Information Technology Reference
In-Depth Information
The RL fractional integral of the GWF is given by [
22
]
t
1
(α)
W
(τ)
d
τ
W
(
−
α)
(
D
−
α
t
t
)
≡
−∞
[
W
(
t
)
]=
(5.35)
(
t
−
τ)
1
−
α
−∞
for 0
<α<
1
,
which, after some not-so-straightforward analysis, yields
∞
1
b
(
2
−
D
+
α)
n
[
W
(
−
α)
(
b
n
t
t
)
=
1
−
cos
(
)
]
.
(5.36)
n
=−∞
It is clear that the fractional integration has shifted the fractal dimension of the GWF
from
D
to
D
, thereby reducing the fractal dimension and making the function
smoother; this is consistent with our intuition about integration.
Similarly the RL fractional derivative of the GWF is given by [
22
]
−
α
t
(τ)
τ
1
(α)
d
dt
W
d
W
(α)
(
)
≡
−∞
D
t
[
(
)
]=
t
W
t
(5.37)
(
−
τ)
α
t
−∞
for 0
<α<
1
,
which, again after some not-so-straightforward analysis, integrates to
∞
1
b
(
2
−
D
−
α)
n
[
W
(α)
(
b
n
t
t
)
=
1
−
cos
(
)
]
.
(5.38)
n
=−∞
Consequently, we see that the fractional derivative has shifted the fractal dimension of
the GWF from
D
to
D
, thereby increasing the fractal dimension and making the
function more erratic; this is consistent with our intuition about derivatives.
Note that we have shown that a fractal function, whose integer derivative diverges,
has a finite fractional derivative. It is therefore reasonable to conjecture that a dynamical
process described by such a fractal function might have fractional differential equations
of motion. This assumption is explored below.
+
α
5.1.2
Deterministic fractional dynamics
Of course, the fractional calculus does not in itself constitute a theory for complex webs
in the physical, social or life sciences; however, such theories are necessary in order to
interpret the fractional derivatives and integrals in the appropriate contexts. We therefore
begin by examining a simple relaxation process described by the rate equation
D
t
[
Q
(
t
)
]+
λ
Q
(
t
)
=
0
,
(5.39)
where
t
determines how quickly the process returns to its
equilibrium state. The solution to this equation is given in terms of its initial value by
>
0 and the relaxation rate
λ
e
−
λ
t
Q
(
t
)
=
Q
(
0
)
,
(5.40)
which is unique. Consequently, everything we can know about this simple network start-
ing from an initial condition is contained in the relaxation rate. Equation (
5.40
) can also
describe a stochastic process if the initial condition is interpreted as the rate
Q
(
0
)
=
λ
,
the rate of occurrence of a random event, so that
Q
(
t
)
is the exponential probability
density and the process being described is Poisson.
