If there is a heat source within the medium, the heat diffusion equation (5.7.2) is altered to

where G is a generation term which can depend on x(vector) and t. Solve this equation in one dimension for the steady-state temperature profile of a metal bar with specific heat per unit mass C/ρV = 400 J/kg-K, heat conductivity K = 1 W/K-cm, and density ρ = 10 g/cm3, if the bar is 1 meter long, with cross-section 10 cm², which is heated everywhere along its length with G = 20 mW/cm³, and clamped at each end to a highly conducting medium with temperature held fixed at T = 300 K. Show that the heat flow out the ends of the bar is equal to the total heat generated in the medium. What is the maximum T reached?

t = DVu + G, (5.7.8)