Page 66 - Mathematical Techniques of Fractional Order Systems

Nonlinear Fractional Order Boundary-Value Problems Chapter | 2 55

2
@ ψðt; E; pÞ
L ϕðt; E; pފ 5 ; ð2:66Þ
½
@t 2
with the property
L½c 0 1 c 1 tŠ 5 0: ð2:67Þ
By choosing H(t) 5 1 and using Eq. (2.25) the solution terms u m is
given as
ðð
m 1:9 2ðn11Þ 2
u m ðt; EÞ 5 χ u m21 ðt; EÞ 1 h D pD ½ϕðt; E; pފðϕðt; E; pÞÞ 2 ψ dtdt
m t
1 c 0 1 c 1 t;
ð2:68Þ
where the constants c 0 and c 1 can be determined by the two boundary
conditions:
0
u ð0; EÞ 5 u m ð0; EÞ 5 0: ð2:69Þ
m
Using the properties of the homotopy derivative (2.7) the mth-order
deformation Eq. (2.68) in simple form becomes
ðð
u m ðt; EÞ 5 χ u m21 ðt; EÞ 1 h R m;n ðu m21 ; t; EÞdtdt 1 c 0 1 c 1 t; ð2:70Þ
m
where
m21 i
X X 1:9 2
R m;23 5 D ½u m212i ðt; Eފu i2k ðt; EÞu i ðt; EÞ 2 ψ ð1 2 χ Þ; ð2:71Þ
t m
i50 k50
and
m21
X 1:9 2
R m;22 5 D ½u m212i ðt; Eފu i ðt; EÞ 2 ψ ð1 2 χ Þ: ð2:72Þ
t m
i50
term of solutions for Eq. (2.70) when n 52 3 are
1 2 2
u 1 ðt; EÞ 52 ht ψ ; ð2:73Þ
2
2:1 2
2
2
u 2 ðt; EÞ 52 0:455hð1:099t 1 ht E Þψ ; ð2:74Þ
2:1 2
2
2 4:1
2
2 2:2 4
2
u 3 ðt; EÞ 5 0:0827hψ 2 ð2 6:0458t 2 11:0043ht E 2 4:9883h t E 1 h t Eψ Þ;
ð2:75Þ
and so on. Also, when n 52 2, the solution of Eq. (2.70) are given by
1
2 2
u 1 ðt; EÞ 52 ht ψ ; ð2:76Þ
2