A metal bar, of length $l=1$ meter and thermal diffusivity $\gamma = 2$, is taken out of a $100^{\circ}$ oven and then fully insulated except for one end, which is fixed to a large ice cube at $0^{\circ}$.
$(a)$ Write down an initial-boundary value problem that describes the temperature $u(t,x)$ of the bar at all subsequent times.
My suggestion: Suppose the bar is insulated at $x=0$, so we have
\begin{align*}
\dfrac{\partial u}{\partial t} = 2\dfrac{\partial^2u}{\partial x^2}, \\
\quad u(t,0) = 0, \quad u_x(t,0) = 0, \\
\quad u(0,x) = f(x) = 100
\end{align*}
Is this correct? Regarding similiar problems with insulation at one end, I sometimes see examples where $u(t,1) = 0$ (or more generally $u(t,l)=0$) is included too. Should I have done the same and why? Or is my answer correct?
Best Answer
If you imposed $u(t, 1) = 0$ it would have meant that the other end of the metal bar is kept at a fixed temperature, which is not what you want. The second condition has to be drawn from the fact the bar is "fully insulated". I would interpret this as an adiabatic condition, i.e., no heat exchange. Therefore, I would impose a Neumann condition to the other end of the bar, i.e., $u_x(t, 1) = 0$. These boundary conditions, together with the initial data $u(0, x) = 100$ for all $x \in [0, 1]$ make the equation well-posed.