the homotopy analysis method for explicit analytical solutions of jaulent-miodek equations

The Homotopy Analysis Method for ExplicitAnalytical Solutions of Jaulent–Miodek EquationsM. M. Rashidi,1 G. Domairry,2 S. Dinarvand1

1Mechanical Engineering Department, Engineering Faculty, Bu-Ali Sina University,Hamedan, Iran

2Mechanical Engineering Department, Mazandaran University, Babol, Iran

Received 16 December 2007; accepted 19 February 2008Published online 13 May 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20358

In this work, the homotopy analysis method (HAM) is applied to obtain the explicit analytical solutions forsystem of the Jaulent–Miodek equations. The validity of the method is verified by comparing the approx-imation series solutions with the exact solutions. Unlike perturbation methods, the HAM does not dependon any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems.Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and controlthe convergence region of the series solution by means of an auxiliary parameter �. Briefly speaking, thiswork verifies the validity and the potential of the HAM for the study of nonlinear systems. © 2008 WileyPeriodicals, Inc. Numer Methods Partial Differential Eq 25: 430–439, 2009

Keywords: homotopy analysis method; Jaulent-Miodek equation; system of nonlinear partial differentialequations

I. INTRODUCTION

Considerable attention has been directed towards the study of nonlinear problems in all areas ofphysics and engineering. Apart from a limited number of these problems, most of them do not havea precise analytical solution, so these nonlinear equations should be solved using approximatemethods.

In 1992, Liao [1] employed the basic ideas of the homotopy in topology to propose a generalanalytic method for nonlinear problems, namely homotopy analysis method (HAM) [2–7]. Onthe basis of homotopy of topology, the validity of the HAM is independent of whether or notthere exist small parameters in the considered equation. Therefore, the HAM can overcome theforegoing restrictions and limitations of perturbation techniques [8]. The HAM also avoids dis-cretization and provides an efficient numerical solution with high-accuracy, minimal calculation,and avoidance of physically unrealistic assumptions. Furthermore, the HAM always provides uswith a family of solution expressions in the auxiliary parameter �, the convergence region and

Correspondence to: M. M. Rashidi, Mechanical Engineering Department, Engineering Faculty, Bu-Ali Sina University,Hamedan, Iran (e-mail: [email protected])

© 2008 Wiley Periodicals, Inc.

HAM METHOD FOR EXPLICIT ANALYTICAL SOLUTIONS 431

rate of each solution might be determined conveniently by the auxiliary parameter �. Besides, theHAM is rather general and contains the homotopy perturbation method (HPM) [7], the Adomiandecomposition method (ADM) [9], and δ-expansion method.

In recent years, this method has been successfully employed to solve many types of non-linear problems in science and engineering. For example, it was applied to the generalizedthree-dimensional MHD flow over a porous stretching sheet by Hayat and Javed [10], to theMHD flow of a second grade fluid in a porous channel by Hayat et al. [11], to the wire coatinganalysis using MHD Oldroyd 8-constant fluid by Sajid et al. [12], to the rotating flow of a thirdgrade fluid in a porous space with Hall current by Hayat et al. [13], to the Burger and regularizedlong wave equations by Rashidi et al. [14], to the MHD flow and heat transfer in a third-orderfluid over a stretching sheet by Sajid et al. [15], to the nonlinear equations arising in heat transferby Abbasbandy [16], to the axisymmetric flow and heat transfer of a second grade fluid past astretching sheet by Hayat and Sajid [17], to the generalized Hirota-Satsuma coupled KdV equa-tion by Abbasbandy [18], to the influence of thermal radiation on MHD flow of a second gradefluid by Hayat et al. [19]. All of these successful applications verified the validity, effectivenessand flexibility of the HAM.

