1). The first three problems deal with the heated rod depicted below. Consider a rod of radius a, thermal conductivity k, and volumetric heat capacity Cp. Initially the rod is at equilibrium with its surroundings at a temperature T0. At time t=0 we have a source of energy per unit volume S which warms things up. Heat is lost to the surroundings governed by a heat transfer coefficient h. So: a. Write down the differential equation and boundary conditions governing this problem. b. Render them dimensionless and show that the only parameter which appears in the dimensionless equations is the Biot number. c. Solve for the asymptotic solution at long times. d. Plot up the temperature profile for Bi=0.1,2, and on the same graph. Note that the Bi= parabola is the same as Poiseuille flow through a tube