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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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  • $\begingroup$ $u$ is temperature, hence $u_{x}$ is temperature gradient $\endgroup$
    – lurscher
    Commented Sep 29, 2017 at 21:31
  • $\begingroup$ typically, temperature gradient is relevant in applications as it induces structural stress in the material and can result in plastic deformations or fracture $\endgroup$
    – lurscher
    Commented Sep 29, 2017 at 21:33

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The local rate of conductive heat flow along the rod is proportional to $u_x$ (the spatial temperature gradient). Your equation $-u_x=u(L,t)-500$ says that the rate of conductive heat flow arriving at x=L must match the rate of heat flow leaving the rod convectively.

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