Page 60 - Mathematical Techniques of Fractional Order Systems

50 Mathematical Techniques of Fractional Order Systems

ð 1
uð1Þ 5 0; uðtÞdt 5 1: ð2:54Þ
0
By applying the iterative formula (2.34) in Eq. (2.40) with initial condi-
tions (2.53) and by choosing H(t) 5 1 then
( 2 )
t
ð
h α21 α E du m ðτ; h; EÞ
u m11 ðt; h; EÞ 5 u m ðt; h; EÞ 2 ðt2τÞ D u m ðτ; h; EÞ 2 dτ dτ:
τ
ΓðαÞ 0 16
ð2:55Þ
According to the conditions (2.53), the initial approximation solution
u 0 (t, E) can be written as
1
2
u 0 ðt; EÞ ðEt 1 2γÞ: ð2:56Þ
2
Starting with the initial approximation solution (2.56), the solution of the
successive approximations (2.55), u m11 (t, h,E), m $ 0 is as follows

t E Eht 21α 2
E
2
u 1 ðt; h; EÞ 5 γ 1 1 ð2:57Þ
2
2 8αð2 1 3α 1 α ÞΓðαÞ
2
t E Eht 21α 2
E
u 2 ðt; h; EÞ 5 γ 1 1 2
2
2 8αð2 1 3α 1 α ÞΓðαÞ
0 1
α 2 2 2α 2
64ð2 1 1 hÞ 8Eht ð2 1 αÞEΓðαÞ 3E h t E Γð2 1 2αÞ
E
Eht 21α 2@ 2 2 2 2 A
αð2 1 3α 1 α Þ Γð3 1 2αÞ a ΓðαÞΓð4 1 3αÞ
;
512ΓðαÞ
ð2:58Þ
and so on. With the help of the remaining two boundary conditions
(2.54), then
ð 1 ð 1
uðtÞdt u m11 ðt; h; EÞdt 5 1; ð2:59Þ
0 0
uð1Þ u m11 ð1; h; EÞ 5 0; ð2:60Þ
By using Eq. (2.60) to delete the unknown parameter γ of the Eq. (2.59)
so that it contains only two unknown parameters E and h. When α 5 3.9, and
according to the Eq. (2.59), the unknown parameter E is a function of the
auxiliary parameter h, which has been plotted in the h-range [0, 2.5] in
Fig. 2.3 for different values of E and m 5 6. Two E-plateaus (two line seg-
ments give constant values of E) can be identified in this Figure for each
case; this means that there are two solutions. From this Figure it is clear that
the valid value of h for the two solutions is one.