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Chapter 9

The Heat Equation

Let u(x, t) denote the temperature at time t and position x, along the length of a thin
rod with uniform cross-sectional area A, density ρ, and length l. The sides of the rod are
perfectly insulated so heat only flows in the x−direction.
It is known from physics that, if k is the thermal conductivity of the material in the
rod, then the heat flow across the face at x = x 0 is equal to −Au x (x 0 , t). The negative
sign is due to the fact that heat flows from areas of higher temperature to areas of lower
temperature, and if u x (x 0 , t) is positive, it means that the temperature is increasing in
the positive x−direction.
We will now assume that the rod is l meters long, totally insulated except for the two
ends at x = 0 and x = l. The density ρ of the rod, its thermal conductivity K, and
specific heat s are all assumed to be constant along its length. Under these conditions
the temperature u(x, t) in the rod will satisfy the heat equation

2
u t (x, t) = c u xx (x, t),

2
where c is the positive constant:
K
2
c = .
sρ
If this equation is written in the form

2
c u xx − u t = 0
2
it can be seen to be a linear second-order pde with constant coefficients A = c , B =
2
C = 0. This means that B − 4AC = 0 and, therefore, this is a parabolic pde.
In order to obtain a unique solution to such an equation, two types of conditions must
be specified.
1. Boundary Conditions.
The temperature u(x, t) must be specified at both ends of the rod, for all values of t > 0.
There are different ways to do this. One way is to specify the temperature at each end
and assume it remains constant for all t > 0. Another condition results if it is assumed
that one or both of the ends are insulated. If, for example, the end at x = l is insulated,
the condition u x (l, t) = 0 for all t > 0 is used to simulate the fact that no heat is flowing

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