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Abstractโ In this paper, we present an algorithm of the
homotopy analysis method (HAM) to obtain symbolic
approximate solutions for linear and nonlinear differential
equations of fractional order. We show that the HAM is
different from all analytical methods; it provides us with a
simple way to adjust and control the convergence region of the
series solution by introducing the auxiliary parameter โ, the
auxiliary function ๐ฏ ๐ , the initial guess ๐๐(๐) and the auxiliary
linear operator . Three examples, the fractional oscillation
equation, the fractional Riccati equation and the fractional
Lane-Emden equation, are tested using the modified algorithm.
The obtained results show that the Adomain decomposition
method, Variational iteration method and homotopy
perturbation method are special cases of homotopy analysis
method. The modified algorithm can be widely implemented to
solve both ordinary and partial differential equations of
fractional order.
Index TermsโAdomian decomposition method, Caputo
derivative, Fractional Lane-Emden equation, Fractional
oscillation equation, Fractional Riccati equation, Homotopy
analysis method.
I. INTRODUCTION
Fractional differential equations have gained importance
and popularity during the past three decades or so, mainly due
to its demonstrated applications in numerous seemingly
diverse fields of science and engineering. For example, the
nonlinear oscillation of earthquake can be modeled with
fractional derivatives, and the fluid-dynamic traffic model
with fractional derivatives can eliminate the deficiency
arising from the assumption of continuum traffic flow. The
differential equations with fractional order have recently
proved to be valuable tools to the modeling of many physical
phenomena [1-8]. This is because of the fact that the realistic
modeling of a physical phenomenon does not depend only on
the instant time, but also on the history of the previous time
which can also be successfully achieved by using fractional
1Department of Mathematics, Zarqa Private University, Zarqa 1311,
Jordan, (email: ajou41@yahoo.com) 2Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa'
Applied University, Salt 19117, Jordan, (email: odibat@bau.edu.jo) 3Department of Mathematics, Mutah University, P. O. Box 7, Al-Karak,
Jordan, (email: shahermm@yahoo.com) 4Department of Mathematics, University of Jordan, Amman 11942,
Jordan, (email: alawneh@ju.edu.jo) *Corresponding author. Email addresses: z.odibat@gmail.com;
odibat@bau.edu.jo, Tel. +962-5-3532519, Fax. +962-5-3530462.
calculus. Most nonlinear fractional equations do not have
exact analytic solutions, so approximation and numerical
techniques must be used. The Adomain decomposition
method (ADM) [9-14], the homotopy perturbation method
(HPM) [15-25], the variational iteration method (VIM)
[26-28] and other methods have been used to provide
analytical approximation to linear and nonlinear problems.
However, the convergence region of the corresponding
results is rather small as shown in [9-28]. The homotopy
analysis method (HAM) is proposed first by Liao [29-33] for
solving linear and nonlinear differential and integral
equations. Different from perturbation techniques; the HAM
doesn't depend upon any small or large parameter. This
method has been successfully applied to solve many types of
nonlinear differential equations, such as projectile motion
with the quadratic resistance law [34], Klein-Gordon
equation [35], solitary waves with discontinuity [36], the
generalized Hirota-Satsuma coupled KdV equation [37], heat
radiation equations [38], MHD flows of an Oldroyd
8-constant fluid [39], Vakhnenko equation [40], unsteady
boundary-layer flows [41]. Recently, Song and Zhang [42]
used the HAM to solve fractional KdV-Burgers-Kuramoto
equation, Cang and his co-authors [43] constructed a series
solution of non-linear Riccati differential equations with
fractional order using HAM. They proved that the Adomian
decomposition method is a special case of HAM, and we can
adjust and control the convergence region of solution series
by choosing the auxiliary parameter โ closed to zero.
The objective of the present paper is to modify the HAM to
provide symbolic approximate solutions for linear and
nonlinear differential equations of fractional order. Our
modification is implemented on the fractional oscillation
equation, the fractional Riccati equation and the fractional
Lane-Emden equation. By choosing suitable values of the
auxiliary parameter โ, the auxiliary function ๐ป ๐ก , the initial
guess ๐ฆ0(๐ก) and the auxiliary linear operator we can
adjust and control the convergence region of solution series.
Moreover, we illustrated for several examples that the
Adomain decomposition, Variational iteration and homotopy
perturbation solutions are special cases of homotopy analysis
solution.
II. DEFINITIONS
For the concept of fractional derivative we will adopt
Caputoโs definition [7] which is a modification of the
Riemann-Liouville definition and has the advantage of
dealing properly with initial value problems in which the
Construction of Analytical Solutions to
Fractional Differential Equations Using
Homotopy Analysis Method
Ahmad El-Ajou1, Zaid Odibat
*2, Shaher Momani
3, Ahmad Alawneh
4
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
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initial conditions are given in terms of the field variables and
their integer order which is the case in most physical
processes [4].
