# Heat Equation Derivation and Mean Value Theorem

Farlow book PDEs for Scientists and Engineers pg. 27 shows derivation for Heat Equation. It starts by stating

Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total heat generated inside $[x,x+\Delta x]$

and writing the conservation equation

$$\textit{Total Heat Inside} [x,x+\Delta x]= cpA \int _{ x}^{x+\Delta x}u(s,t) ds$$

Takes derivative according to time, rewrites the equation

$$\frac{d}{dt} \int _{ x}^{x+\Delta x} c\rho A u(s,t) ds = c\rho A \int _{ x}^{x+\Delta x} u_t(s,t) ds$$

$$= kA [ u_x(x+\Delta x,t) - u_x(x,t)] A \int _{x}^{x+\Delta x} f(s,t) ds$$

At this point he wants to get rid of the integrals, so he uses the mean value theorem which is, for a $a < \xi < b$

$$\int _{ a}^{b} f(x) dx = f(\xi)(b-a)$$

A $\xi$ must exist within the specified interval. He applies it to the equation, and gets

$$c\rho A u_t(\xi_1,t)\Delta x = kA[u_x(x+\Delta x, t) - u_x(x,t)] + Af(\xi_2,t)\Delta x$$

$$x < \xi < x+\Delta x$$

This also makes sense, there are multiple $\xi$'s for two different integrals. However below, he turns two $\xi$'s into one,

$$u_t(\xi,t) = \frac{k}{c\rho} \bigg[ \frac{u_x(x+\Delta x,t) - u_x(x,t)} {\Delta x} \bigg] + \frac{ 1}{c\rho}f(\xi,t)$$

and while $$\Delta x \to 0$$

he gets

$$u_t(x,t) = \alpha^2u_{xx}(x,t) + F(x,t)$$

In this last statement, $\xi$'s are replaced by $x$. So I have three questions: First how did the author combine the two $\xi$'s, second, while $\Delta \to 0$, how he turned them into $x$. I guess it is understandable if the $\Delta x$ becomes infinitesimally small, than whatever's inside can only be $x$? That still does not explain the two $\xi$'s though.

Third question: Net flux of heat across boundary uses $u_x(x+\Delta x,t) - u_x(x,t)$, that is derivative according to space, and takes difference between two sectional endpoints. Why $u_x$? Shouldn't we use simply $u$ here?

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1. In certain contexts it is usual to use the same letter for different quantities. A typical example are inequalities, in which a constant $C$ may take different values. In this case, the author is probably using the same letter $\xi$ to mean "a point between $x$ and $x+\Delta x$", with the understanding that it may take different values in different terms of the equations.
2. Since $x\le\xi\le x+\Delta x$, as $\Delta x\to0$, $\xi\to x$, independently of the real value of $\xi$.
3. Remember that $u$ represents temperature; then $u(x+\Delta x,t)-u(x,t)$ is the chenge of temperature, not the change of heat flux. Newton's law says that the heat flow is proportional to the spatial gradient of temperature, which is $u_x$; hence the expresion for the heat fluxin the interval $[\,x,x+\Delta x\,]$.
Is it also because of #2 he could combine the two $\xi$'s? Since $\Delta x \to 0$, there could be no two different $\xi$ values there. – BB_ML Jan 20 '12 at 10:13
Intuitively (without resorting to the term flux) I guess $u_x$ would mean the heat difference between two infinitesimal particles next to eachtother. Author takes that difference once for $x$, then for $x+\Delta x$, and the difference of these differences is the net change of heat inside the section (minus the generated heat of course). I guess that works because normally the rod would have equal temparature everywhere, $u_x$ would be zero everywhere. – BB_ML Jan 20 '12 at 10:20
The author is not combining two values. Is using the same symbol (in this case the leter $\xi$) to denote two values that are most probably different. It is just a matter of economy of symbols and readability. I know it is confusing at first, but you will get used to it. – Julián Aguirre Jan 20 '12 at 10:47
Economy of symbols, yes, but in this case, changing $\xi_1,\xi_2\to\xi$ really does not increase readability in my opinion. – Willie Wong Jan 20 '12 at 13:36