Recently, a lot of attention has been focused on the studies of linear and nonlinear systemsof partial differential equations (PDEs). Systems of nonlinear partial differential equations arisein many scientific models such as the propagation of shallow water waves and the Brusselatormodel of the chemical reaction-diffusion model. In this work we study the Jaulent-Miodek (JM)equations [20]

ut + uxxx + 3

2vvxxx + 9

2vxvxx − 6uux − 6uvvx − 3

2v2ux = 0,

vt + vxxx − 6vux − 6uvx − 15

2v2vx = 0, (1)

which associate with energy-dependent Schrödinger potential [21–23]. There are many methodsto solve above system, such as F-function method [24], Adomian method [25], tanh method [26],and the variational iteration method [27]. But the motivation of this article is to extend the HAM tosolve system of the Jaulent-Miodek (JM) equations. In this way, we obtain approximate solutionswith high-accuracy and minimal calculation. The layout of the article is as follows: In Section II,the basic concept of the HAM is introduced. In Section III, we extend the application of the HAMto construct approximate solutions for the JM equations. The convergence analysis and numericalexperiments are presented in Section IV.

II. BASIC CONCEPTS OF HAM

Let us consider the following differential equation

N [w(τ)] = 0, (2)

where N is a nonlinear operator, τ denotes independent variable, w(τ) is an unknown function,respectively. For simplicity, we ignore all boundary or initial conditions, which can be treated inthe similar way. By means of generalizing the traditional homotopy method, Liao [5] constructsthe so-called zero-order deformation equation.

(1 − p)L[ϕ(τ ; p) − w0(τ )] = p�N [ϕ(τ ; p)], (3)

Numerical Methods for Partial Differential Equations DOI 10.1002/num

432 RASHIDI, DOMAIRRY, AND DINARVAND

where p ∈ [0, 1] is the embedding parameter, � �= 0 is a non-zero auxiliary parameter, L is anauxiliary linear operator, w0(τ ) is an initial guess of w(τ), ϕ(τ ; p) is a unknown function, respec-tively. It is important, that one has great freedom to choose auxiliary things in HAM. Obviously,when p = 0 and p = 1, it holds

ϕ(τ ; 0) = w0(τ ), ϕ(τ ; 1) = w(τ),

respectively. Thus as p increases from 0 to 1, the solution ϕ(τ ; p) varies from the initial guessw0(τ ) to the solution w(τ). Expanding ϕ(τ ; p) in Taylor series with respect to p, we have

ϕ(τ ; p) = w0(τ ) ++∞∑m=1

wm(τ)pm, (4)

where

wm(τ) = 1

m!∂mϕ(τ ; p)

∂pm

∣∣∣∣p=0

. (5)

If the auxiliary linear operator, the initial guess, and the auxiliary parameter � are so properlychosen, the series (4) converges at p = 1, then we have

w(τ) = w0(τ ) ++∞∑m=1

wm(τ), (6)

which must be one of solutions of original nonlinear equation, as proved by Liao [5]. As � = −1,Eq. (3) becomes

(1 − p)L[ϕ(τ ; p) − w0(τ )] + pN [ϕ(τ ; p)] = 0, (7)

which is used mostly in the homotopy perturbation method, where as the solution obtained directly,without using Taylor series [28, 29].

According to the definition (5), the governing equation can be deduced from the zero-orderdeformation equation (3). Define the vector

�wn = {w0(τ ), w1(τ ), . . . , wn(τ)}.Differentiating equation (3) m times with respect to the embedding parameter p and then settingp = 0 and finally dividing them by m!, we have the so-called mth-order deformation equation

L[wm(τ) − χmwm−1(τ )] = �Rm( �wm−1), (8)

where

Rm( �wm−1) = 1

(m − 1)!∂m−1N [ϕ(τ ; p)]

∂pm−1

∣∣∣∣p=0

, (9)

and

χm ={

0, m ≤ 1,

1, m > 1.

It should be emphasized that wm(τ) for m ≥ 1 is governed by the linear equation (8) withthe linear boundary conditions that come from original problem, which can be easily solved bysymbolic computation software such as Maple and Mathematica.