Definition 1. A real function ๐(๐ฅ), ๐ฅ > 0, is said to be in the
space ๐ถ๐ , ๐ โ โ if there exists a real number ๐ , such
that ๐ ๐ฅ = ๐ฅ๐๐1 ๐ฅ , where ๐1 ๐ฅ โ ๐ถ 0, โ and it is said
to be in the space ๐ถ๐๐ iff ๐ ๐ ๐ฅ โ ๐ถ๐ , ๐ โ โ.
Definition 2. The Riemann-Liouville fractional integral
operator of order ๐ผ > 0 , of a function ๐ ๐ฅ โ ๐ถ๐ , ๐ โฅ โ1,
is defined as:
๐ฝ๐ผ๐ ๐ฅ =1
๐ค ๐ผ ๐ฅ โ ๐ก ๐ผโ1
๐ฅ
0
๐ ๐ก ๐๐ก ,
๐ผ > 0 , ๐ฅ > 0 ,
(1)
๐ฝ0๐ ๐ฅ = ๐ ๐ฅ . (2)
Properties of the operator ๐ฝ๐ผ can be found in [5-8], we
mention only the following:
For ๐ โ ๐ถ๐ , ๐ โฅ โ1, ๐ผ, ๐ฝ โฅ 0 and ๐พ โฅ โ1
๐ฝ๐ผ๐ฝ๐ฝ๐ ๐ฅ = ๐ฝ๐ผ+๐ฝ๐ ๐ฅ = ๐ฝ๐ฝ ๐ฝ๐ผ๐(๐ฅ) , (3)
๐ฝ๐ผ๐ฅ๐พ =๐ค(๐พ + 1)
๐ค(๐ผ + ๐พ + 1)๐ฅ๐ผ+๐พ . (4)
Definition 3. The fractional derivative of ๐(๐ฅ) in the
Caputo sense is defined as:
๐ทโ๐ผ๐ ๐ฅ = ๐ฝ๐โ๐ผ๐ท๐๐ ๐ฅ
=1
๐ค ๐ โ ๐ผ ๐ฅ โ ๐ก ๐ฅโ๐ผโ1
๐ฅ
0
๐(๐) ๐ก ๐๐ก , (5)
for ๐ โ 1 < ๐ผ โค ๐ , ๐ โ โ , ๐ฅ > 0 , ๐ โ ๐ถโ1๐ .
Lemma 1. If ๐ โ 1 < ๐ผ โค ๐, ๐ โ โ and ๐ โ ๐ถ๐๐ , ๐ โฅ โ1,
then
๐ทโ๐ผ ๐ฝ๐ผ๐ ๐ฅ = ๐ ๐ฅ , (6)
๐ฝ๐ผ๐ทโ๐ผ๐ ๐ฅ = ๐ ๐ฅ โ ๐ ๐ 0+
๐ฅ
๐! , ๐ฅ > 0
๐โ1
๐=0
. (7)
III. HOMOTOPY ANALYSIS METHOD
The principles of the HAM and its applicability for various
kinds of differential equations are given in [29-43]. For
convenience of the reader, we will present a review of the
HAM [29, 38]. To achieve our goal, we consider the
nonlinear differential equation.
๐ ๐ฆ ๐ก = 0 , ๐ก โฅ 0 , (8)
where N is a nonlinear differential operator, and ๐ฆ(๐ก) is
unknown function of the independent variable ๐ก.
A. Zeroth- order Deformation Equation
Liao [29] constructs the so-called zeroth-order
deformation equation:
1 โ ๐ ๐ ๐ก; ๐ โ ๐ฆ0 ๐ก = ๐โ๐ป ๐ก ๐ ๐ ๐ก; ๐ , (9)
where ๐ โ 0,1 is an embedding parameter, โ โ 0 is an
auxiliary parameter, ๐ป ๐ก โ 0 is an auxiliary function, is
an auxiliary linear operator, ๐ is nonlinear differential
operator, ๐ ๐ก; ๐ is an unknown function, and ๐ฆ0 ๐ก is an
initial guess of ๐ฆ(๐ก), which satisfies the initial conditions. It
should be emphasized that one has great freedom to choose
the initial guess ๐ฆ0 ๐ก , the auxiliary linear operator , the
auxiliary parameter โ and the auxiliary function ๐ป ๐ก .
