Page 629 - Mathematical Techniques of Fractional Order Systems

600 Mathematical Techniques of Fractional Order Systems

This operator is a notation for taking both the fractional integral and
functional derivative in a single expression defined as (Diethelm and Ford,
2002; Diethelm et al., 2004; Petra ´ˇ s, 2006; Khettab et al., 2017b,c,d)
d
8 q
> q . 0
dt
> q
<
q
D 5 ð20:1Þ
a t
1 q 5 0
>
>
Ð t
: 2q
ðdτÞ q , 0
a
There are some basic definitions of the general fractional integration and
differentiation. The commonly used definitions are Riemann Liouville,
Adams Bashforth Moulton algorithm and the method of Gru ¨nwald Letnikov:
“Numerical evaluation of the fractional derivative of some usual functions.”
The simplest and easiest definition is the Riemann Liouville definition
given as:
RL q 1 d n ð t n2q21 fðτÞdτ
D ftðÞ 5 ð t2τÞ ð20:2Þ
t 0 t Γðn 2 qÞ dt n
t 0
where n is the first integer which is not less than q, i.e., n 2 qÞ , q , n, and
ð
Γ is the Gamma function.
The numerical simulation of a fractional differential equation is not as
simple as that of an ordinary differential equation.
The algorithm which is an improved version of the
Adams Bashforth Moulton (Diethelm and Ford, 2002; Diethelm et al.,
2004) to find an approximation for fractional order systems based on predic-
tor correctors is given (Diethelm et al., 2003). Consider the following dif-
ferential equation
GL q
ð
a D ytðÞ 5 fy tðÞ; tÞ ð20:3Þ
t
ðkÞ
0
where 0 # t # T and y ðÞ 5 y and k 5 0; 1; 2; ... ; m 2 1
ðkÞ
0
Can be expressed as follows
X k
q
q ðÞ
yt k ðÞ 5 fy t k ðÞ; t k Þh 2 c yt k2j ð20:4Þ
ð
j5v j
q
where D t yðtÞ is the Caputo fractional derivative of order q . 0 is defined
0
as:
8 ð t
1
m2q21 ðmÞ
> f ðτÞdτ; m 2 1 , q , m
ðt2τÞ
>
>
< Γðm 2 qÞ 0
q
D yðtÞ 5 ð20:5Þ
0 t d m
>
> yðtÞq 5 m
dt
> m
:
and m is the first integer larger than q.