I am studying heat equation on a 1-D bar. We now that Neumann conditions at both ends leads to a singular matrix (for finite element methods) in equilibrium. Adding an initial condition can lead to unique solution.
It turns out that if we add a Dirichlet boundary condition in equilibrium at one end, we can still have a unique solution (at least when $u_x(0)=u_x(1)=u(1)=0$ and $\int_0^1f(x)dx=0$).
My question is: does it make sense to have both insulated boundary and prescribed temperature at the same end in the real world? How can one simultaneously maintain insulation and constant temperature?
This question might be asked in physics SE, but since I met this problem from a math textbook, I decide to ask it here.