Page 57 - Mathematical Techniques of Fractional Order Systems
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Nonlinear Fractional Order Boundary-Value Problems Chapter | 2 47
3
uðtÞ 5 ð 12tÞ 2 ð2:39Þ
2
It has been shown (Abbasbandy and Shivanian, 2011; Semary and Hassan,
2015; Barletta, 1999; Barletta et al., 2005) by semianalytic and numerical meth-
ods that Eqs. (2.37), (2.38) admit dual solutions for any given E in the interval
(2N,0) , (0, E max )inwhich E max D 228.128. This study generalizes the
boundary value problem (2.37), (2.38) in the fractional order domain as follows:
2
du
E
α
D uðtÞ 5 ð2:40Þ
t
16 dt
where 3 , α # 4 and subject to boundary conditions (2.38). The model
(2.40) with the boundary conditions (2.38) has been solved using PHAM and
power series method (Arqub et al., 2014; Alomari et al., 2013). Alomari
et al. (2013) applied the PHAM explained in Section 2.2.1 by using the aux-
4
@
iliary linear operator L : ½ 5 @t 4 : ½. This section applies the same method for
the model (2.40) but using fractional order linear operator L as follows
α
L½ϕðt; E; pÞ 5 D ½ϕðt; E; pÞ; ð2:41Þ
t
α
where D defined in (2.3) and the operator L has the property
t
α
2
3
D ½c 0 1 c 1 t 1 c 2 t 1 c 3 t 5 0: ð2:42Þ
t
To construct the dual solution of the model (2.40) with conditions (2.38)
can be replaced formally by the following problem
2
du
E
α
D uðtÞ 5 ; u ð0Þ 5 u vð0Þ 5 uð1Þ 5 0; uvð0Þ 5 E; ð2:43Þ
0
0
t
16 dt
and the forcing condition
ð 1
uðtÞdt 5 1: ð2:44Þ
0
From the conditions in Eq. (2.43), the initial approximation guess can be
E
2
u 0 ðt; EÞ 5 ðt 2 1Þ: ð2:45Þ
2
According the linear operator (2.41) and choosing H(t) 5 1, the mth-order
deformation Eq. (2.25) of the problem (2.43) reads
( " 2 #)
α
α
D u m ðt; EÞ 2 χ u m21 ðt; EÞ 5 hD u m21 ðt; EÞ 2 E D m p dϕðt; E; pÞ ;
t m t
16 dt
ð2:46Þ
α
applying the Riemann Liouville integral of order α(J )on Eq. (2.46) and
using the properties of the homotopy derivative (2.7), the mth-order deforma-
tion equation becomes
