Information Technology Reference
In-Depth Information
Hence E satisfies the following two conditions:
E
0
.t /
D
aE
.t /;
(2.13)
E.0/
D
0:
If we multiply both sides of (
2.13
)bye
at
, we observe that
e
at
E
0
.t /
ae
at
E.t/
D
0;
and thus
.e
at
E.t//
0
D
0;
andsowehave
e
at
E.t/
D
E.0/
for all time. Since E.0/
D
0,wehave
E.t/
D
0
for all time, and therefore
q.t/
D
r.t/
for all time, and hence problem (
2.11
) has only one solution.
Stability
Counting rabbits sounds simple, but it is not!
7
That means that we cannot be abso-
lutely sure about the initial state. What happens if our initial estimate is wrong?
Suppose the number of rabbits are modeled by
r
0
.t /
D
ar
.t /;
(2.14)
r.0/
D
r
0
:
Let us also consider the following problem:
q
0
.t /
D
aq
.t /;
(2.15)
q.0/
D
q
0
:
7
Just think about it: They all look almost the same and they run around like crazy. If we are
absolutely positive that the island do not have any rabbits before t
0; we could, of course,
count how many we introduce, but that is an unlikely situation. Probably, we would only be able
to get a rough estimate of the number of rabbits at t
D
0: And this is the typical case for many
parameters introduced in differential equations: They are based on some sort of measurements and
therefore their values are uncertain. The issue of stability is about how this uncertainty affects the
solution. If small perturbations produce gigantic changes, we have an unstable problem. But if
small perturbations in the data produce small changes in the solution, the solution is stable with
respect to perturbations of these data.
D
