Modify Exercise 1.5.5 if the thermal properties depend on r. (Reference Exercise 1.5.5)...

Modify Exercise 1.5.5 if the thermal properties depend on r.

(Reference Exercise 1.5.5)

Assume that the temperature is circularly symmetric: u = u(r, t), where . We will derive the heat equation for this problem. Consider any circular annulus a ≤ r ≤ b.

(a) Show that the total heat energy is

(b) Show that the flow of heat energy per unit time out of the annulus at r = b is . A similar result holds at r = a.

(c) Use parts (a) and (b) to derive the circularly symmetric heat equation without sources: