188
Air Pollution and Turbulence: Modeling and Applications
where M (x, r) = X ⋅ G(x, r) and G(x, r) is the diagonal matrix with components e −d n x .
Further the new unknown arbitrary constant vector ξ is given by ξ = X −1 ⋅ Z (0).
Applying the boundary conditions (Equation 7.13a), we determine the unknown
components of the arbitrary column vector ξ = col(ξ 1 ,ξ 2 ), solving the equation:
11
12
1
1
21
*
22
*
2
2
*
(0)
(0)
(0)
( )
( )
( )
M
M
Z
M L
M L
Z L
ξ
⎛
⎞ ⎛ ⎞ ⎛
⎞
=
⎜
⎟ ⎜ ⎟ ⎜
⎟
ξ
⎝
⎠ ⎝ ⎠ ⎝
⎠
(7.16)
Once these unknown coeffi cients are evaluated, we can construct the analytical solution to problem (Equation 7.6) applying the inverse Laplace transform defi nition.
This procedure yields to the following analytical result:
0
1
( , , )
( , ) ( ) d
2
i
N
rt
n
n
n
i
c x z t
c x r
z e s
i
γ + ∞
= γ − ∞
−
=
Ψ
π ∑ ∫
(7.17)
By analytical, we mean that no approximation is made along the derivation of
solution (Equation 7.17). To overcome the drawback of evaluating the line integral
appearing in Equation 7.17, in the sequel, we report a closed-form solution for this
integral, using the Gaussian quadrature scheme. By this procedure, we get:
=
=
⎛
⎞
−
=
Ψ
⎜
⎟
⎝
⎠
∑ ∑
1
0
( , , )
,
( )
M
N
k
k
k
n
n
k
n
P
P
c x z t
A
c x
z
t
t
(7.18)
where A k and P k are the weights and roots of the Gaussian quadrature scheme tabulated in the book by Stroud and Secrest (1966). Regarding the issue of the adopted
Laplace numerical inversion scheme, it is important to mention that this approach
is exact if the integrand is a polynomial of degree 2M −1 in the 1/r variable. We
are aware of the existence in the literature of methods to invert numerically the
Laplace-transformed functions (Valkó and Abate, 2004; Abate and Valkó, 2004),
but we restrict our attention in the problem considered to the Gaussian quadrature
scheme. The motivation for this choice comes besides the simplicity the good results
achieved.
7.3 PARTICULAR SOLUTIONS
In the sequel, taking advantage of the generality of the discussed solution, we report
simplifi ed solutions for specifi c physical scenarios, readily obtained from Equation
7.17, by just taking limits.
7.3.1 TIME-DEPENDENT FICKIAN FLOW MODEL
From Equation 7.6, we promptly realize that the advection–diffusion equation governed by non-Fickian fl ow is readily obtained by making the parameters β and τ,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
where M (x, r) = X ⋅ G(x, r) and G(x, r) is the diagonal matrix with components e −d n x .
Further the new unknown arbitrary constant vector ξ is given by ξ = X −1 ⋅ Z (0).
Applying the boundary conditions (Equation 7.13a), we determine the unknown
components of the arbitrary column vector ξ = col(ξ 1 ,ξ 2 ), solving the equation:
11
12
1
1
21
*
22
*
2
2
*
(0)
(0)
(0)
( )
( )
( )
M
M
Z
M L
M L
Z L
ξ
⎛
⎞ ⎛ ⎞ ⎛
⎞
=
⎜
⎟ ⎜ ⎟ ⎜
⎟
ξ
⎝
⎠ ⎝ ⎠ ⎝
⎠
(7.16)
Once these unknown coeffi cients are evaluated, we can construct the analytical solution to problem (Equation 7.6) applying the inverse Laplace transform defi nition.
This procedure yields to the following analytical result:
0
1
( , , )
( , ) ( ) d
2
i
N
rt
n
n
n
i
c x z t
c x r
z e s
i
γ + ∞
= γ − ∞
−
=
Ψ
π ∑ ∫
(7.17)
By analytical, we mean that no approximation is made along the derivation of
solution (Equation 7.17). To overcome the drawback of evaluating the line integral
appearing in Equation 7.17, in the sequel, we report a closed-form solution for this
integral, using the Gaussian quadrature scheme. By this procedure, we get:
=
=
⎛
⎞
−
=
Ψ
⎜
⎟
⎝
⎠
∑ ∑
1
0
( , , )
,
( )
M
N
k
k
k
n
n
k
n
P
P
c x z t
A
c x
z
t
t
(7.18)
where A k and P k are the weights and roots of the Gaussian quadrature scheme tabulated in the book by Stroud and Secrest (1966). Regarding the issue of the adopted
Laplace numerical inversion scheme, it is important to mention that this approach
is exact if the integrand is a polynomial of degree 2M −1 in the 1/r variable. We
are aware of the existence in the literature of methods to invert numerically the
Laplace-transformed functions (Valkó and Abate, 2004; Abate and Valkó, 2004),
but we restrict our attention in the problem considered to the Gaussian quadrature
scheme. The motivation for this choice comes besides the simplicity the good results
achieved.
7.3 PARTICULAR SOLUTIONS
In the sequel, taking advantage of the generality of the discussed solution, we report
simplifi ed solutions for specifi c physical scenarios, readily obtained from Equation
7.17, by just taking limits.
7.3.1 TIME-DEPENDENT FICKIAN FLOW MODEL
From Equation 7.6, we promptly realize that the advection–diffusion equation governed by non-Fickian fl ow is readily obtained by making the parameters β and τ,
© 2010 by Taylor and Francis Group, LLC