# Laplace's equation

1. May 23, 2010

### squenshl

1. The problem statement, all variables and given/known data
Consider Laplace's equation uxx + uyy = 0 on the region -inf <= x <= inf, 0 <= y <= 1 subject to the boundary conditions u(x,0) = 0, u(x,1) = f(x), limit as x tends to inf of u(x,y) = 0.
Show that the solution is given by u(x,y) = F-1(sinh(wy)f(hat)/sinh(wy))

2. Relevant equations

3. The attempt at a solution
I used Fourier transforms in x.
I got u(hat)(w,y) = Aeky + Be-ky
In Fourier space:
u(hat)(w,0) = F(0) = 0
u(hat)(w,1) = f(hat)(w)
But u(hat) is a function of y. My question is how do I apply the 3rd boundary condition (as this is the limit as x(not y) tends to inf) to u(hat)

2. May 26, 2010

### squenshl

Is it meant to be limit as y tends to inf not x.

3. May 26, 2010

### gabbagabbahey

You mean $\hat{u}(\omega,y)=Ae^{\omega y}+Be^{-\omega y}$, right?

You already used it. If [tex]\lim_{x\to\pm\infty} u(x,y)\neq 0[/itex], its Fourier transform (from $x$ to $\omega$) might not exist (the integral could diverge).

4. May 27, 2010

### squenshl

Very true. That is what I meant.
But when did I use this boundary condition?

5. May 27, 2010

### gabbagabbahey

When you took the FT of $u(x,y)$, and hence assumed that it existed.