THE Q-HOMOTOPY ANALYSIS METHOD (Q-HAM) - IJAMM

ABSTRACT
THE Q-HOMOTOPY ANALYSIS METHOD (Q-HAM)
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil 1 and S. N. Huseen 2
1 Cairo University, Faculty of Engineering,
Engineering Mathematics Department, Giza, Egypt
Email: magdyeltawil@yahoo.com
2 Thi-Qar University, Faculty of Computer Science and Mathematics,
Mathematics Department, Thi-Qar, Iraq
Email: shn_n2002@yahoo.com
Received 30 November 2011; accepted 27 March 2012
In this paper, a more general method of homotopy analysis method (HAM) is introduced to
solve non-linear differential equations, it is called (q-HAM). The interval of convergence of
HAM, if exists, is increased when using q-HAM. The analysis shows that the series solution
in the case of q-HAM is more likely to converge than that on HAM. The new method is
applied to some nonlinear differential equations to illustrate the method of analysis.
Keywords: Homotopy Analysis Method (HAM), Partial Differential Equations
1 INTRODUCTION
The standard homotopy analysis method (HAM) is an analytic method that provides series
solutions for nonlinear partial differential equations and has been firstly proposed by (Liao
1992). In recent years, this method has been successfully employed to solve many types of
linear and non-linear differential equations in various fields of engineering and science (Liao
1995,1996,1997,2002,2003,2004,2009 ;Wang et al.2007;Bataineh et al.2009;Van Gorder and
Vairavelu 2009). Different from all perturbation techniques ( the techniques are strongly
dependent upon small / large physical parameters (Nayfeh 1981,1985,2000;Lagerstrom
1988;Murdock 1991;Hinch 1991)) and non perturbation techniques, such as the Lyapunov
artificial small parameter method (Lyapunov 1992) , the -expansion method (Karmishin et
al. 1990; Awrejcewicz et al. 1998) , Adomians decomposition method (Rach 1984; Adomain
and Adomain GE 1984,Adomain 1991) and so on which are formally independent of
small/large physical parameters but they can not ensure the convergence of solution series,
the homotopy analysis method has the following advantages:
(i) It is valid even if a given nonlinear problem does not contain any small/large
parameters.
(ii) It can provide us with a convenient way to adjust and control the convergence
region and rate of approximation series.
(iii) It can be employed to efficiently approximate a nonlinear problem by choosing
different sets of base functions.

52
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
In recent years more and more researchers have been successfully applying (HAM) to various
nonlinear problems in science and engineering ,such as, the nonlinear water waves (Tao et al.
2007, groundwater flows (Song and Tao 2007) , time-dependent Emden-Fowler type
equations (Bataineh et al. 2007) , solving high order nonlinear differential equations (Hany N
Hassan and El-Tawil 2010), solving two points nonlinear boundary value problems (Hany N
Hassan and El-Tawil 2010), time-fractional partial differential equations (O Abdulaziz et al.
2008) ,Klein- Gordon equation (Alomari et al. 2008 ,Goursat problems(Syyed Ali
Kazemipour and Ahmad Neyrameh 2010) ,one group neutron diffusion equation(Cavdar
2010) ,algebraic equations (Chin Fugn Yuen et al. 2010 ,integral and integro-differential
equations (Hossein Zadeh et al. 2010), singular higher-order boundary value problems(A
Sami Bataineh et al. 2011).
2 BASIC IDEA OF Q-HOMOTOPY ANALYSIS METHOD (Q-HAM)
Consider the following differential equation:
[ ] (1)
where N is a nonlinear operator, denotes independent variables, is a known
function and is an unknown function.
Let us construct the so-called zero-order deformation equation:
[ ] [ ] (2)
Where , [ ] denotes the so-called embedded parameter , is an auxiliary linear
operator with the property [ ] , is an auxiliary parameter,
denotes a non-zero auxiliary function.
It is obvious that when
respectively. Thus as increases from 0 to
guess to the solution .
(
equation (2) becomes:
) (3)
, the solution varies from the initial
Having the freedom to choose , we can assume that all of them can be
properly chosen so that the solution of equation (2) exists for [ ].
Expanding in Taylor series, one has:
Where:
∑
(4)
(5)

