Math 124A – November 18, 2010
±
Viktor Grigoryan
15
Heat with a source
So far we considered homogeneous wave and heat equations and the associated initial value problems
on the whole line, as well as the boundary value problems on the halfline and the ﬁnite line (for wave
only). The next step is to extend our study to the inhomogeneous problems, where an external heat
source, in the case of heat conduction in a rod, or an external force, in the case of vibrations of a string,
are also accounted for. We ﬁrst consider the inhomogeneous heat equation on the whole line,
±
u
t

ku
xx
=
f
(
x,t
)
,
∞
< x <
∞
,t >
0
,
u
(
x,
0) =
φ
(
x
)
,
(1)
where
f
(
x,t
) and
φ
(
x
) are arbitrary given functions. The right hand side of the equation,
f
(
x,t
) is
called the
source
term, and measures the physical eﬀect of an external heat source. It has units of heat
ﬂux (left hand side of the equation has the units of
u
t
, i.e. change in temperature per unit time), thus
it gives the instantaneous temperature change due to an external heat source.
From the superposition principle, we know that the solution of the inhomogeneous equation can
be written as the sum of the solution of the homogeneous equation, and a particular solution of the
inhomogeneous equation. We can thus break problem (1) into the following two problems
±
u
h
t

ku
h
xx
= 0
,
u
h
(
x,
0) =
φ
(
x
)
,
(2)
and
±
u
p
t

ku
p
xx
=
f
(
x,t
)
,
u
p
(
x,
0) = 0
.
(3)
Obviously,
u
=
u
h
+
u
p
will solve the original problem (1).
Notice that we solve for the general solution of the homogeneous equation with arbitrary initial data
in (2), while in the second problem (3) we solve for a particular solution of the inhomogeneous equation,
namely the solution with zero initial data. This reduction of the original problem to two simpler prob
lems (homogeneous, and inhomogeneous with zero data) using the superposition principle is a standard
practice in the theory of linear PDEs.
We have solved problem (2) before, and arrived at the solution
u
h
(
x,t
) =
Z
∞
∞
S
(
x

y,t
)
φ
(
y
)
dy,
(4)
where
S
(
x,t
) is the heat kernel. Notice that the physical meaning of expression (4) is that the heat
kernel averages out the initial temperature distribution along the entire rod.
Since