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u
t
.0; t /
D
u
t
.1; t /
D
0
for t>0:
(8.9)
Equations (8.8)and(8.9) imply that
Z
1
u
x
.x; t /
dx
D
2
Z
1
0
@
@t
E
2
.t /
D
u
t
.x; t /
dx
0;
0
and thus E
2
is a non-increasing function with respect to time t ,
E
2
.t
2
/
E
2
.t
1
/
for all t
2
t
1
0:
(8.10)
Z
1
u
x
.x; t /
dx
Z
1
0
u
x
.x; 0/
dx
D
Z
1
0
f
x
.x/
dx
for t>0:
(8.11)
0
Consider an object with a non-uniform temperature distribution at time t
D
0.
Assume that the temperature at the boundary of this object is kept constant with
respect to both time and space.
8
We have all experienced how the temperature dis-
tribution in such cases will tend toward a uniform distribution; for sufficiently large
t , a constant temperature throughout the medium is reached. Furthermore, the speed
at which the temperature differences are evened out in the object, will decay with
time. Inequality (8.10) shows, in a somewhat modified form, that this latter property
is fulfilled by our model problem.
8.1.3
Stability
We will now investigate some interesting consequences of the bound (8.7)forthe
solution
u
of our model problem (
8.1
)-(
8.3
). Recall that f represents the tempera-
ture distribution in the medium at time t
D
0. In real-world simulations, such initial
states of the system under consideration will in many cases be based on physical
measurements. These measurements will always contain errors. It is impossible to
measure the temperature at every point in a medium with 100% accuracy. Con-
sequently, it becomes important to investigate whether or not small changes in f
will introduce major changes in the solution
u
of the problem. In mathematics this
is referred to as a question of stability: Is the solution
u
of (
8.1
)-(
8.3
) stable with
respect to perturbations in the initial condition f ?
8
The temperature is constant with respect to the spatial position along the boundary.