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Assume that are so properly chosen such that the series (4) converges at
and:
(
) ∑ (
Defining the vector { }
differentiating equation (2) times with respect to and then setting and finally
dividing them by we have the so-called order deformation equation:
[ ]
where :
and:
{
[ ]
It should be emphasized that for is governed by the linear equation (7) with
linear boundary conditions that come from the original problem. Due to the existence of the
factor (
)
)
, more chances for convergence may occur or even much faster convergence can
be obtained better than the standard HAM. It should be noted that the case of
, standard HAM can be reached.
3 APPLICATIONS
3.1 The nonlinear homogeneous gas dynamics equation
Let
with initial condition
The exact solution of this problem is known to be
This problem was solved by HAM in (Hany Nasr Hassan 2010) .
For q- HAM solution we choose the linear operator :
[ ]
with the property :
[ ]
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
53

54
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
where is constant. Using initial approximation ,
we define a nonlinear operator as
[ ]
We construct the zero order deformation equation:
[ ] [ ]
we can take , and the order deformation equation is :
[ ]
with the initial conditions for
Where:
And:
∑
{
∑
[ ]
Now the solution of equation (10) for becomes
∫
where the constant of integration is determined by the initial conditions (15).
We can obtain components of the solution using q- HAM as follows:
can be calculated similarly. Then the series solution expression by
q- HAM can be written in the form:
∑ (
)
(14)
(15)
(16)
,

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Equation (16) is an approximate solution to the problem (10) in terms of the convergence
parameters . It can be noted that the fraction factor ( )
convergence chances than that of HAM.
55
highly increase the
To find the valid region of , the -curves given by the 10 th order q-HAM approximation at
different values of are drawn in figures ). These figures show the
interval of at which the value of is constant at certain values of .
We choose the horizontal line parallel to as a valid region which provides us
with a simple way to adjust and control the convergence region of the series solution(16).
From these figures , the valid intersection region of for the values of in the
curves becomes larger as increase as in the following Table(1):
10
20
30
40
50
100
region
Table (1): the increase of the convergence interval length with the increase of n
Figures show the comparison among using different values of with
the exact solution (12). Figures show the comparison between of HAM
and of q-HAM using different values of with the exact solution (12), which indicates
that the speed of convergence for q-HAM with is faster in comparison with .
The Absolute errors of the 10 th order solutions q-HAM approximate at using different
values of compared with 10 th order solutions HAM approximate at are
calculated by the formula
| | (17)
Figures show that the series solution obtained by HAM is more accurate at
but at larger the series solutions obtained by q-HAM at converge faster than
(HAM).

56
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
2.5 2.0 1.5 1.0 0.5 0.5 h
Figure : - curve for the HAM (q-HAM; approximation solution of
problem (10) at different values of .
3 2 1
U 10 x, t , n
40
30
20
10
10
20
U 10 x, t , n
30
20
10
U 10 0,1,1
U 10 0.5 ,2,1
U 10 0.7 ,3,1
U 10 0.1 ,3.7 ,1
Figure : - curve for the ( q-HAM; approximation solution of
problem (10) at different values of .
h
U 10 0,1,1.5
U 10 0.5 ,2,1.5
U 10 0.7 ,3,1.5
U 10 0.1 ,3.7 ,1.5

12 10 8 6 4 2
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Figure : - curve for the (q-HAM; approximation solution of problem
(10) at different values of .
25 20 15 10 5
U 10 x, t , n
Figure : - curve for the (q-HAM; approximation solution of
problem (10) at different values of .
30
20
10
U 10 x, t , n
40
30
20
10
10
20
30
h
h
U 10 0,1,5
U 10 0.5 ,2,5
U 10 0.7 ,3,5
U 10 0.1 ,3.7 ,5
U 10 0,1,10
U 10 0.5 ,2,10
U 10 0.7 ,3,10
U 10 0.1 ,3.7 ,10
57

58
solutions
60000
50000
40000
30000
20000
10000
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
2 4 6 8 10 12 14
Figure : Comparison between of HAM (q-HAM; and exact solution of
(10) at with ,
solutions
80000
60000
40000
20000
2 4 6 8 10 12 14
Figure : Comparison between of (q-HAM and exact solution of (10) at
with ,
t
t
Exact
U 10 n 1
U 7 n 1
U 5 n 1
Exact
U 10 n 5
U 7 n 5
U 5 n 5