III. APPLICATION

First we consider the Jaulent–Miodek equations (1), with the initial conditions [26]

u(x, 0) = 1

8λ2

(1 − 4sec h2

[1

2λx

]),

v(x, 0) = λsec h

[1

2λx

], (10)

where λ is arbitrary constant. For application of the homotopy analysis method, we choose thelinear operator

L[ϕ(x, t ; p)] = ∂ϕ(x, t ; p)

∂t, (11)

with the property

L(c) = 0, (12)

where c is constant. From (1), we define a system of nonlinear operators as

N1[ϕ1(x, t ; p), ϕ2(x, t ; p)] = ∂ϕ1(x, t ; p)

∂t+ ∂3ϕ1(x, t ; p)

∂x3+ 3

2ϕ2(x, t ; p)

∂3ϕ2(x, t ; p)

∂x3

+ 9

2

∂ϕ2(x, t ; p)

∂x

∂2ϕ2(x, t ; p)

∂x2− 6ϕ1(x, t ; p)

∂ϕ1(x, t ; p)

∂x

−6ϕ1(x, t ; p)ϕ2(x, t ; p)∂ϕ2(x, t ; p)

∂x− 3

2(ϕ2(x, t ; p))2 ∂ϕ1(x, t ; p)

∂x,

N2[ϕ1(x, t ; p), ϕ2(x, t ; p)] = ∂ϕ2(x, t ; p)

∂t+ ∂3ϕ2(x, t ; p)

∂x3− 6ϕ2(x, t ; p)

∂ϕ1(x, t ; p)

∂x

− 6ϕ1(x, t ; p)∂ϕ2(x, t ; p)

∂x− 15

2(ϕ2(x, t ; p))2 ∂ϕ2(x, t ; p)

∂x. (13)

Using the above definition, we construct the zero-order deformation equations

(1 − p)L[ϕ1(x, t ; p) − u0(x, t)] = p�1N1[ϕ1(x, t ; p), ϕ2(x, t ; p)],(1 − p)L[ϕ2(x, t ; p) − v0(x, t)] = p�2N2[ϕ1(x, t ; p), ϕ2(x, t ; p)]. (14)

Obviously, whenp = 0 andp = 1,

ϕ1(x, t ; 0) = u0(x, t), ϕ1(x, t ; 1) = u(x, t),

ϕ2(x, t ; 0) = v0(x, t), ϕ2(x, t ; 1) = v(x, t).

Differentiating the zero-order deformation equations (14) m times with respect to p, and finallydividing by m!, we have the mth-order deformation equations

L[um(x, t) − χmum−1(x, t)] = �1R1,m(�um−1, �vm−1),

L[vm(x, t) − χmvm−1(x, t)] = �2R2,m(�um−1, �vm−1), (15)



subject to initial conditions

um(x, 0) = 0,

vm(x, 0) = 0, (16)

where

R1,m(�um−1, �vm−1) = ∂um−1(x, t)

∂t+ ∂3um−1(x, t)

∂x3+

m−1∑k=0

[3

2vk(x, t)

∂3vm−1−k(x, t)

∂x3+ 9

2

∂vk(x, t)

∂x

× ∂2vm−1−k(x, t)

∂x2− 6uk(x, t)

∂um−1−k(x, t)

∂x+

k∑n=0

(− 6un(x, t)vk−n(x, t)

×∂vm−1−k(x, t)

∂x− 3

2vn(x, t)vk−n(x, t)

∂um−1−k(x, t)

∂x

)],

R2,m(�um−1, �vm−1) = ∂vm−1(x, t)

∂t+ ∂3vm−1(x, t)

∂x3+

m−1∑k=0

[−6vk(x, t)

∂um−1−k(x, t)

∂x

− 6uk(x, t)∂vm−1−k(x, t)

∂x+

k∑n=0

(−15

2vn(x, t)vk−n(x, t)

∂vm−1−k(x, t)

∂x

)],

(17)

and

χm ={

0, m ≤ 1,1, m > 1.

