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I'm trying to find an analytical solution of Laplace's equation:

[tex]\phi_{xx}[/tex] + [tex]\phi_{tt}[/tex] = 0

with the tricky boundary conditions:

1. [tex]\phi(x=0,|t|>\tau)= 0[/tex]

2. [tex]\phi(x\neq0, |t|>>\tau)=0[/tex]

3. [tex]\phi_{x}(x=0, |t|<\tau)=-1[/tex]

4. [tex]\phi_{t}(x, |t|>>\tau)=0[/tex]

I have the following ansatz(I think that's the correct term):

[tex]\phi(x,t)=\int^{\infty}_{0}A(k)e^{-kx}cos(kt)dk[/tex]

i.e. a fourier integral. It has the form that it has since I don't want the solution blowing up at infinity (I should also add that I'm only interested in [tex]x\geq0[/tex]) and that the solution has to be even in time (this is required by the physics of the problem). My attempts to extract [tex]A(k)[/tex] using standard fourier methods have failed, due to the difficulty of the b.cs.

Can anyone help me come up with an analytical solution to this problem?

PS, it IS possible to solve the problem using conformal mapping, but I'm trying to find another analytical way of solving it, mainly for the purposes of extension to another related problem.