Consider heat flow with convection (see Exercise 1.5.2): (a) Show that the...

Consider heat flow with convection (see Exercise 1.5.2):

(a) Show that the spatial ordinary differential equation obtained by separation of variables is not in Sturm–Liouville form.

*(b) Solve the initial boundary value problem

(c) Solve the initial boundary value problem

Reference Exercise 1.5.2:

For conduction of thermal energy, the heat flux vector is φ = −K₀∇u. If in addition the molecules move at an average velocity V , a process called convection, then briefly explain why φ = −K₀∇u + cρuV . Derive the corresponding equation for heat flow, including both conduction and convection of thermal energy (assuming constant thermal properties with no sources).