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where
s
is the initial density of the substances in the medium, i.e. (
u
1
(0)=
s,u
2
(0)=
s
),
z
(
x,y
)is the spatial nonlinear function of both concentrations.
With
x
=0
,y
= 0 being chosen to be waves center, the constraints on
z
(
x,y
)
are expressed as follows: 0
.
5
<z
(0
,
0)
<
(1
−s
). The process begins when in the
homogeneous mixture of two substances with a certain gel there appears a little
spot of saturated
u
1
.
According to the order of a PDE, the discrete array is a 2-layered one, i.e.
M
=
M
1
∪ M
2
,
M
g
=
{
(
i,j
)
g
:
i,j
=
−n...,
0
,...,n, g
=1
,
2
n
= 150
}
. The
initial arrays
Ω
R
g
(0)=
{
(
s,
(
i,j
)
g
},g
=1
,
2,
s
=0
.
1, except that in the center
of
Ω
R
1
(0)there is a spot 5
×
5 of cells with
u
1
=1.
Fig. 2.
Snapshots of CA evolution simulating the chemical reaction, generating con-
centric waves in 2D space
Boolean discretization of
Ω
R
1
(0)is achieved by setting cell states
v
= 1 with
the probability
P
(
v
g
=1)
=
s
, except the square 5
×
5 in the center of
Ω
R
1
, where
v
= 1 determinately. Border conditions are taken into account according to (18).
Since the second equation in (19)has no diffusion part, there is no need to
perform Boolean discretization of
Ω
R
1
(0).
Each iteration of the CA approximating the PDE system (19)consists of the
following computations.
1)An iteration of BR-diffusion is applied to
Ω
R
1
resulting in
Ω
S
1
(
t
+1).
2)The functions
f
1
and
f
2
are computed for all cells (
i,j
)
∈ M
forming
cellular arrays
Ω
F
1
and
Ω
F
2
. The latter is also used as
Ω
R
2
(
t
+1).
