Analytical Solution to Poisson's Equation

rugabug
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I have the equation del^2 phi =1 for 2-d (x and y) with the boundary condition being 0 along all 4 edges. I've looked in all my math books and can't find how to solve this. If anyone could get me started I would appreciate it.
 
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Let u= phi(x,y)- (1/2)x2. Then del2u= del2phi- 1= 0. The boundary conditions get a little complicated but it can be done as a Fourier series.
 
My experience with Fourier series is extremely limited and your hint doesn't mean anything to me.
 

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