Applied Mathematical Sciences, Vol. 8, 2014, no. 7, 321 - 336
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.39495
A Comparison of Adomian and Generalized
Adomian Methods in Solving the Nonlinear
Problem of Flow in Convergent-Divergent
Channels
Mohamed Rafik Sari
1
Mechanical Engineering Department, University of Skikda
El Hadaiek Road, B. O. 26, 21000 Skikda, Algeria
Mohamed Kezzar
1
Mechanical Engineering Department, University of Skikda
El Hadaiek Road, B. O. 26, 21000 Skikda, Algeria
Rachid Adjabi
2
Mechanical Engineering Department, University Badji Mokhtar of Annaba,
Sidi-Amar, B. O. 12, 23000 Annaba, Algeria
Copyright © 2014 Mohamed Rafik Sari et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
The principal aim of this paper is to compare two analytical methods of
decomposition employed for solving nonlinear problem of flow in
converging-diverging channels, namely the Adomian Decomposition Method
(ADM) and Generalized Decomposition Method (GDM). The comparison
between ADM and GDM is conducted by using the numerical results of the
studied problem as a guide. The results show that GDM method is more accurate
and very efficient than ADM method. Indeed, ADM is a good method in solving
the nonlinear problems, but we notice in solving the nonlinear problem of the
Jeffery-Hamel flow that GDM is powerful, superior than ADM and clearly
constitute an excellent choice for solving nonlinear problems.
322
Mohamed Rafik Sari et al.
Keywords: Converging-Diverging Flow, Adomian Decomposition Method,
Generalized Decomposition Method.
1 Introduction
In these last years the new analytical methods have been extensively used in a
wide research activity. Generally these methods give the approximate solutions to
a large mathematics, physics or engineering applications governing by linear or
nonlinear differential equations [1, 2, 3, 4, 5, 6, 7, 8, 9].
Adomian’s decomposition method discovered by Georges Adomian in 1980s is
one of powerful techniques which provides the solution as a fast convergent series
with elegantly computable terms and does not need linearization or discretization.
The principle of this method is well discussed in [10]. The first proof of
convergence of the series obtained by Adomian method was discussed by
cherruault et al. [11]. Since then, several contributions have been conducted on the
convergence of Adomian method [12, 13, 14, 15, 16]. Many authors were also
discussed the similarity of the Adomian decomposition method to numerical
solutions of initial or boundary value problems such as the Picard method [17, 18]
and the Runge-Kutta method [19, 20].
In literature a large variation of the decomposition methods exists, which can be
also applied for practical solutions of linear or nonlinear ordinary differential
equations, partial differential equations and integral equations. Indeed, these
variations will be referred as modified decomposition [21, 22, 23, 24].
Generalized decomposition method [24] (GDM) is one of powerful modified
decomposition method. This method developed recently by Yong-Chang et al., is
an extension of the decomposition method for solving nonlinear equations.
In this paper, we will make a comparative study in order to test the accuracy of the
ADM and GDM when applied to the nonlinear problem of Jeffery-Hamel flow.
The Jeffery-Hamel flow is a radial two dimensional flow of an incompressible
viscous fluid between two inclined plane walls separated by an angle 2α driven by
a line source or sink. The nonlinear governing equations of this flow, has been
firstly discovered by Jeffery [25] in 1915 and independently by Hamel [26] in
1916. It is worth noting that this flow is one of few exact solutions of the
Navier-Stokes equations and has many applications in areas of aerospace, civil,
mechanical and biomechanical engineering. It is also noticed, that due to the
complexity of mathematical formulation of such types of flow, several techniques
of computations have been used, such as variational, perturbation or numerical.
In literature, several contributions have been conducted in order to solve the
nonlinear problem of Jeffery-Hamel flow. Rosenhead [27] gives on the one hand
the solution in terms of elliptic function and discuss on the other hand the various
mathematically possible types of this flow. Millsaps and pohlhausen [28] have
given the exact solution for the thermal distributions of Jeffery-Hamel flow by
considering a constant temperature at the plane walls. For this contribution the
Solving nonlinear problem of flow
323
numerical solution is also given for diverging and converging channels. Fraenkel
[29] has investigated the laminar flow in symmetrical channels with slightly
curved walls. In this study the resulting velocity profiles of the flow was obtained
as a power series. The nonlinear problem of the Jeffery-Hamel flow in
converging-diverging channels has been well discussed in many textbooks [30, 31,
32]. The two dimensional stability of Jeffery-Hamel flow has been also
extensively studied by many authors [33, 34, 35, 36, 37, 38, 39, 40].
