(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider Laplace's equation u_{xx}+ u_{yy}= 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) = Ae^{ky}+ 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)

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# Laplace's equation

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