ABSTRACT
To derive the heat equation, consider a heat-conducting homogeneous rod, extending from x = 0 to x = L along the x-axis (see Figure 11.1.1). The rod has uniform cross section A and constant density ρ, is insulated laterally so that heat flows only in the x-direction, and is sufficiently thin so that the temperature at all points on a cross section is constant. Let u(x, t) denote the temperature of the cross section at the point x at any instant of time t, and let c denote the specific heat of the rod (the amount of heat required to raise the temperature of a unit mass of the rod by a degree). In the segment of the rod between the cross section at x and the cross section at x+∆x, the amount of heat is
Q(t) =
cρAu(s, t) ds. (11.1.1)
On the other hand, the rate at which heat flows into the segment across the cross section at x is proportional to the cross section and the gradient of the temperature at the cross section (Fourier’s law of heat conduction):
−κA∂u(x, t) ∂x
, (11.1.2)
where κ denotes the thermal conductivity of the rod. The sign in Equation 11.1.2 indicates that heat flows in the direction of decreasing temperature. Similarly, the rate at which heat flows out of the segment through the cross section at x+∆x equals
−κA∂u(x+∆x, t) ∂x
. (11.1.3)
The difference between the amount of heat that flows in through the cross section at x and the amount of heat that flows out through the cross section at x + ∆x must equal the change in the heat content of the segment x ≤ s ≤ x+∆x. Hence, by subtracting Equation 11.1.3 from Equation 11.1.2 and equating the result to the time derivative of Equation 11.1.1,
∂Q
∂t =
cρA ∂u(s, t)
∂t ds = κA
[ ∂u(x+∆x, t)
∂x − ∂u(x, t)
∂x
] . (11.1.4)
function of s, then by the mean value theorem for integrals,∫ x+∆x
∂u(s, t)
∂t ds =
∂u(ξ, t)
∂t ∆x, x < ξ < x+∆x, (11.1.5)
so that Equation 11.1.4 becomes
cρ∆x ∂u(ξ, t)
∂t = κ
[ ∂u(x+∆x, t)
∂x − ∂u(x, t)
∂x
] . (11.1.6)
Dividing both sides of Equation 11.1.6 by cρ∆x and taking the limit as ∆x→ 0,
∂u(x, t)
∂t = a2
∂2u(x, t)
∂x2 (11.1.7)
with a2 = κ/(cρ). Equation 11.1.7 is called the one-dimensional heat equation. The constant a2 is called the diffusivity within the solid.
