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in a medium occupying the closed unit interval Œ0; 1 along the x-axis. Our goal is
to study the behavior of the heat distribution
u
as t increases.
It turns out that by applying basic techniques from mathematical analysis we can
derive an interesting property for a function E
1
.t / defined by
E
1
.t /
D
Z
1
0
u
2
.x; t /
dx
for t
0:
(8.4)
Note that E
1
.t / is a function of time only! At the present stage it might be difficult
to understand the reason for studying this function. However, we will see below that
it turns out to be very useful.
If we multiply the left- and right-hand sides of the diffusion equation (
8.1
)by
u
it follows that
u
t
u
D
u
xx
u
for x
2
.0; 1/; t > 0:
By the chain rule for differentiation, we observe that
@
@t
u
2
D
2
uu
t
;
and hence
1
2
@
@t
u
2
D
u
xx
u
for x
2
.0; 1/; t > 0:
Next, if we integrate this equation with respect to x and apply the rule of
integration by parts,
4
it follows that
u
must satisfy
Z
1
@t
u
2
.x; t /
dx
D
Z
1
1
2
@
u
xx
.x; t /
u
.x; t /
dx
(8.5)
0
0
D
u
x
.1; t /
u
.1; t /
u
x
.0; t /
u
.0; t /
Z
1
0
u
x
.x; t /
u
x
.x; t /
dx
D
Z
1
0
u
x
.x; t /
dx
for t>0;
where the last equality is a consequence of the boundary condition (
8.2
).
Recall that we assumed that
u
is a smooth solution of the diffusion equation. This
implies that we can interchange the order of integration and derivation in (
8.5
), that
is,
Z
1
u
2
.x; t /
dx
D
2
Z
1
0
@
@t
u
x
.x; t /
dx
for t>0;
(8.6)
0
4
Recall that
Z
b
Z
b
u
0
.x/
v
.x/
dx
Œ
u
.x/
v
.x/
a
u
.x/
v
0
.x/
dx
:
D
a
a
