Information Technology Reference
In-Depth Information
Fig. 3.1
Logistic curves
f
a
(
t
) with different continuous parameter
a
[0, 2]. We can see that
f
a
(
t
)
∈
remains be a constant if
a
¼
0, and that
f
a
(
t
) grows more rapid near
x
¼
0if
a
is larger
These discussions are quite important because we can know the phenomena not
only from the form of solutions to differential equations, but also from the differ-
ential equations themselves.
3.5.2 Diffusion in Continuous Spaces
When we discuss the diffusion on flat continuous spaces, Euclidean spaces are
considered to be fundamental metric spaces. Therefore, we can apply the theorems
in geometry and analysis.
Let us consider the diffusion phenomena on Euclidean space
n
. The density in
the diffusion phenomena is a function of time and space, so that we can write
R
n
u
: ð
t
;
x
Þ2
R
R
!
R
:
(3.10)
n
, that is, the
diffusion speeds are different for each point in the space. The diffusion phenomena
obeys the diffusion equation
In general, the diffusion coefficients for density is not uniform in
R
@
@
t
u
ð
t
;
x
Þ¼rð
D
ð
x
Þr
u
ð
t
;
x
ÞÞ;
(3.11)