250000
200000
150000
100000
50000
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
2 4 6 8 10 12 14
Figure : Comparison between of HAM (q-HAM;( )) and (q-HAM
with exact solution of (10) at with
5 10 6
4 10 6
3 10 6
2 10 6
1 10 6
solutions
solutions
5 10 15 20
Figure : Comparison between of HAM (q-HAM and q-HAM(
with exact solution of (10) at with (
,
t
t
Exact
U10 n 1
U10 n 1.5
U10 n 2
U10 n 3
U10 n 4
U10 n 5
Exact
U10 n 1
U10 n 10
U10 n 20
U10 n 30
U10 n 40
U10 n 50
U10 n 100
59

60
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
Figure : The Absolute error of of q-HAM ( for problem (10) at
and using
Figure : The Absolute error of of q-HAM ( for problem (10) at
and using
3.2 The Riccati equation
Let
Absolute Error
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
Absolute Error
30 000
25 000
20 000
15 000
10 000
5000
The exact solution is known to be
1 2 3 4
8 10 12
t
t
A.E n 1
A.E n 40
A.E n 1
A.E n 40
(18)
(19)

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
For q- HAM solution we choose the linear operator :
[ ]
With the property :
[ ]
Where is constant. Using initial approximation
We define a nonlinear operator as :
[ ]
We construct the zero order deformation equation
[ ] [ ]
Where . We can take H , and the order deformation equation is :
[ ]
(21)
With the initial conditions for :
Where:
and
{
∑
[ ]
Now the solution of equation (18) for becomes:
∫
Where the constant of integration is determined by the initial conditions (22).
We now obtain components of the solution using q- HAM as follows:
,
can be calculated similarly.Then the series solution expression by q-
HAM can be written in the form:
(20)
(22)
61

62
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
∑ (
Equation (23) is a family of approximation solutions to the problem (18) in terms of the
convergence parameters .
)
To find the valid region of , the -curves given by the 15 th order q-HAM approximation at
different values of are drawn in figures . From these figures , the valid
intersection region of for the values of in the curves becomes larger as increase
as in the following Table(2).
10
20
30
40
50
100
region
Table (2): the increase of the convergence interval length with the increase of n
U 15 x, n
2.0 1.5 1.0 0.5 0.5 h
10
5
5
U 15 0.1 ,1
U 15 0.4 ,1
U 15 0.6 ,1
U 15 0.8 ,1
Figure : - curves for the HAM (q-HAM; approximation solution of
problem (18) at different values of .
(23)

200 150 100 50
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Figure : - curves for the (q-HAM; approximation solution of
problem (18) at different values of .
Figures show the comparison between of HAM and of q-HAM using
different values of with the exact solution (19), which indicates that the speed of
convergence for q-HAM with is faster in comparison with .
solutions
12
10
8
6
4
2
U 15 x, n
15
10
0.2 0.2 0.4 0.6 0.8 1.0
Figure : Comparison between of HAM(q-HAM and q-HAM( with
exact solution of (18) with( .
5
x
h
U 15 0.1 ,100
U 15 0.4 ,100
U 15 0.6 ,100
U 15 0.8 ,100
Exact
U 15 n 2
U 15 n 1
63

64
solutions
12
10
8
6
4
2
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
0.2 0.2 0.4 0.6 0.8 1.0
Figure : Comparison between of HAM(q-HAM and q-HAM(
with exact solution of (18) with( .
The absolute errors of the 15 th order solutions q-HAM approximate using different values of
compared with 15 th order solutions HAM approximate are shown in figures (15,16).
These figures show that the series solution obtained by HAM is more accurate at
but at larger the series solution obtained by q-HAM at converge faster than
(HAM).
Absolute Error
0.00005
0.00004
0.00003
0.00002
0.00001
0.1 0.2 0.3 0.4 0.5
Figure : The Absolute error of of q-HAM ( for problem (18) at
and using
x
x
Exact
U 15 n 100
U 15 n 1
A.E n 1
A.E n 100

