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:
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