Fourier Series and Partial Differential Equations Lecture Notes

Chapter 3
The heat equation
In this chapter we shall look at the heat equation in one space dimension, learning a
method for its derivation, and some techniques for solving.
3.1 Derivation in one space dimension
A straight rigid metal rod lies along the x-axis. The lateral surface is insulated to prevent
heat loss.
Figure 5
Letρbethemass density perunitlength, cbethe specificheat, T(x,t) bethetemperature
and:
• +q(x,t) be the heat flux from − to +;
• −q(x,t) be the heat flux from + to −.
Consider any interval [a,a+h]:
internal energy =
a+h
ρcT(x,t)dx; (3.1)
net heat flux out of [a,a+h] = q(a+h,t)−q(a,t). (3.2)
By conservation of energy, for every interval [a,a+h],
i.e.
Hence, by Leibniz,
rate of change of internal energy+net heat flux out = 0. (3.3)
1
h
d
dt
a
a+h
ρcT(x,t)dx+[q(a+h,t)−q(a,t)] = 0. (3.4)
a
a+h
a
ρc ∂T
∂t (x,t)dx+
q(a+h,t)−q(a,t)
= 0, (3.5)
h
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