Absolute Error
0.25
0.20
0.15
0.10
0.05
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
0.60 0.65 0.70 0.75 0.80 0.85
Figure : The Absolute error of of q-HAM ( for problem (18) at
and using
It should be noted that the absolute error is highly decreased when modifying the solution by
taking more terms into consideration. Figures (17,18) illustrate this fact.
Absolute Error
0.0008
0.0006
0.0004
0.0002
0.1 0.2 0.3 0.4 0.5
Figure : The Absolute error of ( and of q-HAM at
using respectively.
x
x
A.E n 1
A.E n 100
A.E U 10 n 1
A.E U 10 n 100
A.E U 30 n 100
65

66
Absolute Error
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
Figure : The Absolute error of ( and of q-HAM
at using respectively.
3.3 The logistic growth model
where r and K are positive constants. Here represents the population of the species
at time , and K is the carrying capacity of the environment.
Let :
1.2
1.0
0.8
0.6
0.4
0.2
Then (1) becomes
If then
therefore, the analytical solution of (25) is easily obtained
(such as Bernoulli method and separation of variables)
(26)
For q- HAM solution we choose the linear operator :
[ ]
With the property :
[ ]
0.6 0.7 0.8 0.9
Where is constant.Using initial approximation :
x
A.E U 10 n 1
A.E U 10 n 100
A.E U 30 n 100
(24)
(25)
(27)
(28)

We define a nonlinear operator as :
[ ]
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
We construct the zeroth-order deformation equation:
[ ] [ ]
We can take , and the mth-order deformation equation is :
[ ]
With the initial conditions for :
Where:
And:
{
[ ]
Now the solution of equation (29) for becomes:
∫
∑
2here the constant of integration is determined by the initial conditions (30).
For numerical results we take , therefore (26) becomes:
We now obtain components of the solution using q- HAM as follows:
can be calculated similarly.Then the series solution expression by q-
HAM can be written in the form:
∑ (
Equation (33) is a family of approximation solutions to the problem (24) in terms of the
convergence parameters .
)
(29)
(30)
(31)
(32)
(33)
67

68
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
To find the valid region of ,the -curves given by the 10 th order q-HAM approximation at
different values of are drawn in figures (19 ,20 and 21).This figures show the
interval of where the value of is constant at certain . We choose the
horizontal line parallel to as a valid region which provides us with a simple way
to adjust and control the convergence region.
5 10 14
4 10 14
3 10 14
2 10 14
1 10 14
U 10 , n
0.06 0.04 0.02 0.02 0.04 0.06
Figure : - curves for the HAM (q-HAM; approximation solution of
problem (1) at different values of .
U 10 , n
2000
1500
1000
500
0.06 0.04 0.02 0.02
Figure : - curves for the q-HAM( approximation solution of
problem (1) at different values of .
h
h
U 10 0.1 ,1
U 10 0.6 ,1
U 10 1,1
U 10 2,1
U 10 0.1 ,10
U 10 0.6 ,10
U 10 1,10
U 10 2,10

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.05
Figure : - curves for the q-HAM( approximation solution of
problem (1) at different values of .
Figures (22,23 and 24) show the comparison between of HAM and of q-HAM using
different values of with the solution (32)
solutions
35
30
25
20
15
10
5
U 10 , n
3.0
2.5
2.0
1.5
0.5 1.0 1.5 2.0 2.5 3.0
Figure(22): Comparison between of HAM(q-HAM and q-HAM( and
for logistic growth model (1) with( respectively.
h
U 10 0.1 ,100
U 10 0.6 ,100
U 10 1,100
U 10 2,100
U
U 10 n 2
U 10 n 1
69

70
solutions
3.0
2.5
2.0
1.5
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
0.5 1.0 1.5 2.0 2.5 3.0
Figure(23): Comparison between of HAM(q-HAM and q-HAM( and
for logistic growth model (1) with( respectively.
solutions
3.0
2.5
2.0
1.5
0.5 1.0 1.5 2.0 2.5 3.0
Figure(24): Comparison between of HAM(q-HAM and q-HAM( and
for logistic growth model (1) with( respectively.
The Absolute errors of the 10 th order solutions q-HAM approximate using different values of
compared with 10 th order solutions HAM approximate are calculated by the formula
| | .
U 10 n 10
U 10 n 1
Figures show that the series solution obtained by HAM is divergent at
but, the series solutions obtained by q-HAM becomes more accurate
as increases.
U
U
U 10 n 100
U 10 n 1