According to the auxiliary linear operator and the suitable
initial conditions, when ๐ = 0, we have
๐ ๐ก; 0 = ๐ฆ0 ๐ก , (10)
and when ๐ = 1 , since โ โ 0 and ๐ป ๐ก โ 0 , the
zeroth-order deformation equation (9) is equivalent to (8),
hence
๐ ๐ก; 1 = ๐ฆ ๐ก . (11)
Thus, as ๐ increasing from 0 to 1 , the solution ๐ ๐ก; ๐ various from ๐ฆ0 ๐ก to ๐ฆ(๐ก). Define
๐ฆ๐ ๐ก = 1
๐!
๐๐๐ ๐ก; ๐
๐๐๐ ๐=0
. (12)
Expanding ๐ ๐ก; ๐ in a Taylor series with respect to the
embedding parameter ๐, by using (10) and (12), we have:
๐ ๐ก; ๐ = ๐ฆ0 ๐ก + ๐ฆ๐ ๐ก ๐๐
โ
๐=1
. (13)
Assume that the auxiliary parameter โ , the auxiliary
function ๐ป ๐ก , the initial approximation ๐ฆ0(๐ก) and the
auxiliary linear operator are properly chosen so that the
series (13) converges at ๐ = 1. Then at ๐ = 1, from (11), the
series solution (13) becomes
๐ฆ ๐ก = ๐ฆ0 ๐ก + ๐ฆ๐ ๐ก
โ
๐=1
. (14)
B. High-order Deformation Equation
Define the vector:
๐ฆ ๐ = ๐ฆ0 ๐ก , ๐ฆ1 ๐ก , ๐ฆ2 ๐ก , โฆ , ๐ฆ๐(๐ก) . (15)
Differentiating equation (9) m-times with respect to
embedding parameter ๐ , then setting ๐ = 0 and dividing
them by ๐! , we have, using (12), the so-called mth-order
deformation equation
๐ฆ๐ ๐ก โ ๐๐๐ฆ๐โ1 ๐ก = โ๐ป ๐ก ๐ ๐ ๐ฆ ๐โ1 ๐ก ,
๐ = 1,2, โฆ , ๐ , (16)
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
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where
๐ ๐ ๐ฆ ๐โ1 = 1
(๐ โ 1)!
๐๐โ1๐[๐ ๐ก; ๐ ]
๐๐๐โ1 ๐=0
, (17)
and
๐๐ = 0 , ๐ โค 1
1 , ๐ > 1 . (18)
The so-called mth-order deformation equation (16) is a linear
which can be easily solved using Mathematica package.
IV. AN ALGORITHM OF HOMOTOPY ANALYSIS METHOD
The HAM has been extended in [42,43] to solve some
fractional differential equations. They use the auxiliary linear
operator to be ๐/๐๐ก or ๐ทโ๐ผ , where 0 < ๐ผ โค 1. Also they
fix the auxiliary function ๐ป ๐ก to be 1. In this section, we
present an efficient algorithm of the HAM. This algorithm
can be established based on the assumptions that the
nonlinear operator can involve fractional derivatives, the
auxiliary function ๐ป ๐ก can be freely selected and the
auxiliary linear operator can be considered as = ๐ทโ๐ฝ
, ๐ฝ > 0 . To illustrate its basic ideas, we consider the following
fractional initial value problem
๐๐ผ ๐ฆ ๐ก = 0 , ๐ก โฅ 0 , (19)
where ๐๐ผ is a nonlinear differential operator that may
involves fractional derivatives, where the highest order
derivative is ๐ , subject to the initial conditions
๐ฆ ๐ 0 = ๐๐ , ๐ = 0,1,2, โฆ , ๐ โ 1 . (20)
The so-called zeroth-order deformation equation can be
defined as
1 โ ๐ ๐ทโ๐ฝ ๐ ๐ก; ๐ โ ๐ฆ0 ๐ก = ๐โ๐ป ๐ก ๐ ๐ ๐ก; ๐ ,
๐ฝ > 0 , (21)
subject to the initial conditions
๐ ๐ 0; ๐ = ๐๐ , ๐ = 0,1,2, โฆ , ๐ โ 1 . (22)
Obviously, when ๐ = 0 , since ๐ฆ0 ๐ก satisfies the initial
conditions (20) and = ๐ทโ๐ฝ
, ๐ฝ > 0, we have
๐ ๐ก; 0 = ๐ฆ0 ๐ก , (23)
and, the so-called mth-order deformation equation can be
constructed as
๐ทโ๐ฝ ๐ฆ๐ ๐ก โ ๐๐๐ฆ๐โ1 ๐ก = โ๐ป ๐ก ๐ ๐ ๐ฆ ๐โ1 ๐ก ,
๐ฝ > 0 , (24)
subject to the initial conditions
๐ฆ๐(๐) 0 = 0 , ๐ = 0,1,2, โฆ , ๐ โ 1. (25)
Operating ๐ฝ๐ฝ , ๐ฝ > 0 on both sides of (24) gives the
mth-order deformation equation in the form:
๐ฆ๐ ๐ก = ๐๐๐ฆ๐โ1 ๐ก + โ ๐ฝ๐ฝ ๐ป ๐ก ๐ ๐ ๐ฆ ๐โ1 ๐ก . (26)
V. EXAMPLES
In this section we employ our algorithm of the homotopy
analysis method to find out series solutions for some
fractional initial value problems.
