4.3 A Mechanochemical Model
59
E 1 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a 1 −c1 0
0 . . . 0
−d 1 a 1 −c 1 0 . . . 0
0 −d 1 a 1 −c 1 . . . 0
. . .
. . .
0
0
0 . . . −d 1 a 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
E 2 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a 2 −c 2
. . . 0
−d 2 a 2 −c 2
. . . 0
0 −d 2 a 2 −c 2 . . . 0
. . .
. . .
0
0
0 . . . −d 2 a 2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
L 1 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b 1 c 1
. . . 0
d 1 b 1 c 1
. . . 0
0 d 1 b 1 c 1 . . . 0
. . .
. . .
0 0 0 . . . d 1 b 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
L 2 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b 2 c 2
. . . 0
d 2 b 2 c 2
. . . 0
0 d 2 b 2 c 2 . . . 0
. . .
. . .
0 0 0 . . . d 2 b 2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
K 12 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
k 12 dt 0
. . . 0
0 k 12 dt 0
. . . 0
0
0 k 12 dt 0 . . . 0
. . .
. . .
0
0
0 . . . 0 k 12 dt
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
K 21 =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
k 21 dt 0
. . . 0
0 k 21 dt 0
. . . 0
0
0 k 21 dt 0 . . . 0
. . .
. . .
0
0
0 . . . 0 k 21 dt
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
E =
E 1 0
0 E 2
,
M =
L 1 − K 12
K 21
K 12
L 2 − K 21
(4.43)
Using the boundary conditions P n
0 = P n
2N = 0, the system Eq. 4.41 can be written
in matrix form as (Fig. 4.4):
E.P
n+1
= M.P
n ,
P
n+1
= E
−1 .
M.P
n
P
n
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
P n
1 (1)
P n
2 (1)
.
.
P n
N (1)
P n
1 (2)
P n
2 (2)
.
.
P n
2N (2)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.44)