186
Air Pollution and Turbulence: Modeling and Applications
Recasting Equation 7.10 as a matrix ordinary differential equation, we read
⋅
+
⋅
+
⋅
=
′′
′
1
2
3
( )
( , )
( ) ( , )
( ) ( , ) 0
B x Y x r B x Y x r B x Y x r
(7.11)
where Y (x, r) is the column vector whose components are {c
–
n (x, r)} and the entries
of matrices B 1 , B 2 , and B 3 are respectively given by
=
Ψ
Ψ
+ β Ψ
Ψ
+
′
+ β
Ψ
Ψ
+τ
Ψ
Ψ
′
∫
∫
∫
∫
1 ,
0
0
0
0
( )
( ) ( )d
( ) ( )d
(
) ( ) ( )d
( ) ( )d
h
h
n m
x n
m
x n
m
h
h
x
n
m
x n
m
b
K
z
z z
K
z
z z
K
z
z z r K
z
z z
= − Ψ Ψ
− β Ψ
Ψ
− β ′ Ψ
Ψ
+
′
+
Ψ
Ψ
+ β Ψ
Ψ
+ β
Ψ
Ψ
+
′
′′
′ ′
−τ
Ψ
Ψ
+τ
Ψ
Ψ
′
∫
∫
∫
∫
∫
∫
∫
∫
2 ,
0
0
0
0
0
0
0
0
( )
( ) ( )d
( ) ( )d
( )
( ) ( )d
( ) ( )d
( ) ( )d
(
) ( ) ( )d
( ) ( )d
( ) ( )d
h
h
h
nm
n
m
n
m
n
m
h
h
h
x n
m
x n
m
x
n
m
h
h
n
m
x n
m
b
u
z
z z
u
z
z z
u
z
z z
K
z
z z
K
z
z z
K
z
z z
r u
z
z z r K
z
z z
and
=
Ψ
Ψ
−λ
Ψ
Ψ
−
Ψ
Ψ
+
′ ′
′
− βΨ
Ψ
− β Ψ Ψ
+λ β Ψ Ψ
+
′
′
− β Ψ Ψ
−τ
Ψ
Ψ
+
′ ′
− τ
Ψ′
∫
∫
∫
∫
∫
∫
∫
∫
2
3 ,
0
0
0
2
0
0
0
2
0
0
( )
( ) ( )d
( ) ( )d
( ) ( )d
( ) ( )d
( ) ( )d
( ) ( )d
( ) ( ) ( )d
( ) ( )d
( )
h
h
h
nm
z n
m
n
z n
m
n
m
h
h
h
n
m
n
m
n
n
m
h
h
n
m
n
m
n
b
K
z
z z
K
z
z z
w
z
z z
r
z
z z r
z
z z
w
z
z z
w
z
z z r
z
z z
r w
z Ψ
− Ψ
Ψ
∫
∫
0
0
( )d
( ) ( )d
h
h
m
n
m
z z r
z
z z
To solve Equation 7.11, we proceed likewise the work of Moreira et al. (2006b),
performing a stepwise approximation of the entries of the matrices B 1 (x), B 2 (x),
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