The present study consists of section 2, which develops the nonlinear equations
and boundary conditions governing the Jeffery-Hamel flow. Sections 3 and 4 give
the theory of Adomian and Generalized Adomian methods, respectively. Section 5
applies ADM and GDM in order to obtain the analytical solutions of the studied
problem. Section 6 is a comparison of the methods and section 7 is the conclusion
of the study.
2Governing equations
In this study we consider the flow of an incompressible viscous flow between
nonparallel plane walls (see Fig. 1). This flow known as Jeffery-Hamel is
generated by a line source or sink. We consider that the flow is uniform along
z-direction and we assume purely radial motion, i.e.: for velocity components we
can write: ( , ; 0. In cylindrical coordinates (r, θ, z) the
reduced forms of continuity and Navier-Stokes equations are:
. 0
(1)
.
. +. .
. . . . 0
. (2)
(3)
Where:
: radial velocity ; : density ; : kinematic viscosity ; P : fluid pressure.
According to Eq. (1), we define that the quantities . depends on and we
can write:
. (4)
Fig. 1 Geometry of Jeffery-Hamel flow.
Mohamed Rafik Sari et al.
324
Now by introducing the dimensionless parameters !"#
,
where:
/%
with:
1 ' ' 1
And eliminating the pressure term between Eqs. (2) and (3) and after some
simplifications, we obtain:
′′′ 2)* % ′ 4% ′ 0
(5)
The Reynolds number of flow is introduced as:
.,
)* !"#
(6)
Where fmax is the velocity at the centerline of the channel, and, % is the channel
half-angle.
The boundary conditions of the Jeffery-Hamel flow in terms of are
expressed as follows:
1 , ′ 0
(7), at the centerline of channels.
- 0
(8), at the body of channels.
The third order nonlinear differential equation (5) with boundary conditions (7)
and (8), have been solved analytically using the ADM and GDM and numerically
using fourth order Runge-Kutta and shooting techniques.
3 Fundamentals of Adomian Decomposition Method
Consider the differential equation:
./ )/ 0/ 12
(9)
Where: N is a nonlinear operator, L is the highest ordered derivative and R
represents the remainder of linear operator L.
By considering.4 as an n-fold integration for an nth order of L, the principles
of method consists on applying the operator .4 to the expression (9). Indeed, we
obtain:
.4 ./ .4 1 .4 )/ .4 0/
(10)
The solution of Eq. (10) is given by:
/ 5 .4 1 .4 )/ . 0/
(11)
Where 5 is determined from the boundary or initial conditions.
For the standard Adomian decomposition method, the solution u can be determined
as the infinite series with the components given by:
/ ∑8
(12)
79: /7
And the nonlinear term Nu is given as following:
0/ ∑=8
(13)
79: ;7 /: , / , … . , /7 Where ;7 ′> , called Adomian polynomials has been introduced by George
Adomian [10] by the recursive formula:
AB
8
;7 /: , / , … . , /7 7! @ACB D0EF79: GH /H IJK
C9:
, L 0,1,2, … . . , L
(14)
By substituting the given series (12), (13) into both sides of (11), we obtain the
following expressions:
8
4
4
4 ∑=8
∑8
(15)
79: /7 5 . 1 . ) ∑79: /7 .
79: ;7
Solving nonlinear problem of flow
325
According to Eq. (15), the recursive expression which defines the ADM
components /7 is given as:
(16)
/: 5 .4 1, /7= .4 )/7 ;7 Finally after some iterations, the solution of the studied equation can be given as an
infinite series by :
/ /: / / /M ⋯ /7
(17)
4 Fundamentals of Generalized Decomposition Method
The crux of the Eq. (9) is the nonlinear term Nu. In GDM, Yong-Chang et al. [24]
have developed a newly method of decomposition for the nonlinear term Nu:
(18)
0/ ∑=∞
79: O7 /: , / , … . , /7 Where O7 ′> are function depending on /: , / , … . , /7 . They are given by:
O: 0/: (19)
P
O7 0/: / ⋯ /7 0/: / ⋯ /74 , L Q 1
Here, it is worth noting that the adopted new strategy of decomposition could use
all necessary information which concern the terms /: , / , … . , /7 and the
nonlinear term Nu.
