Page 71 - Mathematical Techniques of Fractional Order Systems

60 Mathematical Techniques of Fractional Order Systems

methods. The purpose of this section is to solve and to show how one can
find out existence of dual solutions for the problem (2.87) in the fractional
order domain. The one-dimensional fractional order Bratu equation is
given by
α
D uðtÞ 1 λe uðtÞ 5 0; ð2:90Þ
t
where 1 , α # 2. To apply the Controlled Picard’s method, suppose that
u (0) 5 E, so the boundary conditions (2.88) become:
0
0
uð0Þ 5 0; u ð0Þ 5 E; ð2:91Þ
with additional forcing condition
uð1Þ 5 0: ð2:92Þ

By applying the iterative formula for Eq. (2.90) with initial conditions
(2.91) and by taking H(t) 5 1, then
h ð t
α
u m11 ðt; h; EÞ 5 u m ðt; h; EÞ 2 ðt2τÞ α21 D ½u m ðτ; h; Eފ 1 λe u m ðτ;h;EÞ dτ
τ
ΓðαÞ 0
ð2:93Þ
and using the property (2.4) then the Eq. (2.93) becomes
h ð t α21
u m11 ðt; h; EÞ 5 ð1 2 hÞu m ðt; h; EÞ 1 hEt 2 ðt2τÞ λe u m ðτ;h;EÞ dτ ð2:94Þ
ΓðαÞ 0
Ð t α21
The term ðt2τÞ λe u m ðτ;h;EÞ dτ in Eq. (2.94) is difficult to obtain the
0
exact value. Therefore, the value of this term is approximated by using com-
posite Simpson method (Atkinson, 1989). This method is as follows:
( )
l21
l
ð t t X X
IðtÞ 5 zðτÞdτ y 0 1 y 2l 1 4y 2i21 1 2y 2i ð2:95Þ
0 6l i21 i51

it
where 2l is the number of subintervals, z 0 5 z(0), z 2l 5 z(t), and z i 5 z i:1
2l
(1)2l 2 1. Then the iterative formula (2.94) to solve the problem (2.90) and
(2.91) becomes:
( )
l l21
th X X
u m11 ðt; hÞ 5 ð1 2 hÞu m ðt; h; EÞ 1 hEt 2 z 0 1 z 2l 1 4z 2i11 1 2z 2i ;
6l
i21 i51
ð2:96Þ

α21
where ðt2τÞ λe u m ðt;h;EÞ it ; ’i:0ð1Þ2l:
ΓðαÞ τ5
2l
By selecting l 5 5 and using the initial solution
α α
u 0 ðt; EÞ 5 Et 2 λt 2 λEt 1 1 which at least satisfies the initial conditions
Γðα 1 1Þ Γðα 1 2Þ