Absolute Error
140
120
100
80
60
40
20
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
0.5 1.0 1.5 2.0 2.5 3.0
Figure(25): The Absolute error of of HAM (q-HAM ( and q-HAM ( for
problem (1) using with
Absolute Error
30
25
20
15
10
5
0.5 1.0 1.5 2.0 2.5 3.0
A.E n 2
A.E n 1
A.E n 10
A.E n 1
Figure(26): The Absolute error of of HAM (q-HAM ( and q-HAM ( for
problem (1) using with .
71

72
Absolute Error
30
25
20
15
10
5
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
Figure(27): The Absolute error of of HAM (q-HAM ( and q-HAM ( for
problem (1) using with
4 CONCLUSION
A more general method (q-HAM) of the homotopy analysis method (HAM) was proposed.
This method provides an approximate solution by taking different values of . The results
show that the convergence region of series solutions obtained by q-HAM is increasing as q is
decreased. The q-HAM improves the performance of HAM and it was shown from the
illustrative examples that the convergence of q-HAM is faster than the convergence of HAM.
REFERENCES
0.5 1.0 1.5 2.0 2.5 3.0
A.E n 100
A.E n 1
Adomain G and Adomain GE(1984).A global method for solution of complex systems.
Mathematical Modelling ;5;521-68.
Adomain G(1991).A review of the decomposition method and some recent results for
nonlinear equations. Computers and Mathematics with Applications ;21:101-27.
Awrejcewicz J Andrianov IV Manevitch LI(1998). Asymptotic approaches in nonlinear
dynamics. Berlin:Springer-Verlag.
AK Alomari Mohd Salmi Md Noorani Roslinda Mohd Nazar(2008). Approximate analytical
solutions of the Klein-Gordon equation by means of the homotopy analysis method, Journal
of Quality Measurement and Analysis 4(1),45-57.
A Sami Bataineh I Hashim S Abbasbandy MSM Noorani(2011). Direct solution of singular
higher-order BVPs by the homotopy analysis method and its modification, World Applied
Sciences Journal 12(6):884-889.
Bataineh AS Noorani MSM Hashim I(2007). Solutions of time-dependent Emden-Fowler
type equations by homotopy analysis method. Physics Letters A ;371:72-82.

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Bataineh AS Noorani MSM Hashim I(2009). On a new reliable modification of homotopy
analysis method. Communications in Nonlinear Science and Numerical Simulation ,14:409-
423.
Chin Fung Yuen Lem Kong Hong Chong Fook Seng(2010). Solving nonlinear algebraic
equation by homotopy analysis method ,proceedings of the Gth IMT-GT conference on
mathematics ,statistics and its applications(ICMSA) University Tunku Abdul Rahman,Kuala
Lumpur,Malaysia.
Hany Nasr Hassan Mohammed(2010),The application of homotopy analysis method in
solving initial and boundary value problems., Ph.D. thesis ,Faculty of engineering, Cairo
University.
Hany N Hassan and Magdy A El-Tawil(2011). A new technique of using homotopy analysis
method for solving high-order nonlinear differential equations. , Math. Meth. Appl. Sci, 34
:728–742.
Hany N Hassan and Magdy A El-Tawil(2011). An efficient analytic approach for solving
two-point nonlinear boundary value problems by homotopy analysis method , Math. Meth.
Appl. Sci. 34: 977–989.
Hany N Hassan and Magdy A El-Tawil(2011). SOLVING CUBIC AND COUPLED
NONLINEAR SCHRÖ DINGER EQUATIONS USING THE HOMOTOPY ANALYSIS
METHOD, Int. J. of Appl. Math and Mech. 7 (8): 41-64
.
Hany N Hassan and Magdy A El-Tawil(2011). MODIFIED HOMOTOPY ANALYSIS
METHOD FOR SECOND ORDER NONLINEAR INITIAL VALUE PROBLEMS, Int. J. of
Appl. Math and Mech. 7 (18): 82-108.
Hinch EJ(1991).Perturbation methods .Cambridge Texts in Applied mathematics. Cambridge
university Press.
H Hossein Zadeh H Jafari and S M Karimi(2010). Homotopy analysis method for solving
integral and integro_differential equations, International Journal of Research and Reviews in
Applied Sciences2(2).February .
Karmishin AV Zhukov AT Kolosov VG(1990). Methods of dynamics calculation and testing
for thin-walled structures ,Moscow:Mashinostroyenie [in Russian].
Lagerstrom PA(1988). Matched asymptotic expansions,Ideas and techniques of applied
mathematical sciences,vol.76.New York:Springer-Verlag.
Liao SJ(1992).The proposed homotopy analysis techniques for the solution of nonlinear
problems. Ph.D. thesis, Shanghai Jiao Tong University.
Liao SJ(1995). An approximate solution technique not depending on small parameters: A
special example .International Journal of non-linear mechanics ;30(2):371-380.
73

