Does heat eq.'s $u$ have meaning or is it just the derivatives of it that are meaningful?
E.g. $du/dx$ could have units $K/m$.
But what about $u$ or e.g. $du/dt$?
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Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x,t)$ is the temperature at point $x$ and time $t$.
So $\frac{\partial u }{\partial t}$ represent the variation of temperature at a particular position and time, with respect to time. Thus, it represents a cooling or a heating, depending on its sign. In response to your comment, yes it can be said that it is the rate of change of temperature with respect to time (and not position!).
answered Sep 24, 2018 at 10:47
untreated_paramediensis_karnikuntreated_paramediensis_karnik
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$\begingroup$ I guess it doesn't matter in what eq the forms are? Their meaning ought to be the same, if they refer to the same $u$, same kind of $dx$ same kind of $dt$ ... $\endgroup$– mavaviljCommented Sep 24, 2018 at 10:47
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$\begingroup$ But what are the derivatives of $u$ then? Rates of change of temperature? $\endgroup$– mavaviljCommented Sep 24, 2018 at 10:48
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$\begingroup$ I've edited my answer regarding the latter. Well, it does matter what u is, because it's a dummy variable. Some people use T instead of u. $\endgroup$ Commented Sep 24, 2018 at 10:51