Laplace equation with boundary condition

[ACG_PhD]

Good afternoon,

I am a PhD student in motions of damaged ships. I am trying to find a solution of Laplace equation inside a box with a set of boundary conditions such that:

∇²$\phi$=0
$\phi$_x=1 when x=-A and x=A
$\phi$_y=0 when y=-B and y=B
$\phi$_z=0 when z=Z_top and z=Z_bot


I have tried different solutions that look like
$\phi$=x+Ʃ(cos(α(x+A))cos(β(y+B))cosh(γ(z-Z_top))

α=n*PI/A n=1...∞
β=m*PI/B m=1...∞
and α²+β²=γ²

I can't find a solution that satisfy everything.
I either satisfy the boundary conditions or Laplace equation but not both.
The problem comes from the need to have a sinh(z)=0 for 2 values...
I can't figure out how to go around the problem.

Any advice would be really nice.

Thanks
Anne

PS: I have the three permutation to find $\phi$_x=1 then $\phi$_y=1 and $\phi$_z=1
I have the same problem in the three cases.

 

[Chestermiller]

Good afternoon,

I am a PhD student in motions of damaged ships. I am trying to find a solution of Laplace equation inside a box with a set of boundary conditions such that:

∇²$\phi$=0
$\phi$_x=1 when x=-A and x=A
$\phi$_y=0 when y=-B and y=B
$\phi$_z=0 when z=Z_top and z=Z_bot


I have tried different solutions that look like
$\phi$=x+Ʃ(cos(α(x+A))cos(β(y+B))cosh(γ(z-Z_top))

α=n*PI/A n=1...∞

β=m*PI/B m=1...∞
and α²+β²=γ²

I can't find a solution that satisfy everything.
I either satisfy the boundary conditions or Laplace equation but not both.
The problem comes from the need to have a sinh(z)=0 for 2 values...
I can't figure out how to go around the problem.

Any advice would be really nice.

Thanks
Anne

PS: I have the three permutation to find $\phi$_x=1 then $\phi$_y=1 and $\phi$_z=1
I have the same problem in the three cases.

Try x+c as the solution