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where the constant is given by
C
=
A
. In general it is not an easy task to obtain the
inverse Fourier-Laplace transform of (
5.161
) for arbitrary indices
C
/
. Recall the
algebra of exponents given for the analysis of such equations by the fractional calculus
[
31
], from which we can conclude that the double inverse transform of (
5.161
) yields a
fractional diffusion equation in space and time. We refer the reader to Magin [
14
] and
Wes t
et al
.[
31
] for a general discussion of fractional diffusion equations.
α
and
β
5.4.4
Scaling solutions
One technique for solving (
5.161
) uses the properties of Fourier transforms and Laplace
transforms to obtain the scaling solutions to this equation and may be implemented
without performing either of the transforms explicitly. First note the scaling of the
E
-dimensional Fourier transform of an analytic spatial function,
f
k
a
∞
∞
d
E
k
d
E
k
(
1
a
E
E
e
−
i
k
·
q
f
E
e
−
i
k
·
a
q
(
)
=
(
)
=
,
f
a
q
k
(5.162)
π
)
π
)
(
2
2
−∞
−∞
and the Laplace transform of an analytic temporal function
∞
∞
g
u
b
e
−
u
t
1
b
e
−
ubt
du
ˆ
g
(
bt
)
=
du
g
ˆ
(
u
)
=
.
(5.163)
0
0
From the last two equations we conclude that the space-time probability density
P
has the Fourier-Laplace transform
P
∗
(
ba
E
(
a
q
,
bt
)
k
/
a
,
u
/
b
)/(
)
, which, on com-
parison with (
5.161
), yields the relationship
ba
E
P
∗
k
a
E
P
∗
b
β/α
k
u
1
u
b
1
a
,
=
a
,
,
(5.164)
whose inverse Fourier-Laplace transform is
P
a
q
t
1
b
E
β/α
P
(
a
q
,
bt
)
=
b
β/α
,
.
(5.165)
The scaling relations (
5.164
) and (
5.165
) can be solved in
E
dimensions by assuming
that the probability density function satisfies a scaling relation of the form
a
q
b
μ
t
μ
1
b
E
μ
t
μ
P
(
a
q
,
bt
)
=
F
(5.166)
α,β
which when inserted into (
5.165
) yields
a
q
b
μ
t
μ
a
q
b
β/α
t
μ
1
b
E
μ
t
μ
1
b
E
β/α
t
μ
F
=
F
.
(5.167)
α,β
α,β
The solution to (
5.167
) is obtained by equating indices to obtain
μ
=
β/α
, resulting in
the scaling solution in the CTRW formalism in
E
dimensions
q
t
β/α
1
t
β/α
P
(
q
,
t
)
=
F
(5.168)
α,β