Example 1. Consider the composite fractional oscillation
equation [18]
๐2๐ฆ
๐๐ก2โ ๐๐ทโ
๐ผ๐ฆ ๐ก โ ๐๐ฆ ๐ก โ 8 = 0 , ๐ก โฅ 0 ,
๐ โ 1 < ๐ผ โค ๐ , ๐ = 1,2 ,
(27)
subject to the initial conditions
๐ฆ 0 = 0 , ๐ฆโฒ 0 = 0 . (28)
First, if we set ๐ผ = 1 , ๐ = ๐ = โ1 , then the equation
(27) becomes linear differential equation of second order
which has the following exact solution
๐ฆ ๐ก = 8 โ 8๐โ๐ก/2 cos 3
2๐ก +
1
3sin
3
2๐ก . (29)
Since, ๐๐ผ ๐ฆ ๐ก =๐2๐ฆ
๐๐ก2 โ ๐๐ทโ๐ผ๐ฆ ๐ก โ ๐๐ฆ ๐ก โ 8 , according
to (12) and (17), we have
๐ ๐ ๐ฆ ๐โ1 = ๐ฆ๐โ1โฒโฒ ๐ก โ ๐๐ทโ
๐ผ๐ฆ๐โ1 ๐ก โ ๐๐ฆ๐โ1 ๐ก
โ8 1 โ ๐๐ . (30)
If we take the auxiliary function ๐ป ๐ก = 1 , and the
parameter ๐ฝ = 2, then the auxiliary linear operator becomes
= ๐2 ๐๐ก2 , (31)
and
๐ฆ๐ ๐ก = ๐๐๐ฆ๐โ1 ๐ก + โ ๐ฝ2 ๐ ๐ ๐ฆ ๐โ1 ๐ก , (32)
subject to the initial conditions
๐ฆ๐ 0 = ๐ฆ๐โฒ 0 = 0 . (33)
(I) If we choose the initial guess approximation
๐ฆ0 ๐ก = 0 , (34)
then we have
๐ฆ1 = โ4โ๐ก2 , (35)
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
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๐ฆ2 = โ4โ 1 + โ ๐ก2 +โ2๐
3๐ก4 +
8โ2๐
๐ค(5 โ ๐ผ)๐ก4โ๐ผ ,
๐ฆ3 = โ4โ 1 + โ 2๐ก2 +2โ2 1 + โ ๐
3๐ก4 โ
โ3๐2
90๐ก6
+16โ2 1 + โ ๐
๐ค 5 โ ๐ผ ๐ก4โ๐ผ โ
16โ3๐๐
๐ค(7 โ ๐ผ)๐ก6โ๐ผ
โ8โ3๐2
๐ค(7 โ 2๐ผ)๐ก6โ2๐ผ .
Moreover, if we set the auxiliary parameter โ = โ1, then the
result is the Variational iteration solution obtained by
Momani and Odibat in [18] and the same homotopy
perturbation solution obtained by Odibat in [21].
(II) If we take the initial guess
๐ฆ0 = 4๐ก2 , (36)
then we will get the approximations
๐ฆ1 = โโ๐
3๐ก4 +
8โ๐
๐ค(5 โ ๐ผ)๐ก4โ๐ผ ,
(37)
๐ฆ2 = โโ 1 + โ ๐
3๐ก4 +
โ2๐2
90๐ก6 โ
8โ 1 + โ ๐
๐ค 5 โ ๐ผ ๐ก4โ๐ผ
+16โ2๐๐
๐ค(7 โ ๐ผ)๐ก6โ๐ผ +
8โ2๐2
๐ค(7 โ 2๐ผ)๐ก6โ2๐ผ ,
๐ฆ3 = โโ 1 + โ 2๐
3๐ก4 +
11โ2 1 + โ ๐2
90๐ก6
โโ3๐3
35 6 4๐ก8 โ
8โ 1 + โ 2๐
๐ค 5 โ ๐ผ ๐ก4โ๐ผ
+32โ2 1 + โ ๐๐
๐ค 7 โ ๐ผ ๐ก6โ๐ผ +
16โ2 1 + โ ๐2
๐ค 7 โ 2๐ผ ๐ก6โ2๐ผ
โ24โ3๐๐2
๐ค 9 โ ๐ผ ๐ก8โ๐ผ โ
24โ3๐2๐
๐ค(9 โ 2๐ผ)๐ก8โ2๐ผ
โ8โ3๐3
๐ค(9 โ 3๐ผ)๐ก8โ3๐ผ .
