Page 54 - Mathematical Techniques of Fractional Order Systems

44 Mathematical Techniques of Fractional Order Systems

M
X
uðt; EÞ U M ðt; h; EÞ 5 u 0 ðt; EÞ 1 u m ðt; EÞ: ð2:27Þ
m51

2.2.2 Controlled Picard’s Method
In 1890, Emile Picard introduced a basic tool for proving the existence of solu-
tions of initial value problems regarding ordinary first order differential equa-
tions namely, the Picard’s method. Recently, this method was used to solve
and analyze the differential and integral equations with different definitions of
the derivative (Azarnavid et al., 2015; El-Sayed et al., 2014; Salahshour et al.,
2015; Vazquez-Leal et al., 2015). Semary et al. (2017a,b) suggested Picard
iterative formula to solve the nonlinear fractional order differential equa-
tions. The advantage for this formula is that it contains an auxiliary
parameter h which proves very effective in controlling the convergence
region of an approximate solution. To drive controlled Picard’s iterative
formula for Eq. (2.1), multiply h and H(t) to its both sides to become in
the following form:
α
F½t; uðtÞ; αފ 5 HðtÞhD uðtÞ 1 fðt; uðtÞÞ 5 0: ð2:28Þ
t
α
Adding and subtracting D uðtÞ from the Eq. (2.28) to become in the form:
t
α
a
D uðtÞ 1 F½t; uðtÞ; አ2 D uðtÞ 5 0: ð2:29Þ
t t
α
Applying the Riemann Liouville integral of order α (J ) on the problem
(2.29) and after using of the property (2.4), the integrated form of Eq. (2.29)
can be written as follows:
n21 k
X t α α
uðtÞ 5 u ð0Þ 2 J Ft; uðtÞ; አ2 D uðtÞ : ð2:30Þ
ðkÞ
½
k! t
k50
Applying Picard method to the integral Eq. (2.30), the solution can be
reconstructed as follows:
n21 k ð t
α
X t 1
u m11 ðtÞ 5 d k 2 ðt2τÞ α21 fF τ; u m ðτÞ; አ2 D u m ðτÞgdτ;
½
k! ΓðαÞ τ
k50 0
ð2:31Þ
(k)
where d k 5 u (0), k 5 0, 1, ..., n 2 1 and m $ 0. From Eq. (2.28), the iter-
ative formula (2.31) becomes:
n21 k ð t
X t 1 α21 α α α
u m11 ðtÞ 5 d k 2 ðt2τÞ HðτÞhD u m ðτÞ 1 f τ; u m ðτފ dτ 1 J D u m ðtÞ;
½
τ
t
k! ΓðαÞ 0
k50
ð2:32Þ