Obviously, the solution of the mth-order deformation equations (15) for m ≥ 1 becomes

um(x, t) = χmum−1(x, t) + �1L−1[R1,m(�um−1, �vm−1)],

vm(x, t) = χmvm−1(x, t) + �2L−1[R2,m(�um−1, �vm−1)]. (18)

For simplicity, we suppose �1 = �2 = �.We choose the initial approximations

u0(x, t) = u(x, 0) = 1

8λ2

(1 − 4sec h2

[1

2λx

]),

v0(x, t) = v(x, 0) = λsec h

[1

2λx

], (19)

where λ is arbitrary constant. From (18) and (19), we now successively obtain

u1(x, t) = − 1

4λ5

�tsec h2

[1

2λx

]tan h

[1

2λx

],

u2(x, t) = − 1

32λ5

�tsec h4

[1

2λx

](λ3

�t(−2 + cos h[λx]) + 4(1 + �) sin h[λx]),

u3(x, t) = − 1

192λ5

�tsec h4

[1

2λx

] (12λ3

�(1 + �)t cos h[λx] + (24(1 + �)2



+ λ6�

2t2) sin h[λx] − 6λ3�t

(4 + 4� + λ3

�t tan h

[1

2λx

])),

u4(x, t) = 1

1536λ5

�tsec h2

[1

2λx

] (−288λ3

�(1 + �)2t − 2λ9�

3t3 − 15λ9�

3t3sec h4

[1

2λx

]

− 48(1 + �)(8(1 + �)2 + λ6�

2t2) tan h

[1

2λx

]+ 3λ3

�tsec h2

[1

2λx

]

×(

144(1 + �)2 + 5λ6�

2t2 + 48λ3�(1 + �)t tan h

[1

2λx

])), (20.1)

and

v1(x, t) = 1

4λ4

�tsec h

[1

2λx

]tan h

[1

2λx

],

v2(x, t) = 1

64λ4

�tsec h3

[1

2λx

](λ3

�t(−3 + cos h[λx]) + 8(1 + �) sin h[λx]),

v3(x, t) = 1

768λ4

�tsec h3

[1

2λx

] (24λ3

�(1 + �)t cos h[λx] + (96(1 + �)2 + λ6�

2t2) sin h[λx]

− 12λ3�t

(6 + 6� + λ3

�t tan h

[1

2λx

])),

v4(x, t) = 1

6144λ4

�tsec h

[1

2λx

] (576λ3

�(1 + �)2t + λ9�

3t3 + 24λ9�

3t3sec h4

[1

2λx

]

+ 48(1 + �)(32(1 + �)2 + λ6�

2t2) tan h

[1

2λx

]+ 4λ3

�tsec h2

[1

2λx

]

×(

−288(1 + �)2 − 5λ6�

2t2 − 72λ3�(1 + �)t tan h

[1

2λx

])). (20.2)

Therefore, the five-term approximate solutions are given by

u(x, t) =4∑

i=0

ui(x, t),

v(x, t) =4∑

i=0

vi(x, t). (21)

IV. CONVERGENCE ANALYSIS AND NUMERICAL EXPERIMENTS

The series solutions of the functions u(x, t) and v(x, t) are given in Eq. (21). The convergenceof these series and rate of the approximation for the homotopy analysis method strongly dependsupon the value of the auxiliary parameter �, as pointed out by Liao [5]. In general, by means of theso-called �-curve, it is straightforward to choose a proper value of � to control the convergenceof the approximation series. To find the range of admissible values of �, �-curves of ut(1, 0),and vt (1, 0) obtained by the five-term approximation of the HAM are plotted in Fig. 1. From this



FIG. 1. The �-curves of ut (1, 0) and vt (1, 0) obtained by the five-term approximation of the HAM, whenλ = 0.5.

figure, the valid regions of � correspond to the line segments nearly parallel to the horizontalaxis.

