Page 51 - Mathematical Techniques of Fractional Order Systems
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Nonlinear Fractional Order Boundary-Value Problems Chapter | 2 41
m r 1 r 2 r σ22
σ
m
5. D ðϕ Þ 5 P u m2r 1 P u r 2 2r 1 P u r 2 2r 3 ... P u r σ22 2r σ21 r σ21 ;
u
r 1 50 r 2 50 r 3 50 r σ21 50
where m $ 2 and σ $ 2 are positive integer numbers. The above properties
are proved by Liao (2009). To illustrate the procedures of the HAM, let N [u
(t)] 5 0 denote a nonlinear differential equation. The HAM is based on a
kind of continuous mapping u(t)-ϕ(t, p), constructed from the so-called the
zero-order deformation equation as follows
ð1 2 pÞL½ϕðt; pÞ 2 u 0 ðtÞ 5 HðtÞh pN½ϕðt; pÞ; ð2:9Þ
where L is an auxiliary linear operator, u 0 (t) is an initial guess, H(t) is the
auxiliary function, and h is the nonzero control parameter. When p 5 0, the
zero-order deformation Eq. (2.9) becomes:
L½ϕðt; 0Þ 2 u 0 ðtÞ 5 0; ð2:10Þ
which gives ϕ(t,0) 5 u 0 (t). And when p 5 1, since H(t)6¼0 and h6¼0
Eq. (2.9) is equivalent to:
N ϕðt; 1Þ 5 0; ð2:11Þ
½
which is exactly the same as the original equation N [u(t)] 5 0, provided ϕ(t,
1) 5 u(t). So, as the homotopy parameter p increases from 0 to 1, the solution
ϕ(t, p) of the zero-order deformation Eq. (2.9) varies from the initial guess u 0 (t)
to the exact solution u(t) of the nonlinear differential equation N [u(t)]5 0. By
using Taylor theorem, ϕ(t, p) expands in a power series of p as follows
N
X
m
ϕðt; pÞ 5 ϕðt; 0Þ 1 u m ðtÞp ; ð2:12Þ
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m
where u m (t) 5 D ϕ(t, p). From ϕ(t,0) 5 u 0 (t) then Eq. (2.12) becomes:
N
X m
ϕðt; pÞ 5 u 0 ðtÞ 1 u m ðtÞp : ð2:13Þ
m51
When choosing the linear operator, the initial guess, the auxiliary func-
tion, the control parameter, and from ϕ(t,1) 5 u(t) then the power series
ϕ(t, p) (2.13) converges at p 5 1, and the so-called homotopy series solution
is given by:
N
X
uðtÞ 5 u 0 ðtÞ 1 u m ðtÞ: ð2:14Þ
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which satisfies the original equation N [u(t)] 5 0, as proved by Liao (2009).
The terms u m (t) are unknown functions and determined by the so-called
high-order deformation equations. Differentiating the zero-order deformation
Eq. (2.9) m times with respective to the homotopy parameter p and then
dividing it by m! and finally setting p 5 0, then the term u m (t) is given by
m
21
u m ðtÞ 5 χ u m21 1 hL fHðtÞD ðpN½ϕðt; pÞÞg; ð2:15Þ
m
