Page 15 - Mathematical Techniques of Fractional Order Systems

Variable Order Fractional Derivatives and Bone Remodeling Chapter | 1 5

α(t)
α(t)
D f (t) D f (t)
t
t
f (t) f (t)
FIGURE 1.1 Left: block diagram for Eq. (1.6); right: block diagram for Eq. (1.7).

1
8
> 2
> t if tA½0; 1½
2
>
>
>
>
>
>
t if tA½1; 2½
>
>
1 3
>
>
> 2
> t 2
> if tA½2; 3½
>
> 2 2
>
>
>
>
t if tA½3; 4½
>
<
D αðtÞ fðtÞ 5 1 2 ð1:7Þ
t 2 5if tA½4; 5½
2
>
>
>
>
>
>
t if tA½5; 6½
>
>
>
1 11
>
> 2
> t 2
> if tA½6; 7½
>
> 2 2
>
>
>
>
t if tA½7; 8½
>
>
>
...
:
This result corresponds to an integral that, whenever it is not used, does
not grow. When the 21 order branch is used again, the integral restarts at
the same value it had when used the last time. In other words, the operator
remembers that, for some past time intervals, the value of the order was not
the current one. Notice that this memory of past values of the order is inde-
pendent of the memory of past values of the differentiated function, which is
always present for order 21 (and not for order 0).
Section 1.2.2 will introduce mathematical definitions of variable-order
derivatives with and without memory of past values of the order. Before
that, it is important to assert that both cases (in Eqs. 1.6 and 1.7) make sense:
it is not that one definition is correct and the other not. And these results can
be found in practice. The most obvious way is the use of electronic compo-
nents, such as operational amplifiers and switches. The corresponding block
diagrams are given in Fig. 1.1.
1.2.2 Defining Variable Order Derivatives
To generalize differentiation and integration notions of order nAℕ to orders of
αAR, there are several alternative definitions: those of Gru ¨nwald Letnikoff,
Riemann Liouville, Caputo, Atangana Baleanu, etc. Here the first two are
α
addressed. Notice that, save only for orders α 5 1; 2; 3; ...,operator D is non-
local, as it always depends on the integration limits c and t. That is why it has