Consider a thin one-dimensional rod without sources of thermal energy whose lateral surf...

Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is not insulated.

(a) Assume that the heat energy flowing out of the lateral sides per unit surface area per unit time is w(x, t). Derive the partial differential equation for the temperature u(x, t).

(b) Assume that w(x, t) is proportional to the temperature difference between the rod u(x, t) and a known outside temperature γ(x, t). Derive

where h(x) is a positive x-dependent proportionality, P is the lateral perimeter, and A is the cross-sectional area.

(c) Compare (1.2.15) with the equation for a one-dimensional rod whose lateral surfaces are insulated, but with heat sources.

(d) Specialize (1.2.15) to a rod of circular cross section with constant thermal properties and 0° outside temperature.

*(e) Consider the assumptions in part (d). Suppose that the temperature in the rod is uniform [i.e., u(x, t) = u(t)]. Determine u(t) if initially u(0) = u0.