What's the meaning of Neumann BCs in heat conduction problems?
Such as given here:
https://web1.eng.famu.fsu.edu/~dommelen/pdes/style_a/svbex.html
Why does one specify $u_x=g_0$ at the end of the rod. Rather than $u=g_0$?
Is it because it's assumed to "change", whereas the start of rod is thought to "have an initial value"?
You can think of Neumann BCs as fixing the heat flux and Dirichlet BCs, of course, fix the temperature.
You could generally specify either; they're just different problems. For example, $\partial_x u =0$ on the endpoints would imply perfect insulation (no heat flux). On the other hand, you could specify $u=0$ which would say that the temperature is fixed at zero at the endpoints of the rod.