Page 519 - Mathematical Techniques of Fractional Order Systems

506 Mathematical Techniques of Fractional Order Systems

method for solving a wide range of problems whose mathematical models
involve algebraic, differential, biological models and chaotic systems. The
definition of Gru ¨nwald Letnikov derivative has been used in numerical
analysis to discretize the fractional differential equations. The technique has
many advantages over the classical techniques, and provides an efficient
numerical solution.
The Caputo fractional derivative (Gorenflo and Mainardi, 1997) of order
α is defined as:
α
d fðtÞ
α
D fðtÞ 5
dt α
8 ð t m
1 f ðτÞ
> dτ
> m 2 1 , α , m
> α2m11 ð17:1Þ
> Γðm 2 αÞ
< 0 ðt2τÞ
5 ;
d m
>
>
> fðtÞ α 5 m
> m
: dt
where m is the first integer greater than α and Γð:Þ is the gamma function
defined by:
ð N
2t z21
ΓðzÞ 5 e t dt; Γðz 1 1Þ 5 zΓðzÞ: ð17:2Þ
0
Consider the fractional order differential equation
α
D xðtÞ 5 fðt; xÞ: ð17:3Þ
Gru ¨nwald Letnikov method of approximation (Hussian et al., 2008)is
defined as follows:
t=h
α 2α X j α
D xðtÞ 5 lim h ð21Þ xðt 2 jhÞ; ð17:4Þ
h-0 j
j50
where h is the step size. This equation can be discretized as follows:
n11
X α
c xðt 2 jhÞ 5 fðt n ; xðt n ÞÞ; j 5 1; 2; 3; ... ð17:5Þ
j
j50
α
where t n 5 nh and c are the Gru ¨nwald Letnikov coefficients defined as:
j
1 1 α
α
α
c 5 1 2 c α ; j 5 1; 2; 3; ...; c 5 h 2α : ð17:6Þ
j j21 0
j
The NSFD discretization technique is based on replacing the step size h
by a function φðhÞ (Hussian et al., 2008; Moaddy et al., 2012) and applying
it with (17.5) to solve (17.3).