Solving Laplace's Equation: Problem With Boundary Conditions

[Logarythmic]

I have a problem solving

$$\nabla^2 T(x,y,z) = 0$$
$$T(0,y,z)=T(a,y,z)=0$$
$$T(x,0,z)=T(x,b,z)=T_0 \sin{\frac{\pi x}{a}$$
$$T(x,y,0)=T(x,y,c)=const.$$

I use separation of variables and get

$$X_n (x) = \sin{\frac{n \pi x}{a}$$
$$Y_n (y) = \cosh{\sqrt{\frac{n^2 \pi^2}{c^2} + \frac{n^2 \pi^2}{a^2}}y} + \sinh{\sqrt{\frac{n^2 \pi^2}{c^2} + \frac{n^2 \pi^2}{a^2}}y}$$
$$Z_n (z) = \cos{\frac{n \pi z}{c}$$
$$T(x,y,z) = \sum_{n=1}^\infty a_n X_n (x) Y_n (y) Z_n (z)$$

where I have used the boundary conditions for x and z. Is this correct?
If it is, I'm having problems to wrap this up. I suppose I can use the condition for T(x,0,z) to get the constants. My calculations gives me

$$a_n = \frac{T_0}{\cos{\frac{\pi z}{c}}}$$

but then I can't get it toghether with the condition for T(x,b,z)...
Any ideas?

 

[dextercioby]

Well, first of all, the "y" part of the solution must be periodic, but I'm afraid sinh & cosh are not...The same with the "z" & "x" part.

Daniel.

 

[Logarythmic]

So the X- and the Z-part are correct, but not the Y-part?

 

[Meir Achuz]

That is easiest to do as 4 separate problems, each having 5 sides grounded.
Then add the 4 solutions.

 

[Logarythmic]

Sorry, I do not understand.