# Homework Help: PDE Harmonic Function help

1. Nov 15, 2011

### Lawrencel2

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$_{0}$$\sum$$\frac{1-cos(\alpha_{n})}{\alpha_{n}(h+cos^{2}\alpha_{n})}e^{-\alpha_{n}x}sin(\alpha_{n}y)$

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

$u(x,y)= X(x)Y(y)$

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?
$Y(y)=C(e^{\alpha_{n}y}-e^{-\alpha_{n}y}), Y(0)=0, X(x)=C_{1}cos(\alpha_{n}x)+C_{2}sin(\alpha_{n}x), \frac{-\alpha_{n}}{h}=tanh(\alpha_{n})$

Last edited: Nov 15, 2011
2. Nov 15, 2011

### Lawrencel2

Re: PDE Harmonic Function help!!

no help?