Page 50 - Mathematical Techniques of Fractional Order Systems
P. 50
40 Mathematical Techniques of Fractional Order Systems
α 1 ð t
D uðtÞ 5 ðt2τÞ n2α21 ðnÞ ð2:3Þ
u ðτÞdτ; n 2 1 , α , n:
t
Γðn 2 aÞ 0
α
The important property of the Caputo fractional derivative operator D is
t
(Podlubny, 1999)
n21 t i
α
α
J D uðtÞ 5 uðtÞ 2 X u ð0Þ ; ð2:4Þ
ðiÞ
t i!
i50
α
where J is the Riemann Liouville fractional integral operator and defined
as
α
J uðtÞ 5 1 ð t ðt2τÞ α21 uðτÞdτ: ð2:5Þ
ΓðαÞ 0
2.2.1 Predictor Homotopy Analysis Method
Liao (1992) proposed a powerful and easy semianalytic tool for nonlinear
problems, namely the HAM. The homotopy is a continuous transformation
from one function to another (Liao, 2012, 2003). The homotopy between
two continuous functions f (t) and g(t) from a topological space X to topolog-
ical space Y is defined to be continuous function ℋ(t, p)
ℋðt; pÞ:X 3 ½0; 1-Y; ð2:6Þ
where p is called the homotopy parameter and p A [0, 1] such that if t A X
then ℋ(t,0) 5 f (t) and ℋ(t,1) 5 g(t). The mth-order homotopy derivative
of ℋ(t, p) is defined by
m
m
D ℋðt; pÞ 5 1 @ ℋðt; pÞ p50 ð2:7Þ
m! @p m
where m $ 0 is an integer. And has the following properties, for homotopy
series
N N
X i X i
ϕðt; pÞ 5 u i ðtÞp ; φðt; pÞ 5 v i ðtÞp ; ð2:8Þ
i50 i50
the following are held:
m
m
j
1. D (ϕ) 5 u m and D (p ϕ) 5 D m2j ϕ 5 u m2j .
2. Let L to be a linear operator independent of the homotopy parameter p
m
m
then D (Lϕ) 5 L [D (ϕ)]
m
m
3. D ðϕφÞ 5 P u m2k v k :
k50
4. If f and g are functions independent of homotopy parameter p then
m
D ðfφ 1 gϕÞ 5 fv m 1 gu m :
