This is regarding the heat equation.
If a bar is fully insulated what is the temperature distribution u(x, t), 0 < x < L, for all time t > 0, if the bar is initially heated to a uniform temperature T∗ : u(x, 0) = T∗ > 0, 0 < x < L? Is it just u(x,t) = T* since the bar will be the same temperature for all time because it's fully insulated?
What if both ends of the bar are no longer insulated and only the side of the bar is insulated. The ends are quenched in ice to maintain u(0, t) = u(L, t) = 0 for all t > 0. What is the ultimate temperature distribution u(x, ∞), 0 < x < L, if the bar is initially heated to a uniform temperature T∗? Not sure what to do for this one, by 'side of the bar' do they mean the length of the bar not including the ends? Please help!