Analytical Solution to Poisson's Equation

[rugabug]

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.

 

[HallsofIvy]

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.

 

[rugabug]

My experience with Fourier series is extremely limited and your hint doesn't mean anything to me.