Product solutions for unsteady heat conduction in solids 

(a) In Example 12.1-2 the unsteady state heat conduction equation is solved for a slab of thickness 2b. Show that the solution to Eq. 12.1-2 for the analogous problem for a rectangular block of finite dimensions 2a, 2b, and 2c may be written as the product of the solutions for three slabs of corresponding dimensions in which Θ(y/b, at/b²) is the right side of Eq. 12.1-31. 

(b) Prove a similar result for cylinders of finite length; then rework Problem 12A.4 without the assumption that the cylinder is infinitely long.

T - T(x, y, z, t) T - To at at a'