For Generalized Decomposition Method, the solution u can be determined as an
infinite series with the components given by:
/ ∑8
(20)
79: /7
And the nonlinear term Nu is given as following:
0/ ∑=8
(21)
79: O7 /: , / , … . , /7 By substituting the given series (20), (21) into both sides of (11), we obtain the
following expressions:
8
4
4
4 ∑=8
∑8
(22)
79: /7 5 . 1 . ) ∑79: /7 .
79: O7
According to Eq. (22), the recursive formula which defines the GDM components
/7 is given as:
/: 5 .4 1, /7= .4 )/7 O7 (23)
Finally after some iterations, the solution of the studied equation can be given by:
/ /: / / /M ⋯ /7
(24)
5 Application of ADM and GDM to the nonlinear problem of
Jeffery-Hamel flow
Considering the Eq. (9), Eq. (5) can be written as:
. 2)* % ′ 4% ′
AR
Where the differential operator L is given by: . ASR.
(25)
The inverse of operator . is expressed by .4 and can be represented as:
S V V
.4 T: TW TW ∎ XXX
(26)
The application of Eq. (26) on Eq. (25) and considering the boundary conditions (7)
Mohamed Rafik Sari et al.
326
and (8), we obtain:
S
(27)
0 ′ 0 ′′ 0 .4 0/
′
′
Where:
0/ 2)* % 4% (28)
′
′′
The values of 0, 0 and 0 depend on boundary condition for
convergent-divergent channels. In order to make difference between convergent
and divergent channels, the boundary conditions are taken in different manner [41].
Indeed, on the one hand for divergent channels, we assume that the body of
channels is given by 0 and by considering a symmetric condition in the center
of channels; the solution is studied between 0 0 at the body of channels and
1 1 at the centerline of channels. On the other hand, for convergent channels,
the center of channels is given by 0 and consequently the solution varies from
1 0 at the body of channels to 0 1 at the centerline of channels.
For convergent channel the boundary conditions are expressed as following:
0 1 , Y 0 0
(29), at the centerline of channels.
1 0
(30), at the body of channels.
But for divergent channels, the boundary conditions are:
At the body of channels: 0 0.
At the centerline of channels: 1 1.
(31)
(32)
5.1 Analytical solution by ADM
5.1.1 Convergent Channel
By applying the boundary conditions (29), (30) and considering YY 0 Z, we
obtain:
4
∑8
(33)
79: 7 : +. 0/ S
Where :
: 1 Z (34)
The terms of Adomian polynomials, are defined by applying Adomian
decomposition method as following:
;: 2ZRe% 4Z% Z Re%M
(35)
]
]
M
; 4Z% ZRe % M ZRe% M M Z% ^ M Z Re % _ ` Z Re% M _
_
M
M
a
M
M
_
_ Z Re % (36)
By using algorithm (16), the first few components of the solution in convergent
channel are:
ZRe%^ ` Z% ^ : Z Re%`
(37)
ZRe % `
]:
b R cd , S ef
Z% ^ `
^_
ZRe% M ` ^_
Z% ^ ` _`:
Z Re % ] Z Re% M ] :]:: (38)
Finally, the solution for convergent channel is given by Adomian decomposition
method as:
]:
Solving nonlinear problem of flow
327
: ⋯ 7 1 Z
`
^
Z Re% Z% S
`
ZRe % :
`
]:
]
Z Re % Z Re% M ]
_`:
]:
^_
b R cd , Sef
:]::
ZRe%^ Z% ^ M `
ZRe% ⋯
^_
`
Z% ^ ` (39)
The value of constant c is obtained by solving Eq. (39) using Eq. (30).
