Page 52 - Mathematical Techniques of Fractional Order Systems

42 Mathematical Techniques of Fractional Order Systems

where χ m is given by

0; m # 1
χ 5 ð2:16Þ
m 1; m . 1:
The HAM has been successfully employed to solve many types of nonlin-
ear equations in science and engineering such as time-dependent
Emden Fowler type equations (Bataineh et al., 2007), the modified KdV-
type equations (Liu and LiZ, 2009), the Boussinesq problem (Hassan and
El-Tawil, 2011b), the Fitzhugh Nagumo equation (Gorder, 2012), and the
Sturm Liouville problems (Abbasbandy and Shirzadi, 2011). Liao (2003)
proved that the HAM logically contains Adomian decomposition method
(ADM) in general. The homotopy perturbation method (HPM) is a special
case from the HAM when setting H(t) 5 1 and h 52 1(Hayat and Sajid,
2007; Sajid et al., 2007). Motsa et al. (2010) suggested modifying the HAM,
the so-called spectral HAM (SHAM). This modified method is based on
solving the higher order deformation equations by using the Chebyshev pseu-
dospectral method and the auxiliary linear operator L is defined in terms of
the Chebyshev spectral collocation differentiation matrix described by Don
and Solomonoff (Don and Solomonoff, 1995). Semary and Hassan (2016)
introduced the HAM to solve linear and nonlinear differential equations in q
calculus. Abbasbandy and Shivanian (2011) suggested the predictor HAM
(PHAM) to predict the multiplicity of solutions of nonlinear differential
equations. And used it for finding the multiple solutions of nonlinear frac-
tional differential equation with boundary conditions (Alomari et al., 2013).
The core point in the PHAM for solving fractional differential Eqs. (2.1, 2.2)
is to divide the boundary condition (2.2) to equivalent boundary conditions
with unknown parameter E as follows

@u
ℬ u; E; 5 0; uðaÞ 5 b; ð2:17Þ
0
@n
where u(a) 5 b is the forcing condition that arises from original boundary
conditions (2.2). By applying the HAM on the problem (2.1) and with the
condition ℬ u; E; @u 5 0. Then the general zero-order deformation equation

0
@n
and the corresponding boundary conditions as follows:
α
ð1 2 pÞL½ϕðt; E; pÞ 2 u 0 ðt; Eފ 5 HðtÞhpfD ϕðt; E; pÞ 1 fðt; ϕðt; E; pÞÞg; ð2:18Þ
t

@ϕðt; E; pÞ
ℬ ϕðt; E; pÞ; E; 5 0; ð2:19Þ
0
@n
where u 0 (t, E) is initial approximation guess of the exact solution u(t, E)
@u
which satisfies the boundary conditions ℬ u; E; 5 0. The general zero-
0
@n
order deformation Eq. (2.18) satisfies
ϕðt; E; 0Þ 5 u 0 ðt; EÞ; ϕðt; E; 1Þ 5 uðt; EÞ; ð2:20Þ