138 Chapter 2 The Heat Equation
c ρκk=ρκc
Material (gcal◦C)(cmg 3 )(scmcal◦C)(cms^2 )
Aluminum 0. 21 2. 70. 48 0. 83
Copper 0. 094 8. 90. 92 1. 1
Steel 0. 11 7. 80. 11 0. 13
Glass 0. 15 2. 60. 0014 0. 0036
Concrete 0. 16 2. 30. 0041 0. 011
Ice 0. 48 0. 92 0. 004 0. 009
Table 1 Typical values of constantsofu(x,t∗), witht∗a fixed time. If a portion of the graph is shaped likeU,Jor
backwardsJ, the graph is concave there — that is,∂^2 u/∂x^2 is positive. Then by
the heat equation,∂u/∂tmust be positive as well. Vice versa, when the graph
is convex,∂^2 u/∂x^2 and hence∂u/∂tmust be negative. Thus, a solution of the
heat equation tends to straighten out.
This equation alone is not enough information to completely specify the
temperature, however. Each of the functions
u(x,t)=x^2 + 2 kt,
u(x,t)=e−ktsin(x)satisfies the partial differential equation, and so do their sum and difference.
Clearly this is not a satisfactory situation either from the mathematical or
physical viewpoint; we would like the temperature to be uniquely determined.
More conditions must be placed on the functionu. The appropriate additional
conditions are those that describe the initial temperature distribution in the
rod and what is happening at the ends of the rod.
Theinitial conditionis described mathematically as
u(x, 0 )=f(x), 0 <x<a,
wheref(x)is a given function ofxalone. In this way, we specify the initial
temperature at every point of the rod.
Theboundary conditionsmay take a variety of forms. First, the temperature
at either end may be held constant, for instance, by exposing the end to an ice-
water bath or to condensing steam. We can describe such conditions by the
equations
u( 0 ,t)=T 0 , u(a,t)=T 1 , t> 0 ,whereT 0 andT 1 may be the same or different. More generally, the temperature
at the boundary may be controlled in some way, without being held constant.
Ifx 0 symbolizes an endpoint, the condition is
u(x 0 ,t)=α(t), (6)