5.1.2 Divergent Channel
By applying the boundary conditions (31), (32) and assuming that g Y 0 and
h YY 0, the solution is given as following:
4
∑8
(40)
79: 7 : +. 0/
Where:
S
: g h (41)
For divergent channel by applying Adomian algorithm, the terms of Adomian
polynomials are expressed by:
(42)
;: 4g% 2g Re% 4h% 3ghRe% h Re%M
: ]
_ M
^ M M
^ M
^
; 4g% 4h% 8g% M g Re% M h% ` g Re % `
a
6ghRe% M ^ : g hRe % _ _ h Re% M _ _ gh Re % ` h M Re % a (43)
By using algorithm (16), the first few components of the solution in divergent
channel are:
M g% M g Re%^ ` h% ^ : ghRe%_ : h Re%`
(44)
g% M h% ^ g% ^ _ g Re% M ` h% ^ ` gM Re % a M
`
_
]
^_
_
_
l mcd , Sn
lm cd , So
m R cd , Sef
ghRe% M a ]: h Re% M ] :]:: M_
MM`:
:]:
(45)
Finally, the solution for divergent channel is given by Adomian decomposition
method as:
: ⋯ 7 g h
S
M g% M g Re%^ h% ^ : ghRe%_ : h Re%` M g% M ` h% ^ _ g% ^ _ `
g Re% M ` ^_ h% ^ ` _ gM Re % a M_ ghRe% M a ]
lm cd , So
m R cd , Sef
l mcd , Sn
MM`:
h Re% M ] ⋯
(46)
:]:
:]::
The quantities a and b can be determined by solving obtained equation (46) using
Eqs. (31) and (32).
]:
5.2 Analytical solution by GDM
5.2.1 Convergent Channel
By applying the boundary conditions (29), (30) and considering YY 0 Z, we
Mohamed Rafik Sari et al.
328
obtain:
4
∑8
79: 7 : +. 0/
(47)
S
Where :
: 1 Z (48)
The terms of Jn’s polynomials and solution, are defined by applying Generalized
decomposition method as following:
O: 2ZRe% 4Z% Z Re%M
ZRe%^ ` Z% ^ : Z Re%`
]
]
M
(49)
(50)
`
O M ZRe % M M ZRe% M M M Z% ^ M _ Z Re % _ _ Z Re% M _ b q cdR , R S ee Z M Re % a ] Z ReM % M a p Z Re % ^ a p Z Re% _ a a Z M ReM % M p _
Z M Re % ^ p M`
F ZRe % ` ]:
b R cd , Sef
:]::
b q cdR ,R Seq `:]::
::
ZRe% M ` ^_
b cdR ,R S ef
p`:
(51)
Z Re % ] Z Re% M ]
_`:
]:
b cd, s Sef
b R cdR ,R Se
b R cd ,q Se
Z% ^ ` ^_
b cd ,q Sef
M^:
M^:
p_:^:
^a_:
(52)
Finally, the solution for convergent channel is given by Generalized
decomposition method as:
: ⋯ 7 1 Z
b R cd , S ef
S
ZRe%^ ` Z% ^ Z Re%` ]: ZRe % ` ^_ ZRe% M ` ^_ Z% ^ ` _`: Z Re % ] :
Z Re% M ] ]:
b R cd ,q Se
^a_:
:]::
b q cdR ,R S eq
`:]::
b cdR ,R Sef
p`:
⋯
b cd , q Sef
M^:
b cd,s Sef
M^:
b R cdR ,R Se
p_:^:
(53)
The value of constant c is obtained by solving Eq. (53) using Eq. (30).
