Page 333 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 11 323

Definition 1: Let nAN be such that n 2 1 , α , n, the Riemann Liouville
definition (Podlubny, 1998) for fractional derivative of order α for any func-
tion say f(t) is given as:
α
α
a D fðtÞ 5 d fðtÞ α
t
dðt2aÞ
1 d n ð t ð11:2Þ
5 ðt2qÞ n2α21 fðqÞdðqÞ
Γðn 2 αÞ dt n 0
where n is the first integer such that, n 2 1 # α , n and Γ denotes the
Gamma function which is defined as:
ð N
Γ xðÞ 5 t x21 2t ð11:3Þ
e dt
0
Definition 2: The Grunwald Letnikov definition (Podlubny, 1998) for
derivative of fractional order α, of any function f(t) is defined as under:
α
d fðtÞ
α
a D fðtÞ 5 α
t
dðt2aÞ
0 2 31
2α N ð11:4Þ
X j α t 2 a
t2a
5 lim N-N ð21Þ f @ t 2 j 4 5A
N j N
j51
This definition is considered as simplest and easiest definition to be used
in fractional calculus.
n
Definition 3: Let fAC ; nAN. The (left-sided) Caputo definition
21
(Podlubny, 1998) of derivative of fractional order α is defined as:
8 ð t n
1 f ðτÞ
>
> dτ; :n 2 1 , α , n
α112n
>
> Γðn 2 αÞ 0 ðt2τÞ
<
α
0 D 5 n ð11:5Þ
t
d f ðtÞ
>
> ; :α 5 n
>
dt
> n
:
where n is the smallest integer, greater than α.
The Grunwald Letnikov and the Riemann Liouville definitions are
equivalent for the functions fðtÞ having n continuous derivatives for t $ 0such
that n 2 1 # α , n. In terms of Laplace transform, the Riemann Liouville
fractional integral and derivative is as follows:
α α
Lf 0 D fðtÞg 5 S FðsÞ:α # 0: ð11:6Þ
t