sep 23, 2012 #2

    LCKurtz

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    Well, so far you have ##X_n(x) = \sin(nx)## and ##Y_n(y) = \sinh ny## so your potential solution is $$
    u(x,y)=\sum_{n=1}^\infty b_n\sin(nx)\sinh(ny)$$ which satisfies 3 of the four boundary conditions. You still have all the ##b_n## to use. So you need$$
    u(x,2\pi)=\sum_{n=1}^\infty b_n\sin(nx)\sinh(2n\pi)=1$$I'm guessing you know how to do that.
     
    Last edited: Sep 23, 2012
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