Page 17 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 7

It seems reasonable to think that, if a memory of past values of the order
α is intended, then, in the summation of Eq. (1.8a), and in the integral of 8c,
for each time instant, the value of the order that was available at that time
instant should be used, as in Eq. (1.10).

k αðt 2 khÞ
t2c fðt 2 khÞ
h bc ð21Þ
X k
c D αðtÞ fðtÞ 5 lim ð1:10aÞ
t
h-0 1 h αðt2khÞ
k50
8
2αðτÞ21
Ð ðt2τÞ
> t 2
> fðτÞ dτ; if αðtÞAR
c
>
>
> Γð2 αðτÞÞ
>
<
c D αðtÞ fðtÞ 5 fðtÞ; if αðtÞ 5 0 ð1:10bÞ
t
> dαðtÞe
> d
> 1
> c D αðtÞ2dαðtÞe fðtÞ; if αðtÞAR
> t
dt
> dαðtÞe
:
This time, however, the GL definition of Eq. (1.10a) and the RL defini-
tion of Eq. (1.10b) do not provide the same result. The RL definition of
Eq. (1.10b) will for instance correspond to the result of Eq. (1.7); the GL
definition will not. In any case, these definitions are called type-2 variable
order derivatives in Lorenzo and Hartley (2002a); Vale ´rio and Sa ´ da Costa
(2011b); Vale ´rio and Sa ´ da Costa (2013), and type-B variable order deriva-
tives in Sierociuk et al. (2015a,b).
Another way of obtaining a memory of past values of the order is using,
in the RL definition, the order at time t 2 τ, that appears in the numerator of
the kernel. In the GL definition what corresponds to the difference between t
(the current time) and τ (the time at which the function appears in the defini-
tion) turns out to be t 2 ðt 2 khÞ 5 kh, thereby resulting in a GL definition
including a discrete time convolution. The expressions are those of
Eq. (1.11), which are again not equivalent. They are called type-3 variable
order derivatives in Lorenzo and Hartley (2002a); Vale ´rio and Sa ´ da Costa
(2011b); Vale ´rio and Sa ´ da Costa (2013), and type-C variable order deriva-
tives in Sierociuk et al. (2015a,b).

k αðkhÞ
t2c fðt 2 khÞ
h bc ð21Þ
X k
c D αðtÞ fðtÞ 5 lim ð1:11aÞ
t
h-0 1 h αðkhÞ
k50
8
2αðt2τÞ21
Ð ðt2τÞ
> t 2
> fðτÞ dτ; if αðtÞAR
c
>
>
> Γð2 αðt 2 τÞÞ
>
<
c D αðtÞ fðtÞ 5 fðtÞ; if αðtÞ 5 0 ð1:11bÞ
t
d
> dαðtÞe
>
> 1
> c D αðtÞ2dαðtÞe fðtÞ; if αðtÞAR
> t
dt
> dαðtÞe
: