Consider the following problems of conduction heat transfer within a solid body. For each problem, write the entire system of governing equations and boundary conditions. Assume constant physical properties in

(a) and (b), but not in (c). Do not make any assumptions about the nature of the solution, i.e. write it for the general three-dimensional time-dependent case. In each problem, write the solution twice: in a coordinate-free form and using an appropriate coordinate system.

a) A cylindrical metal rod, the ends of which are maintained at constant temperatures \(T_{1}\) and \(T_{2}\) and the sidewall is perfectly thermally insulated.

b) A metal cuboid with volumetric Joule heating \(\dot{Q}(\boldsymbol{x}, t)\) generated by electric currents. The walls are exposed to air of temperature \(T_{\text {air }}\). The heat transfer coefficient \(h\) is a known constant.

c) A plastic spherical shell with time-variable spatially uniform heat flux per unit area \(\dot{q}(t)\) applied at the inner surface. The outer surface is cooled by water of constant temperature \(T_{w}\). The heat transfer coefficient \(h\) at the outer surface is a known constant. In this problem, assume that the density and specific heat of the plastic are constant but the thermal conductivity varies significantly as a function of temperature \(\kappa=\kappa(T)\).