Page 55 - Mathematical Techniques of Fractional Order Systems

Nonlinear Fractional Order Boundary-Value Problems Chapter | 2 45

Using the property (2.4) of Caputo fractional order derivative, then
Eq. (2.32) becomes:

n21 k ð t
X t 1 α21 α
u m11 ðtÞ 5 d k 2 ðt2τÞ HðτÞhD u m ðτÞ 1 f τ; u m ðτފ dτ
½
τ
k! ΓðαÞ
k50 0
n21 k
X t
1 u m ðtÞ 2 u ðkÞ :
m k!
k50
ð2:33Þ
The successive approximation u m (t) must satisfy the initial conditions, for
that u ðkÞ 5 d k and the iterative formula (2.33) becomes:
m
h ð t α21 α
u m11 ðtÞ 5 u m ðtÞ 2 ðt2τÞ HðτÞ D u m ðτÞ 1 f τ; u m ðτފ dτ: ð2:34Þ
½
τ
ΓðαÞ 0
The important property (2.4) for Caputo fractional order derivative is
right for the integer order case (α 5 n) and it is proved easily by using inte-
gration by parts. So, it should be emphasized that the iteration formula
(2.34) is suitable to solve the problem (2.1) for integer and fractional orders.
n21
t
Starting by an initial approximation u 0 ðtÞ 5 P d k k! k or which satisfies at least
k50
the initial conditions for the problem. Therefore, the approximate solution
u m11 (t) for the nonlinear differential Eq. (2.1) can be obtained with the
@u
conditions ℬ u; E; @n 5 0 after being converted to initial conditions as
0
(k)
u (0) 5 d k , k 5 0, 1, ..., n 2 1. The VIM is one of the famous semianalytics
techniques used to solve nonlinear differential equations (Semary and
Hassan, 2015; He, 1999; Wazwaz, 2009). To solve the integer order differen-
tial Eq. (2.1) by the VIM (He, 1999; Wazwaz, 2009), one can construct an
iteration formula for the problem (2.1) when α 5 n as follows:
ð t
n
u m11 ðtÞ 5 u m ðtÞ 1 λðτÞfD u m ðτÞ 1 f½τ; u m ðτފgdτ; ð2:35Þ
τ
0
n
where D u m ðτÞ 5 u ðτÞ and λ(τ) is a general Lagrange multiplier and it is
ðnÞ
τ
m
equal to λ(τ) 52(t 2 τ) (n21) / n 2 1! (Wazwaz, 2009).
Remark: The Controlled Picard iterative formula (2.34) leads to a varia-
tional iterative formula generated by the variational iteration method
(2.35) when h 5 H(t) 5 1, α 5 n,and the general Lagrange multiplier
(n21)
λ(τ) 52(t 2 τ) /Γ(n).

2.2.3 The Prediction of Solutions Multiplicity
The approximate solutions (2.27) and (2.34) outputs of PHAM and con-
trolled Picard’s method still have two unknown parameters, namely h and E,