Here, if we put โ = โ1 in (37), then we will get the Adomian
decomposition solution obtained by Momani and Odibat in
[18].
(III) If we take the initial guess
๐ฆ0 = 3๐ก2 ,
(38)
then we will get the approximations
๐ฆ1 = โโ๐ก2 โโ๐
4๐ก4 โ
6โ๐
๐ค(5 โ ๐ผ)๐ก4โ๐ผ ,
(39)
๐ฆ2 = โโ๐ก2 โโ๐
4๐ก4 โ
6โ๐
๐ค 5 โ ๐ผ ๐ก4โ๐ผ
+โ2๐ก2โ2๐ผ 360๐2๐ค 7 โ ๐ผ ๐ก4 + โฏ .
(VI) If we replace ๐ป ๐ก = 1 by ๐ป ๐ก = ๐ก1/2 in (III), then
we have
๐ฆ1 = โ8โ
15๐ก
52 โ
4โ๐
21๐ก
92
โ24โ๐
9 โ 2๐ผ (7 โ 2๐ผ)๐ค(3 โ ๐ผ)๐ก
92โ๐ผ ,
๐ฆ2 = โ8โ
15๐ก
52 โ
4โ๐
21๐ก
92
โ24โ๐
9 โ 2๐ผ 7 โ 2๐ผ ๐ค 3 โ ๐ผ ๐ก
92โ๐ผ +
โ2
(4)1+3๐ผ๐ก3โ2๐ผ โฆ .
By means of the so-called โ-curves [29], Fig.1 shows that
the valid region of โ is the horizontal line segment. Thus, the
valid regions of โ for the HAM of Eq. (27) at different values
of ๐ผ are shown in Fig.1.
Fig. 2.(a)~(f) shows the approximate solutions for Eq. (27)
obtained for different values of ๐ผ , โ , ๐ป ๐ก and ๐ฆ0(๐ก) using
HAM. Fig. 2.(a) shows that the best solution results when we
use the initial guess ๐ฆ0 = 3๐ก2. Fig. 2.(b) shows that the best
solution results when we use the auxiliary parameter โ =โ0.7. Fig. 2.(c) shows that the best solution results when we
use the auxiliary function ๐ป ๐ก = 1. In Fig. 2.(d)~(f), we
compare the approximate solutions for different values of โ
and ๐ป ๐ก . As shown in Fig. 2.(a)~(f), we can observe that by
choosing a proper value of the auxiliary parameter โ, the
auxiliary function ๐ป ๐ก , the auxiliary linear operator and
the initial guess ๐ฆ0(๐ก) we can adjust and control convergence
region of the series solutions. Moreover, we can observe that
convergence region increases as โ goes to left end point of
the valid region of โ , but this may decreases the agreement
with the exact solution. Choosing a suitable auxiliary
function ๐ป ๐ก with the left end point of the valid region of โ
scrimps the disagreement with the exact solution. Fig.3. show
the โresidual errorโ for 15th order approximation for Eq. (27)
at ๐ผ = 1 obtained for different values of โ and ๐ป ๐ก .
Fig.1. The โ-curves of ๐ฆโฒ which are corresponding to the15th-order
approximate HAM solution of Eq.(27) when ๐ป ๐ก = ๐ก1/2 ,
๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2. Dotted line: ๐ผ = 0.5, dash.dotted
line: ๐ผ = 1, solid line: ๐ผ = 1.5.
2.5 2.0 1.5 1.0 0.5 0.0h
0.5
1.0
1.5
2.0
2.5
3.0
3.5
y'
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
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(a) ๐ผ = 1 , ๐ป ๐ก = 1 , โ = โ1 , โ = ๐2 ๐๐ก2 .
Dashed line: ๐ฆ0 = 0 , dash.dotted line: ๐ฆ0 = 4๐ก2, dotted line: ๐ฆ0 =3๐ก2, solid line: Exact solution
(b) ๐ผ = 1 , ๐ป ๐ก = 1 , ๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2 .
Dashed line: โ = โ1 , dash.dotted line: โ = โ0.7 , dotted
line: โ = โ0.3 , solid line: Exact solution
(c) ๐ผ = 1 , โ = โ0.7 , ๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2 .
Dashed line: ๐ป ๐ก = ๐ก, dash.dotted line: ๐ป ๐ก = ๐ก1/2, dotted line:
๐ป ๐ก = 1 , solid line: Exact solution.
(d) ๐ผ = 1 , ๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2 .
