Differential Equations for Engineers · Boundary-value problems and PDE separation methods
In the cube 0<x<π, 0<y<π, 0<z<1, solve u_(xx)+u_(yy)+u_(zz)=0 with zero Dirichlet data…
Problem
In the cube $0<x<\pi$, $0<y<\pi$, $0<z<1$, solve $u_{xx}+u_{yy}+u_{zz}=0$ with zero Dirichlet data on the five faces other than $z=1$ and with $u(x,y,1)=\sin(2x)\sin(3y)$.
Hint
Expand in Dirichlet sines in $x$ and $y$; the $z$-factors are hyperbolic.
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