Page 16 - Mathematical Techniques of Fractional Order Systems

6 Mathematical Techniques of Fractional Order Systems

memory of past values of fðtÞ (Vale ´rio and Sa ´ da Costa, 2013). Recall that, for
constant differentiation orders, the Gru ¨nwald Letnikoff (GL) definition is
given by Eq. (1.8a), and the Riemann Liouville (RL) definition is given by
Eq. (1.8c).

t2c
h bc
P k α
ð21Þ fðt 2 khÞ
α k50 k
c D fðtÞ 5 lim α ð1:8aÞ
t
h-0 1 h
8
Γðα 1 1Þ
;
> if α; k; ðα 2 kÞARZ 2
>
>
Γðk 1 1ÞΓðα 2 k 1 1Þ
>
>
α 5 < k
>
2
k > ð21Þ Γðk 2 αÞ ; if αAZ XkAZ 1 ð1:8bÞ
> 0
> Γðk 1 1ÞΓð2 αÞ
>
>
> 2 2
0;
: if ðkAZ 3ðk 2 αÞAℕÞXα=2Z
2α21
8
Ð ðt2τÞ
> t 2
> fðτÞ dτ; if αAR
>
c
>
> Γð2 αÞ
>
<
α
c D fðtÞ 5 fðtÞ; if α 5 0 ð1:8cÞ
t
d
> dαe
>
> 1
> c D α2dαe fðtÞ; if αAR
> t
dt
> dαe
:
The fractional order of integrals and derivatives can be a function of time
or some other variable (Lorenzo and Hartley, 2002b). Here we will express
it a function of time; the other case is a straightforward generalization. If no
memory of past values of the order is intended, the definition must only
include its current value:
t2c
h bc
P k αðtÞ
ð21Þ k fðt 2 khÞ
c D αðtÞ f ðtÞ 5 lim k50 ð1:9aÞ
t
h-0 1 h αðtÞ
2αðtÞ21
8
Ð ðt2τÞ
t
> fðτÞ dτ; if αðtÞAR 2
>
c
>
>
> Γð2 αðtÞÞ
>
>
<
c D αðtÞ fðtÞ 5 fðtÞ; if αðtÞ 5 0 ð1:9bÞ
t
>
d
>
> dαðtÞe
> 1
> αðtÞ2dαðtÞe fðtÞ; if αðtÞAR
> c D
> t
: dt dαðtÞe
These GL and RL formulations without memory of α are equivalent (if
the function is well-behaved enough so that both formulations can be
applied). They are called type-1 variable order derivatives in Lorenzo and
Hartley (2002a); Vale ´rio and Sa ´ da Costa (2011b); Vale ´rio and Sa ´ da Costa
(2013), and type-A variable order derivatives in Sierociuk et al. (2015a,b).