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In-Depth Information
E
2
.t /
D
Z
1
0
u
x
.x; t /
dx
:
Our analysis of E
2
is similar to that of E
1
. However, instead of multiplying the
diffusion equation (
8.1
)by
u
, as we did in the analysis leading to inequality (8.7),
we will now multiply it by
u
t
and integrate with respect to x. Consequently,
u
must
satisfy the equation
Z
1
u
t
.x; t /
dx
D
Z
1
0
u
xx
.x; t /
u
t
.x; t /
dx
:
0
Integration by parts leads to
Z
1
u
t
.x; t /
dx
D
Œ
u
x
.x; t /
u
t
.x; t /
0
Z
1
0
u
x
.x; t /
u
tx
.x; t /
dx
D
u
x
.0; t /
u
t
.0; t /
u
x
.1; t /
u
t
.1; t /
Z
1
0
0
u
x
.x; t /
u
xt
.x; t /
dx
;
where we have used the basic property that we can change the order of differ-
entiation
7
of
u
with respect to x and t . The chain rule for differentiation implies
that
@
@t
u
x
D
2
u
x
u
xt
;
and hence
Z
1
u
t
.x; t /
dx
D
u
x
.0; t /
u
t
.0; t /
u
x
.1; t /
u
t
.1; t /
Z
1
0
u
x
.x; t /
u
xt
.x; t /
dx
0
Z
1
1
2
@
@t
u
x
.x; t /
dx
:
D
u
x
.0; t /
u
t
.0; t /
u
x
.1; t /
u
t
.1; t /
0
As above, we can interchange the order of integration and differentiation and thereby
conclude that
Z
1
u
x
.x; t /
dx
D
Z
1
0
1
2
@
@t
u
t
.x; t /
dx
u
x
.0; t /
u
t
.0; t /
C
u
x
.1; t /
u
t
.1; t /:
0
(8.8)
Note that (8.8) contains the derivatives
u
t
.0; t / and
u
t
.1; t / of
u
with respect
to time t at the boundary of the solution domain .0; 1/. What do we know about
these quantities? They are not present in our model problem (
8.1
)-(
8.3
). However,
according to the boundary condition (
8.2
),
u
.0; t / and
u
.1; t / are equal to zero for
all t>0. Thus,
u
.0; t / and
u
.1; t / are constant with respect to time and we therefore
conclude that
7
We assumed that
u
is smooth.