As mentioned earlier, the HAM is rather general and contains the homotopy perturbationmethod (HPM) [7] and Adomian decomposition method (ADM) [9]. In actual, the results of theHPM and ADM can be obtained as a special case of the HAM, when � = −1. We should notethat if one chooses a good enough initial guess and good enough auxiliary linear operator, one canget accurate approximations by only a few terms with � = −1. However, even if the initial guessand auxiliary linear operator are not good enough but reasonable, one can still get convergentresults by properly choosing the auxiliary parameter �. Furthermore, in some cases, for a strongnonlinearity of governing equation, � = −1 might not give convergent series solution and weshould adjust the convergence of the series solution by means of the auxiliary parameter �. Theseadvantages verify the validity and potential of the HAM for the study of nonlinear problems.

TABLE I. The absolute errors for u(x, t) and v(x, t) obtained by the five-term approximation of the HAMfor � = −1, when λ = 0.1.

t

x 0.1 1 10 50 100

|uExact − uHAM| 0.1 4.3668 E – 19 4.3668 E – 19 6.5052 E – 18 2.4492 E – 14 1.0139 E – 121 4.3668 E – 19 4.3668 E – 19 5.4210 E – 17 1.7777 E – 13 5.9089 E – 12

10 4.3668 E – 19 4.3668 E – 19 1.4745 E – 16 4.5784 E – 13 1.4516 E – 1150 2.1684 E –19 0.0000 E – 00 1.9516 E – 18 6.5765 E – 15 2.1010 E – 13

100 2.1684 E – 19 0.0000 E – 00 2.1684 E – 19 7.3075 E – 17 2.3384 E – 15|vExact − vHAM| 0.1 1.3878 E – 17 0.0000 E – 00 2.7756 E – 17 1.0987 E – 13 4.5483 E – 12

1 0.0000 E – 00 1.3878 E – 17 2.3592 E – 16 7.9969 E – 13 2.6589 E – 1110 1.3878 E – 17 0.0000 E – 00 9.2981 E – 16 2.9088 E – 12 9.2661 E – 1150 6.9389 E – 18 3.4694 E – 18 3.4694 E – 18 2.0612 E – 14 6.5081 E – 13

100 1.3010 E – 18 6.5052 E – 19 1.7347 E – 18 3.3825 E – 15 1.0802 E – 13



FIG. 2. The behavior of: (a) u(x, t), (c) v(x, t) obtained by the five-term approximation of the HAM for� = −0.9; (b) u(x, t), (d) v(x, t) obtained by the exact solutions (22), when λ = 0.5.

To demonstrate the efficiency of the HAM for JM equations, we compare approximate solutionsof u(x, t) and v(x, t), with exact solutions [26]

u(x, t) = 1

8λ2

(1 − 4sec h2

[1

2λ

(x + 1

2λ2t

)]),

v(x, t) = λsec h

[1

2λ

(x + 1

2λ2t

)], (22)

where λ is arbitrary constant. Table I shows the absolute errors for differences between the exactsolutions (22) and the approximate solutions (21) obtained by the HAM, at some points. Besides,the behavior of the exact and approximate solutions are illustrated in Fig. 2. A very good agree-ment between the results of the HAM and exact solutions is observed, which confirms the validityof the HAM.

V. CONCLUSIONS

In this work, HAM was used for finding the approximate solutions of the JM equations. TheHAM yields a very rapid convergence of the solution series in most cases, usually only a fewiterations leading to very accurate solutions. The HAM provides us with a convenient way tocontrol the convergence of approximation series which is a fundamental qualitative difference in



analysis between HAM and other methods. Thus the auxiliary parameter � plays an important rolewithin the frame of HAM which can be determined by the so-called �-curve. This work verifiesthe validity and potential of HAM for the studies of nonlinear systems. Briefly speaking, Liao’sHAM is a universal one which can solve various kinds of nonlinear equations.

References

1. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D.Thesis, Shanghai Jiao Tong University, 1992.

2. S. Liao, A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J FluidMech 385 (1999), 101–128.

3. S. J. Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int J NonlinearMech 34 (1999), 759–778.

4. S. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over astretching sheet, J Fluid Mech 488 (2003), 189–212.

5. S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, Chapman and Hall/CRCPress, Boca Raton, 2003.

6. S. Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput 147 (2004),499–513.

7. S. J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method,Appl Math Comput 169 (2005), 1186–1194.

8. M. Sajid, T. Hayat, and S. Asghar, Comparison between the HAM and HPM solutions of thin film flowsof non-Newtonian fluids on a moving belt, Nonlinear Dyn 50 (2007), 27–35.

