Page 58 - Mathematical Techniques of Fractional Order Systems

48 Mathematical Techniques of Fractional Order Systems

( m21 )
α
α
u m ðt; EÞ 5 χ u m21 ðt; EÞ 1 J hD u m21 ðt; EÞ 2 E X u ðt; EÞu 0 m212i ðt; EÞ
0
m
i
t
16
i50
3
2
1 c 0 1 c 1 t 1 c 2 t 1 c 3 t :
ð2:47Þ
and the constants c i , i: 0, 1, 2, 3 can be determined by the boundary
conditions:
0
0
u ð0; EÞ 5 uv m ð0; EÞ 5 uv ð0; EÞ 5 u m ð1; EÞ 5 0: ð2:48Þ
m m
Using the software of Wolfram Mathematica, starting with u 0 (t, E) thus
the first two terms of Eqs. (2.47, 2.48) are
Ehð2 1 1 t 1 αÞE 2
2
u 1 ðt; EÞ 52 ; ð2:49Þ
2
8αð2 1 3α 1 α ÞΓðαÞ
EhE 2 212α
u 2 ðt; EÞ 5 ðEhð2 1 1 t Þð2 1 αÞ E Γð3 1 αÞ
64αð1 1 αÞð2 1 αÞΓðαÞΓð3 1 2αÞ
21α
2 8ð1 1 hÞð2 1 1 t ÞΓð3 1 2αÞÞ;
ð2:50Þ
in this way can be obtained the mth-order approximate solution for the prob-
lem (2.43)
M
X
uðtÞ U M ðt; h; EÞ 5 u m ðt; EÞ; ð2:51Þ
m50
The homotopy solution (2.51) still contains two unknowns parameters,
namely h and E.So Eq. (2.36) and with the help of the forcing condition
Ð 1
0 uðtÞdt 5 1, becomes
ð 1 ð 1
uðtÞdt U M ðt; h; EÞdt 5 1; ð2:52Þ
0 0
by selecting some values of E and according to the above Eq. (2.52) in
Figs. 2.1 and 2.2, the h 2 E curves has been plotted in the range [ 2 2.5, 0]
and when E 5 20 and E 52 20. The curve gives two lines that are parallel
to the h-axes for each value of fractional order parameter α. This means the
model (2.40) has two solutions in these cases and so is in full agreement
with the solutions result reported by Arqub et al. (2014) and Alomari et al.
(2013). On other hand, to apply the controlled Picard’s method for the model
(2.40), the model boundary conditions (2.38) can be replaced as follows
u ð0Þ 5 u vð0Þ 5 0; uvð0Þ 5 E; uð0Þ 5 γ; ð2:53Þ
0
0
where γ and E are the unknown parameters and the two forcing conditions