# Dealing with insulation at one end of the Heat Equation

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?

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.
• Well, you have heat flowing inside the bar, so $u_x$ is nonzero in the interior Oct 8, 2019 at 12:34