Page 595 - Mathematical Techniques of Fractional Order Systems

Control and Synchronization Chapter | 19 565

k
The initial conditions are defined as, f aðÞ 5 0; ðk 5 0; 1; ... ; n 2 1Þ and
m 2 1 , q , m.
Integer order derivatives and integrals can be very clearly interpreted
geometrically and physically. But, the physical and geometric significance of
fractional calculus is not straightforward and is a topic of research. With the
help of the concept of transformed time and projection of integration on dif-
ferent axes, Podlubny (2002) has made an effort to explain the physical and
geometrical significance of fractional calculus. Some basic Podlubny (2002)
properties of fractional calculus are given as follows:
α
1. For α 5 n; where n is integer, the operation D fðtÞ gives the same result
as classical differentiation of integer order n.
α
0
2. For α 5 0 the operation D fðtÞ is the identity operator: D ftðÞ 5 fðtÞ.
3. Fractional differentiation and fractional integration are linear operation:
α α α
D λftðÞ 1 μgtðÞð Þ 5 λD ftðÞ 1 μD gtðÞ:
β
β
α
α
4. The additive index law D D ftðÞ 5 D D ftðÞ 5 D α1β ftðÞ holds under
some reasonable constraints on the function ftðÞ.
5. The Leibniz’s rule for fractional differentiation is given as:
N
α
α X k ðÞ α2k
D φðtÞftðÞð Þ 5 φ ðtÞD fðtÞ:
k
k50
6. The Laplace transform of fractional order derivative is defined as:
n21
k
a α X a2k21
L D ftðÞ 5 s L ftðފ 2 s D ftðÞ ; n 2 1 , α , n:
½
a t a t t50
k51
19.2.1 Numerical Solution of Fractional Order Differential
Equations
A number of approximation techniques have been proposed in literature for
the solution of FODEs. One of the common numerical methods for the solu-
tion of nonlinear FODEs has been derived from the GL definition (Vinagre
et al., 2003; Dorcak, 1994; Petras, 2011). The explicit numerical approxima-
th
tion of the q order derivative at the points kh ðk 5 1; 2; ...Þ is given as:
k k
q 2q X j q 2q X ðqÞ
D ftðÞ h ð 21Þ ft k2j 5 h c ft k2j ð19:5Þ
k 2 L m =h kh j j
j50 j50
where, L m 5 ‘memory length’, h 5 time step of calculation. c ðqÞ
j
ðj 5 0; 1; ... ; kÞ are the binomial coefficients and are given as

1 1 q
c ðqÞ 5 1; c ðqÞ 5 1 2 c ðqÞ ð19:6Þ
0 j j21
j