Originally Posted by

**Maccaman** A long cylinder of material with a with a rectangular cross-section occupies the region 0 < x < a, 0 < y < b in space. The faces at x = 0 and x = a are held at temperature u=0 , and the face at y = b is thermally insulated. The face at y = 0 is held at temperature $\displaystyle u = u_0 $ (const.) until a steady temperature u(x,y) is reached in the cylinder. Show that

$\displaystyle u(x,y) = \frac{4u_0}{\pi}\sum_{m=1}^{\infty} \ \frac{sin ([2m -1] \pi x/a) \ cosh ([2m - 1] \pi [y -b]/a)}{[2m -1] \ cosh([2m - 1] \pi b/a)} $

for 0 < x < a, 0 < y < b.