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PDE Harmonic Function help

  1. Nov 15, 2011 #1
    PDE Harmonic Function help!!

    1. The problem statement, all variables and given/known data
    A bounded harmonic function u(x,y) in a semi-infinite strip x>0, 0<y<1 is to satisfy the boudary conditions:
    u(x,0)=0, uy(x,1)=-hu(x,1), u(0,y)=u0,
    Where h (h>0) and u0 are constants. Derive the expression:
    u(x,y)=2hu[itex]_{0}[/itex][itex]\sum[/itex][itex]\frac{1-cos(\alpha_{n})}{\alpha_{n}(h+cos^{2}\alpha_{n})}e^{-\alpha_{n}x}sin(\alpha_{n}y)[/itex]

    where [itex]tan\alpha_{n}=\frac{-\alpha_{n}}{h}[/itex] ([itex]\alpha>0[/itex]
    2. Relevant equations
    [itex]u_{xx}+u_{yy}=0[/itex]

    [itex] u(x,y)= X(x)Y(y) [/itex]

    3. The attempt at a solution
    I Cant seem to even come close to the answer.
    Maybe some hints as to how i should approach this?
    [itex]Y(y)=C(e^{\alpha_{n}y}-e^{-\alpha_{n}y}), Y(0)=0,
    X(x)=C_{1}cos(\alpha_{n}x)+C_{2}sin(\alpha_{n}x),

    \frac{-\alpha_{n}}{h}=tanh(\alpha_{n}) [/itex]
     
    Last edited: Nov 15, 2011
  2. jcsd
  3. Nov 15, 2011 #2
    Re: PDE Harmonic Function help!!

    no help?
     
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