Dotted line: ๐ป ๐ก = 1 , โ = โ0.7, dash.dotted line: ๐ป ๐ก = ๐ก1/2 ,โ = โ0.3 , solid line: Exact solution.
(e) ๐ผ = 0.5 , ๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2 .
Dotted line: ๐ป ๐ก = 1 , โ = โ0.7, solid line: ๐ป ๐ก = ๐ก1/2 , โ = โ0.3 .
(f) ๐ผ = 1.5 , ๐ฆ0 = 3๐ก2 , โ = ๐2 ๐๐ก2 .
Dotted line: ๐ป ๐ก = 1 , โ = โ0.7, solid line : ๐ป ๐ก = ๐ก1/2 , โ = โ0.3 .
Fig. 2. Approximate solutions for Eq. (27)
Example 2. Consider the fractional Riccati equation [17]
๐ทโ๐ผ๐ฆ ๐ก + ๐ฆ2 ๐ก โ 1 = 0 , 0 < ๐ผ โค 1 , ๐ก โฅ 0 , (40)
subject to the initial condition
๐ฆ 0 = 0 . (41)
According to (17), we have
๐ ๐ ๐ฆ ๐โ1 = ๐ทโ๐ผ๐ฆ๐โ1 ๐ก + ๐ฆ๐ ๐ก ๐ฆ๐โ1โ๐ ๐ก
๐โ1
๐=0
โ 1 โ ๐๐ ,
(42)
(I) If we choose the initial guess approximation
๐ฆ0 ๐ก = ๐ก , (43)
0 1 2 3 4 5 6 7t
2
4
6
8
10
12
14
y
2 4 6 8 10 12t
5
10
15
y
0 2 4 6 8 10 12t
2
4
6
8
10
12
14
y
0 2 4 6 8 10 12t
2
4
6
8
10
12
14
y
0 2 4 6 8 10 12t
2
4
6
8
10
12
14
y
0 2 4 6 8 10t
2
4
6
8
10
12
14
y
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
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Fig. 3. The residual error for Eq. (27) when ๐ผ = 1 , ๐ฆ0 = 3๐ก2, โ = ๐2 ๐๐ก2 . Solid line: ๐ป ๐ก = 1 , โ = โ0.7 , dotted
line: ๐ป ๐ก = ๐ก1/2 , โ = โ0.3 .
the parameter ๐ฝ = 1 and ๐ป ๐ก = 1, then according to (26)
we find
๐ฆ1 = โโ ๐ก โ๐ก3
3โ
๐ก2โ๐ผ
๐ค 3 โ ๐ผ
๐ฆ2 = โโ ๐ก โ๐ก3
3โ
๐ก2โ๐ผ
๐ค 3 โ ๐ผ
โโ2 2๐ก3
3โ
2๐ก5
15+
๐ก2โ๐ผ
๐ค 3 โ ๐ผ โ
2๐ก4โ๐ผ
๐ค 4 โ ๐ผ โ
๐ก3โ2๐ผ
๐ค 4 โ 2๐ผ ,
๐ฆ3 = โโ ๐ก โ๐ก3
3โ
๐ก2โ๐ผ
๐ค 3 โ ๐ผ
โโ2 4๐ก3
3โ
4๐ก5
15+
2๐ก2โ๐ผ
๐ค 3 โ ๐ผ โ
2๐ก3โ2๐ผ
๐ค 4 โ 2๐ผ
โ 6
๐ค 3 โ ๐ผ +
6
๐ค 4 โ ๐ผ โ
2๐ค 5 โ ๐ผ
๐ค 4 โ ๐ผ 2
๐ค 4 โ ๐ผ ๐ก4โ๐ผ
๐ค 5 โ ๐ผ
+โ3 ๐ก3
3โ
2๐ก5
5+
17๐ก7
315โ
4๐ก4โ๐ผ
๐ค 4 โ ๐ผ +
2
3๐ค 3 โ ๐ผ +
4
๐ค 4 โ ๐ผ +
16
๐ค 6 โ ๐ผ
๐ค 6 โ ๐ผ ๐ก4โ๐ผ
๐ค 7 โ ๐ผ
+ 1
๐ค 3 โ ๐ผ 2+
2
๐ค 4 โ 2๐ผ +
2๐ค 5 โ ๐ผ
๐ค 4 โ ๐ผ ๐ค 5 โ 2๐ผ
ร๐ค 5 โ 2๐ผ ๐ก5โ2๐ผ
๐ค 6 โ 2๐ผ โ
๐ก3โ2๐ผ
๐ค 4 โ 2๐ผ +
๐ก4โ3๐ผ
๐ค 5 โ 3๐ผ .
Now, if we take โ = โ1 , then we have the homotopy
perturbation solution obtained by Odibat and Momani in
[17].
