Agriculture Reference
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where
D
is the fractional dispersion coefficient (
L
a
T
-1
) and the superscript
a is the order of fractional differentiation,
0
<α≤
. It is worth noting
that the classic CDE is recovered when the fractional order of FCDE is set
to 2. Fractional derivatives are integro-differential operators defined as
(Podlubny, 1999):
2
α
k
x
∂
∂
c
1
∂
∂
∫
k
−α−
1
=
Γ−α
(
x
−ξ ξξ
)
c
(,)
td
(4.38)
α
k
x
(
k
)
x
−∞
α
k
k
∞
∂
∂−
c
x
−
Γ−α
(1)
(
∂
∂
∫
k
−α−
1
=
(
ξ− ξξ
x
)
c
(,)
td
(4.39)
α
k
()
k
)
x
x
where
α>
, Γ is the gamma function, and
k
is the smallest integer num-
ber larger than α. If the value of α is a whole number, fractional derivatives
reduce to ordinary derivatives. Some properties of fractional derivative are
given in the appendix of Benson et al. (2000a).
Benson et al. (2000b) applied the FCDE to two cases: a laboratory sandbox
transport experiment and the Cape Cod field-scale transport experiment. The
fractional order for the laboratory data was found to be 1.55, whereas the Cape
Cod bromide plumes could be modeled using an FCDE with an order of 1.65
to 1.8. Pachepsky, Benson, and Rawls (2000) simulated scale-dependent solute
transport in soils using the. They also presented a comparison between the
FCDE and ADE based on statistical analysis. They found that the FCDE could
describe transport processes of a solute tracer better. Benson et al. (2000b)
also provided a method to estimate the fractional order a and the fractional
dispersion coefficient
D
separately. However, in our opinion, a simultane-
ous estimation of both α and
D
is more appropriate. Specific reasons will be
discussed later in this chapter. Benson et al. (2000b) estimated the fractional
order based on the relationship between measured apparent dispersivity and
distance from the sandbox experiment. Pachepsky, Benson, and Rawls (2000)
justified the application of the FCDE based on the scale-dependent transport
phenomenon. Zhou and Selim (2003) argued that the order of FCDE α cannot
be associated with scale-dependent transport directly. In fact, the relation-
ship between α and scale-dependent dispersivity is not theoretically sup-
ported and application of the FCDE needs to be justified.
0
4.3.3 Comparison between CDE and FCDE
As indicated above, the classical ADE predicts a linear increase of the vari-
ance of travel distance with time or mean travel distance, whereas the FCDE
predicts a nonlinear increase of the variance of travel distance. Therefore,
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