# PDE Boundary conditions for a bar

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!

• try to write your formula using latex – zeraoulia rafik Oct 19 '15 at 14:36
• it is good to ask this question in SP – zeraoulia rafik Oct 19 '15 at 14:36
• So first, the insulation along the sides of the bar is required to use the heat equation. Without that you would have an additional term in the PDE corresponding to heat flux along the length of the bar. That said, in the first case, you have the heat equation with the boundary condition $u_x(0)=u_x(L)=0$. You've guessed that $u=T^*$ is the equilibrium. Check that in fact $u(x,t)=T^*$ for all $t$. The second case is similar, except that there is actually a time evolution occurring. – Ian Oct 19 '15 at 14:39
• Er...sorry, I meant "corresponding to heat entering and escaping the system as a whole at points along the length of the bar". The point is that it would change the PDE, not just the boundary condition, because there would be an effect on the interior of the domain. – Ian Oct 19 '15 at 16:11