(II) If we chose the initial guess approximation
๐ฆ0 ๐ก = ๐ก1/2 , (44)
the auxiliary linear operator
โ = ๐ทโ๐ฝ
, ๐ฝ > 0 , (45)
and the auxiliary function
๐ป ๐ก = ๐ก๐พ , ๐พ > โ1 , (46)
then (26) gives
๐ฆ1 = โ๐ก๐ฝ+๐พ โ๐ค 1 + ๐พ
๐ค 1 + ๐พ + ๐ฝ +
๐ค 2 + ๐พ ๐ก
๐ค 2 + ๐พ + ๐ฝ
+ ๐๐ค
32
+ ๐พ โ ๐ผ ๐ก12โ๐ผ
2๐ค 32
โ ๐ผ ๐ค 32
+ ๐พ โ ๐ผ + ๐ฝ ,
๐ฆ2 = โ๐ก๐ฝ+๐พ โ๐ค 1 + ๐พ
๐ค 1 + ๐พ + ๐ฝ +
๐ค 2 + ๐พ ๐ก
๐ค 2 + ๐พ + ๐ฝ
+ ๐๐ค
32
+ ๐พ โ ๐ผ ๐ก12โ๐ผ
2๐ค 32
โ ๐ผ ๐ค 32
+ ๐พ โ ๐ผ + ๐ฝ +
โ2๐ก2๐ฝ+2๐พโ1
2 โฆ ,
โฎ
and so on. As pointed above, the valid region of โ is a
horizontal line segment. Thus, the valid region of โ for the
HAM solution of Eq. (40) at ๐ผ = 0.5 , ๐พ = โ0.5 , ๐ฝ = 1 is
โ1.15 < โ < โ0.1 as shown in Fig. 4.
Fig. 5.(a)~(d) show the approximate solution for Eq. (40)
obtained for different values of โ, ๐ฝ, ๐ป(๐ก) and ๐ฆ0(๐ก) using
HAM. As observed in Fig. 2.(a)~(f), we can notice that the
convergence region can be adjusted and controlled by
choosing proper values of the auxiliary parameter โ , the
auxiliary function ๐ป ๐ก , the auxiliary linear operator and
the initial guess ๐ฆ0(๐ก).
Fig. 4. The โ-curves of ๐ฆโฒ & ๐ฆโฒโฒ which are corresponding to the
HAM solution of Eq. (40) when ๐ผ = 0.5 , ๐ฆ0 = ๐ก1/2 , ๐พ = โ0.5 ,๐ฝ = 1 . Dotted curve: 15th-order approximation of ๐ฆโฒ ,
solid line: 15th-order approximation of ๐ฆโฒโฒ .
0 2 4 6 8 100.03
0.02
0.01
0.00
0.01
1.5 1.0 0.5h
20
10
10
20
y', y
"
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
______________________________________________________________________________________
(a) ๐ผ = 0.5 , โ = ๐ทโ
๐ผ , ๐ป ๐ก = 1 , ๐ฆ0 = ๐ก๐ผ /๐ค 1 + ๐ผ .
Dash. dotted line: โ = โ1 , dotted line: โ = โ0.5 , solid line:
โ = โ0.2 .
(b) ๐ผ = 0.5 , โ = ๐ทโ
๐ผ , ๐ป ๐ก = 1 , โ = โ0.2 .
Dash. dotted line: ๐ฆ0 = ๐ก, dotted line: ๐ฆ0 = ๐ก๐ผ /๐ค 1 + ๐ผ , solid
line: ๐ฆ0 = ๐ก1/2 .
(c) ๐ผ = 0.5 , ๐ฆ0 = ๐ก1/2 , ๐ป ๐ก = 1 , โ = โ0.2 .
Dash. dotted line: โ = ๐ทโ0.5, dotted line: โ = ๐ทโ
0.75, solid line: โ =๐ทโ .
(d) ๐ผ = 0.5 , ๐ฆ0 = ๐ก1/2 , โ = ๐ทโ = ๐ ๐๐ก , โ = โ0.2 .
Dash. dotted line: ๐ป ๐ก = 1 , dotted line: ๐ป ๐ก = ๐กโ1/4 , solid
line: ๐ป ๐ก = ๐กโ1/2 .
