Biomedical Engineering Reference
In-Depth Information
The common formulation for the fractional integral can be derived directly from
a traditional expression of the repeated integration of a function. This approach is
commonly referred to as the Riemann-Liouville approach:
.....
t
0
t
1
τ)
n
−
1
f(τ)dτ
f (τ ) d
τ.
...
.dτ
n
=
(t
−
(A.11)
−
!
(n
1
)
0
n
which demonstrates the formula usually attributed to Cauchy for evaluating the
n
th
integration of the function
f(t)
. For the abbreviated representation of this formula,
we introduce the operator
J
n
such as shown in:
t
1
J
n
f(t)
τ)
n
−
1
f(τ)dτ
=
f
n
(t)
=
(t
−
(A.12)
(n
−
1
)
!
0
Often, one will also find another operator,
D
−
n
, used in place of
J
n
. While they
represent the same formulation of the repeated integral function, and can be seen as
interchangeable, one will find that the us of
D
−
n
may become misleading, especially
when multiple operators are used in combination. For direct use in (
A.11
),
n
is
restricted to be an integer. The primary restriction is the use of the factorial which
in essence has no meaning for non-integer values. The gamma function is, however,
an analytic expansion of the factorial for all reals, and thus can be used in place of
the factorial as in (
A.2
). Hence, by replacing the factorial expression for its gamma
function equivalent, we can generalize (
A.12
) for all
α
∈
R
,asshownin:
t
1
(α)
J
α
f(t)
τ)
α
−
1
f(τ)dτ
=
f
(t)
=
(t
−
(A.13)
∞
0
It is also possible to formulate a definition for the fractional-order derivative
using the definition already obtained for the analogous integral. Consider a dif-
ferentiation of order
α
=
1
=
2;
α
∈
R
+
. Now, we select an integer
m
such that
m
−
1
<α<m
. Given these numbers, we now have two possible ways to define the
derivative. The first definition, which we will call the Left Hand Definition is
D
α
f(t)
=
dt
m
1
α)
0
τ)
(α
+
1
−
m)
dτ
m
−
d
m
f(τ)
1
<α<m
(m
−
f(n)
=
(t
−
(A.14)
d
m
dt
m
f(t),
α
=
m
The second method, referred to here as the Right Hand Definition, is
D
α
∗
α)
t
f
(m)
1
(m
f(t)
:=
τ)
(α
+
1
−
m)
dτ
m
−
1
<α<m
f(n)
=
−
0
(A.15)
(t
−
d
m
dt
m
f(t),
α
=
m
Unlike the Riemann-Liouville approach, which derives its definition from the
repeated integral, the Grunwald-Letnikov formulation approaches the problem from
the derivative side. For this, we start from the fundamental definition a derivative:
f(x
+
h)
−
f(x)
f
(x)
=
lim
h
(A.16)
h
→
0