5.2.2 Divergent Channel
By applying the boundary conditions (31), (32) and assuming that g Y 0 and
h YY 0, the solution is given as following:
4
∑8
(54)
79: 7 : +. 0/
Where:
: g h
S
(55)
For divergent channel by applying GDM algorithm, the terms of Jn’s polynomials
and solution are expressed by:
Solving nonlinear problem of flow
329
O: 4g% 2g Re% 4h% 3ghRe% h Re%M
M g% M g Re%^ ` h% ^ : ghRe%_ : h Re%`
O 8g% ^ :
M
]
_
g Re% M M M h% ^ M ` gM Re % ^ 6ghRe% M ^ `
]
a
g hRe % _ h Re% M _ g Re% _ _ gh Re % ` :
_
M
_
a M
^
g Re % ^ ` ghRe% _ ` hM Re % a g^ ReM % M a p
p
_
]
M^ M
g hRe % ^ a p h Re% _ a ^: gM hReM % M ] ^ gh Re % ^ ]
^_
a
lmR cdR ,R Sef
m q cdR ,R See
g h ReM % M p hM Re % ^ p ::
::
]:
M`
(58)
g% ^ _
_
l mcd , Sn
(56)
(57)
h% ^ ` gM Re % a ghRe% M a ^_
_
M_
lm cd , So
M ]
_ ]
h
Re%
g
Re%
`^] gM Re % ^ p MM`:
]:
`
:]:
m R cd , S ef
l q cdR ,R Sef
al mcd , q Sef
m cd, sS ef
ghRe% _ p :]:: p`: `::
M^:
M^
l R mcdR , R See
lm cd ,q S ee
al m cdR ,R Se
m R cd ,q Se
lm R cdR ,R SeR
M::
m q cdR , R Seq
`:]::
]
g Re% M ` Mp`:
Ma`::
^a_:
]a::
(59)
Finally, the solution for divergent channel is given by Generalized decomposition
method as:
: ⋯ 7 g h
S
M g% M g Re%^ h% ^ : ghRe%_ : h Re%` _ g% ^ _ ] g Re% M ` ^_ h% ^ ` `
gM Re % a M_ ghRe% M a _
g Re% _ ] `
l q cdR ,R Sef
:]:
al mcd , q Sef
p`:
al m cdR ,R S e
Ma`::
lm cd , So
`::
m R cd ,q Se
^a_:
l mcd , S n
MM`:
^ p
]: h Re% M ] `^] gM Re % M^ ghRe% _ p m cd,s Sef
M^:
lm R cdR , R SeR
]a::
l R mcdR , R See
M::
m q cdR ,R Seq
`:]::
m R cd , Sef
lm cd ,q S ee
Mp`:
⋯
:]::
(60)
The quantities a and b can be determined by solving obtained equation (60) using
Eqs. (31) and (32).
6 Comparisons of the GDM and ADM
In this study we are particularly interested to the nonlinear problem of the
Jeffery-Hamel flow. The third order nonlinear differential equation (5) with
boundary conditions (7) and (8), have been solved analytically for some values of
the governing parameters )* and %by using ADM and GDM techniques.
Mohamed Rafik Sari et al.
330
a) Convergent Channel
b) Divergent Channel
Fig. 2 Comparison between GDM, ADM and Numerical results
for Re=72, α=3°.
a) Convergent Channel
b) Divergent Channel
Fig. 3 Comparison between GDM, ADM and Numerical results for Re=300,
α=3°.
With intention to compare the two adopted analytical methods ADM and GDM,
we use the numerical results of the studied problem as a guide. Indeed, the
numerical solution of the third order nonlinear differential equation governing the
Jeffery-Hamel flow is obtained by using fourth order Runge-Kutta method.
The comparison between ADM and GDM is depicted in (Figs. 2-6). Indeed, (Figs.
2-5) show the velocity profiles in convergent-divergent channels, but (Fig. 6)
shows the variation of skin friction in divergent channel, related to )* %. According
to the obtained results for velocity profiles (Figs. 2-5) in convergent-divergent
channels, we notice that GDM and ADM techniques are in good agreement with the
numerical treatment.
Solving nonlinear problem of flow
a)
331
Convergent Channel
b) Divergent Channel
Fig. 4 Comparison between GDM, ADM and Numerical results for Re=268, α=5°.
Diverging Channel (α = 5°)
ADM
GDM
Numerical
Re
a
b
a
b
Cf
40
80
120
160
1.95211
-1.76685
1.23557
0.543702
0.540035
2.32756
-0.0823697
3.63423
1.99492
-1.97462
1.994920
1.28636
0.351419
1.286405
0.592567
2.16681
0.5927929
-0.0321302
3.50547
-0.031620387
Table 1 Constant of divergent channel (skin friction) when α=3° (Analytical and
numerical results).
In order to show the accuracy of the GDM and ADM, the error between analytical and
numerical solutions is presented in tables 2 and 3 for converging-diverging channels.
Obtained results show that GDM method is closer to the numerical solution and has a high
precision than ADM method. In general, results with six decimal places are sufficient to
give a match between GDM and ADM.
a) Convergent Channel
b) Divergent Channel
Fig. 5 Comparison between GDM, ADM and Numerical results for Re=102, α=5°.