Fig. 5. Approximate solutions for Eq. (40)
Example 3. Consider the Lane-Emden fractional differential
equation [13, 21
๐ทโ๐ผ๐ฆ +
2
๐ก๐ฆโฒ + ๐ฆ3 โ 6 + ๐ก6 = 0 , ๐ก โฅ 0 ,
๐ โ 1 < ๐ผ โค ๐ , ๐ = 1,2 ,
(47)
subject to initial conditions
๐ฆ 0 = 0 , ๐ฆโฒ 0 = 0 . (48)
Hence, according to (17), we have
๐ ๐ ๐ฆ ๐โ1 = ๐ทโ๐ผ๐ฆ๐โ1 ๐ก +
2
๐ก๐ฆ๐โ1
โฒ ๐ก
+ ๐ฆ๐โ1โ๐ ๐ก ๐ฆ๐ ๐ก ๐ฆ๐โ๐ ๐ก
๐
๐ =0
๐โ1
๐=0
โ 6 + ๐ก6 1 โ ๐๐ .
(49)
In view of the modified homotopy analysis method, if we set
๐ฆ0 = 0 , ๐ป ๐ก = 1 , ๐ฝ = 2 , (50)
then the mth-order deformation equation (26) gives
๐ฆ1 = โโ ๐ก2 3 + ๐ก6
56 ,
Fig. 6. The โ-curves of ๐ฆโฒโฒ which are corresponding to the
5th-order approximate HAM solution of Eq. (47) for different
values of ๐ผ . Dash dotted line: ๐ผ = 2, dotted line: ๐ผ = 1.75 , solid
line: ๐ผ = 1.5 .
0 1 2 3 4 5t
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y
0 1 2 3 4 5 6t
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y
0 1 2 3 4 5 6t
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y
0 2 4 6 8 10t
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y
1.0 0.8 0.6 0.4 0.2 0.0h
0.5
1.0
1.5
2.0
2.5
3.0
y"
IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01
(Advance online publication: 13 May 2010)
______________________________________________________________________________________
(a) ๐ผ = 2 , ๐ฆ0 = 0 , ๐ป ๐ก = 1 , ๐ฝ = 2.
Dotted line: โ = โ0.5, dash.dotted line: โ = โ0.35, dash.dot dotted
line: โ = โ0.1, solid line: exact solution.
(b) ๐ฆ0 = 0 , ๐ป ๐ก = 1 , ๐ฝ = 2 , โ = โ0.35.
Dash. dotted line: ๐ผ = 1.5 , dotted line: ๐ผ = 1.75, solid line: ๐ผ = 2 .
Fig. 7. Approximate solutions for Eq. (47)
๐ฆ2 = โโ ๐ก2 3 + ๐ก6
56
โ6โ2๐ก4โ๐ผ
๐ค 11 โ ๐ผ 120๐ก6 +
๐ค ๐ผ โ 4
๐ค ๐ผ โ 10 + โ ๐ก2 1176 + ๐ก6 ,
โฎ
and so on. Moreover, if we replace the initial guess
๐ฆ0 = 0 by ๐ฆ0 = ๐ก2, then we have ๐ฆ๐ = 0, โ ๐ โฅ 1, hence,
๐ฆ ๐ก = ๐ก2 is the exact solution.
The valid region of โ for the HAM solution of Eq. (47) at
๐ผ = 2 is โ0.6 < โ < โ0.15 , at ๐ผ = 1.75 is โ0.6 < โ <โ0.15 and at ๐ผ = 1.5 is โ0.6 < โ < โ0.15 as shown in
Fig. 6.
Fig.7.(a) shows the approximate solution for Eq. (47) at
๐ผ = 2 obtained for different values of โ using HAM. As the
previous examples, the convergence region of the series
solution increases as โ goes to the left end point of its valid
region. Fig. 7.(b) shows the HAM solution for Eq. (47) at
different values of ๐ผ obtained for โ = โ0.35.
VI. DISCUSSION AND CONCLUSIONS
In this work, we carefully proposed an efficient algorithm
of the HAM which introduces an efficient tool for solving
linear and nonlinear differential equations of fractional order.
The modified algorithm has been successfully implemented
to find approximate solutions for many problems. The work
emphasized our belief that the method is a reliable technique
to handle nonlinear differential equations of fractional order.
As an advantage of this method over the other analytical
methods, such as ADM and HPM, in this method we can
choose a proper value for the auxiliary parameter โ , the
auxiliary function ๐ป ๐ก , the auxiliary linear operator and
the initial guess ๐ฆ0 to adjust and control convergence region
of the series solutions.
There are some important points to make here. First, we
can observe that the convergence region of the series solution
increases as โ tends to zero. Second, choosing a suitable
auxiliary function ๐ป ๐ก or initial approximation ๐ฆ0 ๐ก may
accelerate the rapid convergence of the series solution and
may increase the agreement with the exact solution. Third,
for a certain value of โ , choosing a suitable auxiliary linear
operator โ = ๐ทโ๐ฝ
, ๐ฝ > 0 may increase the convergence
region. Finally, generally speaking, the proposed approach
can be further implemented to solve other nonlinear problems
in fractional calculus field.
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(Advance online publication: 13 May 2010)
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