The Heat Equation

CHAPTER

2

2.1 Derivation and Boundary Conditions

As the first example of the derivation of a partial differential equation, we
consider the problem of describing the temperature in a rod or bar of heat-
conducting material. In order to simplify the problem as much as possible, we
shall assume that the rod has a uniform cross section (like an extrusion) and
that the temperature does not vary from point to point on a section. Thus, if
we use a coordinate system as suggested in Fig. 1, we may say that the temper-
ature depends only on positionxand timet.
The basic idea in developing the partial differential equation is to apply the
laws of physics to a small piece of the rod. Specifically, we apply the law of
conservation of energy to a slice of the rod that lies betweenxandx+ x
(Fig. 2).
The law of conservation of energy states that the amount of heat that enters
a region plus what is generated inside is equal to the amount of heat that leaves
plus the amount stored. The law is equally valid in terms of rates per unit time
instead of amounts.
Now letq(x,t)be the heat flux at pointxand timet. The dimensions ofq
are^1 [q]=H/tL^2 ,andqis taken to be positive when heat flows to the right.
TherateatwhichheatisenteringtheslicethroughthesurfaceatxisAq(x,t),
whereAis the area of a cross section. The rate at which heat is leaving the slice
through the surface atx+ xisAq(x+ x,t).

(^1) Square brackets are used to symbolize “dimension of.”H=heat energy,t=time,T=
temperature,L=length,m=mass, and so forth.
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