Chapter 2 The Heat Equation 137
should be recognized as a difference quotient. If we allow xto decrease, this
quotient becomes, in the limit,
limx→ 0 q(x+^ x ,tx)−q(x,t)=∂∂qx.The limit process thus leaves the law of conservation of energy in the form
−∂q
∂x+g=ρc∂u
∂t. (2)
We are not finished, since there are two dependent variables,qandu,inthis
equation. We need another equation relatingqandu.ThisrelationisFourier’s
law of heat conduction, which in one dimension may be written
q=−κ∂u
∂x.
In words, heat flows downhill (qis positive when∂u/∂xis negative) at a rate
proportional to the gradient of the temperature. The proportionality factorκ,
called thethermal conductivity,maydependonxif the rod is not uniform and
also may depend on temperature. However, we will usually assume it to be a
constant.
Substituting Fourier’s law in the heat balance equation yields
∂
∂x
(
κ∂u
∂x)
+g=ρc∂u
∂t. (3)Note thatκ,ρ,andcmay all be functions. If, however, they are independent
ofx,t,andu,wemaywrite
∂^2 u
∂x^2 +g
κ=ρc
κ∂u
∂t. (4)The equation is applicable where the rod is located and after the experiment
starts: for 0<x
and is called thethermal diffusivity. Table 1 shows approximate values of these
constants for several materials.
For some time we will be working with the heat equation without genera-
tion,
∂^2 u
∂x^2 =
1
k∂u
∂t,^0 <x<a,^0 <t, (5)
which,toreview,issupposedtodescribethetemperatureuin a rod of length
awith uniform properties and cross section, in which no heat is generated and
whose cylindrical surface is insulated.
Some qualitative features can be obtained from the partial differential equa-
tion itself. Suppose thatu(x,t)satisfies the heat equation, and imagine a graph