Mohamed Rafik Sari et al.
332
On the other hand, as it is shown in Fig. 6 and table 1 for skin friction in diverging
channel, it is worth noting that GDM results are very closer to the numerical results.
Indeed, we conclude that GDM method is more reliable and efficient than ADM
method. Generally, GDM and ADM give the solution as an infinite fast series.
Fig. 6 Skin friction versus Reα for divergent channels (α=5°).
Re=102
η
Numerical
ADM
GDM
Re=268
0
1
1
1
Error of
ADM
0
Error of
GDM
0
Numerical
ADM
GDM
1
1
1
Error of
ADM
0
Error of
GDM
0
0.2
0.986715
0.986714
0.986639
0.000001
0.000076
0.997176
0.997297
0.997176
0.020121
0.000000
0 .4
0.936995
0.936702
0.936992
0.000293
0.000003
0.982574
0.983331
0.982571
0.000757
0.000003
0.6
0.815384
0.814819
0.815375
0.000565
0.000027
0.924963
0.928266
0.924952
0.003303
0.000011
0.8
0.544735
0.544153
0.544718
0.000582
0.000017
0.708745
0.720642
0.708711
0.011897
0.000034
1
0
0
0
0
0
0
0
0
0
0
Table 2 the comparison between ADM and GDM for velocity distribution in
convergent channel when: α=5°.
In Fig. 7, we notice also that GDM method has a fast convergence for series
solutions than the ADM method. Indeed, the solution obtained by GDM in
converging-diverging channels for a fixed Reynolds number (Re=190) when
% 3°, converges quickly: at the third iteration for convergent flow and at the
five iteration for divergent flow. On the other hand, the obtained solution by
ADM converges after 5 iterations for the convergent channel but in divergent
channel after 9 iterations.
Solving nonlinear problem of flow
333
Re=102
η
Numerical
ADM
Error of
GDM
0
Numerical
ADM
GDM
0
Error of
ADM
0
0
0
0
0
0.2
0.106412
0.098823
0.106383
0.007589
0.000029
-0.177053
0.4
0.324432
0.314567
0.32438
0.009865
0.000052
0.6
0.617131
0.609694
0.617074
0.007437
0.8
0.88723
0.884622
0.887195
1
1
1
0
1
GDM
Re=268
0
Error of
ADM
0
Error of
GDM
0
-0.182964
-0.177572
0.005911
0.000519
-0.112344
-0.121612
-0.113207
0.009268
0.000863
0.000057
0.20424
0.193795
0.203251
0.010445
0.000989
0.002608
0.000035
0.710777
0.703969
0.710262
0.006808
0.000515
0
0
1
1
1
0
0
Table 3 the comparison between the ADM and GDM results for velocity in
divergence channel when: α=5°.
The ADM method is a good technique in solving the nonlinear problems, but we
notice in solving the nonlinear equation of the Jeffery-Hamel flow that the GDM
method is powerful, superior to the ADM method and clearly constitute an
excellent choice for solving nonlinear problems.
a) Convergent Channel
b) Divergent Channel
Fig. 7 Velocity profiles by GDM, ADM and Numerical methods after iteration
for Re=190 and α=3°.
7 Conclusion
In this paper, the Adomian decomposition and Generalized decomposition
methods have been successfully used in solving the nonlinear problem of the flow
of an incompressible viscous fluid in converging-diverging channels. It is clearly
shown that GDM and ADM techniques are powerful in obtaining the solution of
the studied problem. The comparison of ADM and GDM with the numerical
solution used as a guide showed that GDM method is more accurate, reliable and
efficient than ADM method from a computational viewpoint. Indeed, the new
adopted decomposition strategy for the nonlinear term Nu which could use all
necessary information concerning the terms of series elements /: , / , … . , /7 and
334
Mohamed Rafik Sari et al.
the nonlinear term Nu, is one of the major advantages of the accuracy of GDM
method. On the other hand, GDM has a fast and better convergence than ADM.
Finally, the Generalized Decomposition Method is highly recommended and
clearly constitute an excellent choice to solve nonlinear problems.
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Received: September 15, 2013