Consider a uniform bar of length L which has been insulated both ends. If the inital temperature is given by u(x,0)=f(x) (some general distribution), and there is a constant, nonzero source of thermal energy.
a)Demonstrate mathematically that the equilibrium temperature cannot be found.
I know that the energy from the internal source cannot escape through the boundaries and so the temperature will continue to increase at all locations for all time never reaching equilibrium. However, I cannot demonstrate it. Please help.
b)Suggest what kind of boundary condition change would allow for an equilibrium to be reached.
Solution is $\frac{\partial u}{\partial x}(0,t) = A$ and $\frac{\partial u}{\partial x}(0,t)=B$ but I couldn't understand why. Could you please explain. Thank you.