9. F. M. Allan, Derivation of the Adomian decomposition method using the homotopy analysis method,Appl Math Comput 190 (2007), 6–14.

10. T. Hayat and T. Javed, On analytic solution for generalized three-dimensional MHD flow over a porousstretching sheet, Phys Lett A 370 (2007), 243–250.

11. T. Hayat, N. Ahmed, M. Sajid, and S. Asghar, On the MHD flow of a second grade fluid in a porouschannel, Comput Math Appl 54 (2007), 407–414.

12. M. Sajid, A. M. Siddiqui, and T. Hayat, Wire coating analysis using MHD Oldroyd 8-constant fluid, IntJ Eng Sci 45 (2007), 381–392.

13. T. Hayat, S. B. Khan, M. Sajid, and S. Asghar, Rotating flow of a third grade fluid in a porous spacewith Hall current, Nonlinear Dyn 49 (2007), 83–91.

14. M. M. Rashidi, G. Domairry, and S. Dinarvand, Approximate solutions for the Burger and regularizedlong wave equations by means of the homotopy analysis method, Commun Nonlinear Sci Numer Simul,in press, 2007 Oct 17, (Epub ahead of print).

15. M. Sajid, T. Hayat, and S. Asghar, Non-similar analytic solution for MHD flow and heat transfer in athird-order fluid over a stretching sheet, Int J Heat and Mass Transfer 50 (2007), 1723–1736.

16. S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heattransfer, Phys Lett A 360 (2006), 109–113.

17. T. Hayat and M. Sajid, Analytic solution for axisymmetric flow and heat transfer of a second grade fluidpast a stretching sheet, Int J Heat Mass Transfer 50 (2007), 75–84.

18. S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota-Satsumacoupled KdV equation, Phys Lett A 361 (2007), 478–483.

19. T. Hayat, Z. Abbas, M. Sajid, and S. Asghar, The influence of thermal radiation on MHD flow of asecond grade fluid, Int J Heat Mass Transfer 50 (2007), 931–941.



20. M. Jaulent and J. Miodek, Nonlinear evolution equations associated with energy-dependent Schrödingerpotentials, Lett Math Phys 1 (1976), 243–250.

21. H. T. Ozer and S. Salihoglu, Nonlinear Schrödinger equations and N = 1 superconformal algebra,Chaos Solitons Fractals 33 (2007), 1417–1423.

22. J. L. Zhang, M. L. Wang, and X. R. Li, The subsidiary elliptic-like equation and the exactsolutions of the higher-order nonlinear Schrödinger equation, Chaos Solitons Fractals 33 (2007),1450–1457.

23. J. M. Zhu and Z. Y. Ma, Exact solutions for the cubic-quintic nonlinear Schrödinger equation, ChaosSolitons Fractals 33 (2007), 958–964.

24. E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematicalphysics, Chaos Solitons Fractals 16 (2003), 819–839.

25. J. B. Chen and X. G. Geng, Decomposition to the modified Jaulent–Miodek hierarchy, Chaos SolitonsFractals 30 (2006), 797–803.

26. A.-M. Wazwaz, The tanh-coth and the sech methods for exact solutions of the Jaulent–Miodek equation,Phys Lett A 366 (2007), 85–90.

27. D. D. Ganji, M. Jannatabadi, and E. Mohseni, Application of He’s variational iteration method tononlinear Jaulent–Miodek equations and comparing it with ADM, J Comput Appl Math 207 (2007),35–45.

28. J. H. He, Homotopy perturbation method for solving boundary value problems, Phys Lett A 350 (2006),87–88.

29. J. H. He, Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B 20 (2006),1141–1199.


the homotopy analysis method for explicit analytical solutions of jaulent-miodek equations

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