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I'm studying partial differential equations and I am struggling to understand the heat equation $u_t =ku_{xx}$. I understand the physical interpretation of $u_{t}$, but the meaning of $u_{x}$ eludes me.

This question arises from the following problem:

Consider the 1-D heat equation in a rod of length $L$ with diffusion constant $k$. Suppose the left endpoint is fixed at 100°, while the right endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant $K=1$) with an outside medium which is 500°. The initial temperature distribution in the rod is given by $f(x)$.

While setting up the PDE, I interpreted the boundary condition at the right end as $u_{t}(L,t)=u(L,t)-500$ but the answer in the back of the text book has $-u_{x}(L,t)=u(L,t)-500$. Why is that?

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  • $\begingroup$ @fosho but what does it mean? $\endgroup$ – Zachary F Sep 29 '17 at 21:11
  • $\begingroup$ The title says "heat" and maybe there's the issue - $u$ actually describes temperature, not heat; one should not mix those up. $\endgroup$ – Nick Pavlov Sep 29 '17 at 21:37
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$u_x$ is the spatial derivative of temperature, that is the gradient of temperature along the rod. Newton's law of cooling cannot say what happens to temperature directly, but only how heat (that is energy) is transferred at the boundary. It says that the rod will gain/lose energy through the endpoint at a rate that is proportional to the temperature difference between that endpoint and the environment. If you think of the boundary itself (or an infinitesimal region of the rod around it), the energy stored in it is also infinitesimal, so any gain/loss to the outside is compensated by an equal loss/gain from the rest of the rod. So the energy flow (energy transferred per unit time through a cross-section of the rod) is the quantity which is proportional to $u(L, t) - 500$.

In any material that obeys the heat equation, that energy flow, at any point, is proportional to $-u_x$. [This is actually the more basic law from which the heat equation is derived: the difference between the heat flows through two points that are a small distance apart "becomes" the second derivative $u_{xx}$ in it. Physically, it is also a measure of the net rate of energy gain for that segment of the rod, which in turn is proportional to its rate of temperature increase $u_t$.]

So in your example, the energy flow at the right boundary at time $t$ is $-u_x(L, t)$ and by what I wrote in the first paragraph, it must be proportional to $u(L, t) - 500$. How exactly all the various proportionality constants are defined in all these equations is a matter of convention. I guess what your textbook calls $K$ is exactly the proportionality constant in that last relation.

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