Consider Laplaces equation 2u = 2u x2 + 2u y2 + 2u z2 = 0 in a right cylinder whose base

Chapter 7, Problem 7.3.6

(choose chapter or problem)

Consider Laplace’s equation

\(\nabla^{2} u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\)

in a right cylinder whose base is arbitrarily shaped (see Fig. 7.3.3). The top is z = H, and the bottom is z = 0. Assume that

\(\begin{aligned}
\frac{\partial}{\partial z} u(x, y, 0) & =0 \\
u(x, y, H) & =\alpha(x, y)
\end{aligned}\)

and u = 0 on the “lateral” sides.

(a) Separate the z-variable in general.

(b) Solve for u(x, y, z) if the region is a rectangular box, \(0<x<L, 0<y<W\), \(0<z<H\).