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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}[itex]\phi[/itex]=0

[itex]\phi[/itex]_{x}=1 when x=-A and x=A

[itex]\phi[/itex]_{y}=0 when y=-B and y=B

[itex]\phi[/itex]_{z}=0 when z=Z_{top}and z=Z_{bot}

I have tried different solutions that look like

[itex]\phi[/itex]=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 α^{2}+β^{2}=γ^{2}

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 [itex]\phi[/itex]_{x}=1 then [itex]\phi[/itex]_{y}=1 and [itex]\phi[/itex]_{z}=1

I have the same problem in the three cases.

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# Laplace equation with boundary condition

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