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and consequently, identifying 1
−
α
=
μ
−
1, we can rewrite (
5.9
)as
d
dt
0
(
D
μ
−
2
t
t
)
=−
(
3
−
μ)ϒ
[
0
(
t
)
]
.
(5.13)
Equation (
5.13
) can also be expressed as
D
3
−
μ
3
−
μ
0
(
[
0
(
)
]=−
λ
),
t
t
(5.14)
t
where we have introduced the dissipation constant
1
/(
3
−
μ)
.
λ
=[
(
3
−
μ)ϒ
]
(5.15)
However, we note that 3
−
μ>
1for
μ<
2 and therefore the fractional derivative is
greater than unity and we write
3
−
μ
=
1
+
ε.
(5.16)
The solution to (
5.14
) can be obtained using Laplace transforms, as we explain later.
The Laplace transform of the fractional derivative can be written, see page 160 of West
et al
.[
31
],
D
1
+
ε
t
u
u
ε
u
0
(
)
.
LT
[
0
(
t
)
];
=
u
)
−
0
(
0
(5.17)
f
We use throughout the notation
(
u
)
to denote the Laplace transform of the func-
tion
f
(
t
)
,
∞
f
e
−
ut
(
u
)
=
dt f
(
t
)
,
(5.18)
0
and take the Laplace transform of the fractional dynamical equation (
5.14
) so that, using
(
5.17
), we obtain
u
1
+
ε
0
(
u
ε
0
(
1
+
ε
0
(
u
)
−
0
)
=−
λ
u
),
which, on rearranging terms, gives
u
λ
1
+
ε
1
u
1
0
(
u
)
=
0
(
0
).
(5.19)
1
+
ε
+
(
u
/λ)
1
The asymptotic-in-time form of the solution to the fractional differential equation can
be obtained from the small
u
form of (
5.19
),
u
λ
1
+
ε
1
u
0
0
(
lim
u
u
)
=
0
(
0
),
(5.20)
→
whose inverse Laplace transform yields
)
≈
0
(
0
)
0
(
t
+
ε
.
(5.21)
1
(λ
t
)
The velocity autocorrelation function therefore decays asymptotically as an inverse
power law in time with an index given by 3
−
μ.