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spectrum that is observed in turbulent fluid flow where the spatial eddies give rise to an
inverse power-law energy spectrum of the form
k
−
5
/
3
.
It is possible to generalize stochastic differential equations even further by intro-
ducing a second fractional exponent into (
5.102
) to obtain
2
α/
2
Y
(
−
)
γ
+
λ
α,γ
(
x
)
=
ξ(
x
),
(5.116)
where we now index the solution with subscripts for each of the exponents. Here the new
index is restricted to the interval 0
<γ
≤
1 and defines a Riesz fractional derivative in
terms of its Fourier transform to be [
31
]
FT
[
(
−
)
γ
F
2
γ
F
(
x
)
;
k
]=|
k
|
(
k
).
(5.117)
As Lim and Teo [
12
] point out, the operator in (
5.116
) can be regarded as a shifted Riesz
fractional derivative, which has the formal series expansion
α/
2
α/
2
∞
2
(
−
)
γ
+
λ
λ
α
−
2
j
(
−
)
γ
j
=
,
(5.118)
j
j
=
1
so that its Fourier transform is
k
α/
2
α/
2
F
∞
2
(
−
)
γ
+
λ
λ
α
−
2
j
2
γ
j
F
FT
(
x
)
;
=
|
k
|
(
k
).
(5.119)
j
j
=
1
Consequently, the solution to (
5.116
) can be expressed by the integral
ξ(
e
−
i
k
·
x
k
2
γ
+
λ
1
k
)
2
α/
2
d
2
k
Y
α,γ
(
x
)
=
(5.120)
2
(
2
π)
R
2
and this function is well defined as an ordinary random field only when
αγ >
1 since it
2
norm remains finite over the entire domain. To see this
restriction on the power-law indices, consider the covariance
is only in this case that the
R
x
)
=
x
)
,
C
α,γ
(
x
,
Y
α,γ
(
x
)
Y
α,γ
(
(5.121)
so that, on substituting (
5.120
)into(
5.121
) and using the delta-correlated property of
the Gaussian field,
e
−
i
k
·
(
x
)
x
−
2
x
)
=
d
2
k
C
α,γ
(
x
,
k
2
γ
+
λ
2
α
.
(5.122)
(
2
π)
2
R
2
The covariance is therefore isotropic,
x
)
=
x
),
C
α,γ
(
x
,
C
α,γ
(
x
−
(5.123)
x
.
and the variance is obtained when
x
However, from the integral we determine by
direct integration that the variance diverges as
k
=
→∞
when
αγ <
1
.
Consequently,
the solution
Y
α,γ
(
1 and the traditional methods
for its analysis no longer apply. Consequently, the concept of a random field must be
generalized to include those without central moments. But this is not the place to make
such generalizations.
x
)
is not just a random field when
αγ <
