Page 67 - Mathematical Techniques of Fractional Order Systems

56 Mathematical Techniques of Fractional Order Systems

2
2:1 2
2
u 2 ðt; EÞ 52 0:455hð1:099t 1 ht E Þψ ; ð2:77Þ
2 2:2 2
2
2
2:1
2 4:1 2
u 3 ðt; EÞ 5 0:04134hψ ð2 12:0917t 2 22:0087ht E 2 9:9768h t E 1 h t ψ Þ;
ð2:78Þ
and so on. In this way can be obtained mth-order approximate solution for
the problem (2.63) with initial condition (2.64) as follows
M
X
uðtÞ U M ðt; h; E; nÞ 5 u m ðt; EÞ: ð2:79Þ
m50
According to the frame of the controlled Picard’s method the iterative
formula (2.34), for Eqs. (2.63), (2.64) by choosing H(t) 5 1 is given by
h ð t α21 α
2
u m11 ðt; h; EÞ; u m ðt; h; EÞ 2 ðt2τÞ D ½u m ðτ; h; Eފðu m ðt; EÞÞ 2ðn11Þ 2 ψ dτ
t
ΓðαÞ 0
ð2:80Þ
Starting with the initial approximation solution u 0 (t, E) 5 E which satisfies
the initial condition (2.64), the successive approximations u m11 (t, h, E) for
n 52 3is
1:9 2
u 1 ðt; h; EÞ 5 E 1 0:5472ht ψ ; ð2:81Þ
u 2 ðt; h; EÞ 5 1:0945ht ψ 2 0:5472h t E ψ 2 0:01292h t ψ 6
2 1:9 2 2
4 5:7
1:9 2
ð2:82Þ
3 3:8 4
1 Eð1 2 0:11212h t ψ Þ;
and for n 52 2, the first two solutions of Eq. (2.80) are given by
1:9 2
u 1 ðt; h; EÞ 5 E 1 0:5472 ht ψ ; ð2:83Þ
4
2 1:9 2
1:9 2
3 3:8
u 2 ðt; h; EÞ 5 1:0944ht ψ 2 0:05606h t ψ 1 Eð1 2 0:54724h t ψ Þ;
ð2:84Þ
and similarly for other iterations. With the help of the remaining boundary
conditions (2.65), then the relation between the parameters h and E for the
two methods is given by
uð1Þ U M ð1; h; E; nÞ 5 1; uð1Þ u m11 ð1; h; E; nÞ 5 1: ð2:85Þ

It has been shown in Magyari (2008) and Abbasbandy et al. (2009) that
when n 52 2, the model (2.61) admits dual solutions for 0 , ψ #
0.765152. To show that the number of solutions is continuous for the model
in the fractional order domain the value of ψ variation about 0.7 is selected.
Fig. 2.7A and B shows the h 2 E curve of Eq. (2.85), where two values of E
are obtained which reflect the dual solutions in this case. It is clear from
Fig. 2.7B the two line segment where E constant ended when ψ 5 0.74, this