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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
t
)
The velocity autocorrelation function therefore decays asymptotically as an inverse
power law in time with an index given by 3
μ.
 
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