74
Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
M. A. El-Tawil and S. N. Huseen
Liao SJ(1996).What’s the common ground of all numerical and analytical tichniques for
nonlinear problems.Communications in nonlinear science & numerical simulation; (4):26-30.
Lioa SJ(1997). Homotopy analysis method :A new analytical technique for nonlinear
problems. Communications in nonlinear science & numerical simulation; 2(2):95-100.
Lioa SJ(2002). A direct boundary element approach for unsteady non-linear heat transfer
problems. Engineering analysis with boundary element method ;26:55-59.
Lioa SJ(2003). Beyond perturbation : Introduction to the homotopy analysis method .CRC
Press,Chapman & Hall.
Loia SJ(2004). On the homotopy analysis method: for nonlinear problems .Applied
mathematics and computation ; 147:499-513.
Liao SJ(2009). Notes on the homotopy analysis method : Some definitions and theorems .
Communications in nonlinear science & numerical simulation ,14:983-997.
Lyapunov AM ( 1992). General problem on stability of motion .Taylor & Francis ,London
[English translation].
Murdock JA(1991).Perturbation :theory and methods.New York:John Wiley&Sons.
Nayfeh AH(1981). Introduction to perturbation techniques ,New York; John Wiley &Sons.
Nayfeh AH(1985). Problems in perturbation ,New York; John Wiley &Sons.
Nayfeh AH(2000). Perturbation methods. New York: John Wiley&Sons.
O Abdulaziz I Hashim and A Saif(2008). Series Solutions of Time-Fractional PDEs by
Homotopy Analysis Method, Hindawi Publishing Corporation Differential Equations and
Nonlinear Mechanics Volume 2008, Article ID 686512, 16 pages, doi:10.1155/686512.
Rach R(1984). On the Adomain method and comparisons with Picards method . Journal of
Mathematical Analysis and Applications ;10:139-59.
Seyyed Ali Kazemipour and Ahmad Neyrameh(2010).Analytical study on Goursat
problems,world applied sciences Journal;10(8) 867-870.
Song H and Tao L(2007). Homotopy analysis of 1 D unsteady, nonlinear groundwater flow
through porous media. Journal of Coastal Research ;50:292-5.
Sukran Cavdar(2010). Homotopy analysis method for one group Neutron diffusion equation,
5 th International Ege Energy Symposium and Exhibition (IEESE-5) 27-30 June,Pamukkale
University , Denizli, Turkey.
Tao L Song H Chakrabarti S(2007). Nonlinear progressive waves in water of finite depth-an
analytic approximation,Coastal Engineering ;54:825-34.

Int. J. of Appl. Math and Mech. 8 (15): 51-75, 2012.
The q-Homotopy Analysis Method
Van Gorder RA. and Vajravelu K(2009). On the selection of auxiliary functions, operators
,and convergence control parameters in the application of the homotopy analysis method to
nonlinear differential equations:A general approach. Communications in nonlinear science &
numerical simulation; 14:4078-4089
Wang H Zou L Zhang H(2007).Applying homotopy analysis method for solving differential
difference equation. Physics Letters A; 369:77-84.
75