2.BasicIdeaoftheVIM 1.Introduction LanXu andEricW.M.Lee VariationalIterationMethodfortheMagnetohydrodynamicFlowoveraNonlinearStretchingSheet ResearchArticle

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Volume 2013, Article ID 573782,5pages http://dx.doi.org/10.1155/2013/573782

Research Article

Variational Iteration Method for the Magnetohydrodynamic Flow over a Nonlinear Stretching Sheet

Lan Xu

^1,2

and Eric W. M. Lee

³

1National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou, Jiangsu 215123, China

2Nantong Textile Institute of Soochow University, 58 Chong-chuan Road, Nantong, Jiangsu 226018, China

3Department of Civil and Architectural Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong

Correspondence should be addressed to Eric W. M. Lee; [email protected] Received 10 December 2012; Accepted 22 February 2013

Academic Editor: de Dai

Copyright © 2013 L. Xu and E. W. M. Lee. 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.

The variational iteration method (VIM) is applied to solve the boundary layer problem of magnetohydrodynamic flow over a nonlinear stretching sheet. The combination of the VIM and the Pad´e approximants is shown to be a powerful method for solving two-point boundary value problems consisting of systems of nonlinear differential equations. And the comparison of the obtained results with other available results shows that the method is very effective and convenient for solving boundary layer problems.

1. Introduction

It is well known that most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential equations. Recent advances of partial differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. There are many tra- ditional and recently developed methods to give numerical and analytical approximate solutions of nonlinear differential equations such as Euler method, Runge-Kutta method, Taylor series method, Adomian decomposition method [1], Variational iteration method [2,3], Hankel-Pad´e method [4], DTM-Pad´e method [5], homotopy perturbation method [6], and Hamiltonian method [7].

In this paper, we consider the model proposed by authors in [1] describing the problem of the boundary layer flow of an incompressible viscous fluid over a nonlinear stretching sheet. The boundary layer flow is often encountered in many engineering and industrial processes. Such processes include the aerodynamic extrusion of plastic sheets, hot rolling, glass fiber production, and so on [1, 4, 5]. And various aspects of the stretching flow problem were discussed by various

investigators. Chiam [8] analyzed the MHD flow of a viscous fluid bounded by a stretching surface with power law velocity.

He presented the numerical solution of the boundary value problem by utilizing the Runge-Kutta shooting algorithm with Newton iteration. Here, we aim to solve the MHD flow caused by a sheet with nonlinear stretching. The approximate solution of the nonlinear problem is obtained by the variational iteration method.

The variational iteration method [2] is a type of Lagrange multiplier method to find analytical solutions. The method gives the possibility to solve many kinds of non linear equations. In this method, general Lagrange multipliers are introduced to construct correction functional for the problems. The multipliers can be identified optimally via variational theory. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximation [9].

2. Basic Idea of the VIM

The basic idea was systematically illustrated and discussed in [9,10]. To illustrate the basic idea of the VIM, we consider the

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following general nonlinear system:

𝐿 [𝑢 (𝑡)] + 𝑁 [𝑢 (𝑡)] = 𝑔 (𝑡) , (1) where𝐿,𝑁, and𝑔(𝑡)are the linear operator, the nonlinear operator, and a given continuous function, respectively. The basic character of the method is to construct a correction functional for the system, which reads

𝑢_𝑛+1(𝑡) = 𝑢_𝑛(𝑡) + ∫^𝑡

0𝜆 (𝑠) [𝐿𝑢_𝑛(𝑠) + 𝑁̃𝑢_𝑛(𝑠) − 𝑔 (𝑠)] 𝑑𝑠, (2) where 𝜆 is a Lagrange multiplier which can be identified optimally via the variational theory. The subscript𝑛indicates the𝑛th approximation, and̃𝑢_𝑛denotes a restricted variation, that is,𝛿̃𝑢_𝑛= 0.

3. Problem Statement and Governing Equations

We consider the magnetohydrodynamic (MHD) flow of an incompressible viscous fluid over a stretching sheet at𝑦 = 0. The fluid is electrically conducting under the influence of an applied magnetic field𝐵(𝑥)normal to the stretching sheet. The induced magnetic field is neglected. The resulting boundary layer equations are as follows [1]:

𝜕𝑢

𝜕𝑥+ 𝜕V

𝜕𝑦 = 0, (3)

𝑢𝜕𝑢

𝜕𝑥+V𝜕𝑢

𝜕𝑦 =]𝜕²𝑢

𝜕𝑦² −𝜎𝐵²(𝑥)

𝜌 𝑢, (4)

where𝑢andVare the velocity components in the𝑥and𝑦 directions, respectively,]is the kinematic viscosity,𝜌is the fluid density, and𝜎is the electrical conductivity of the fluid.

In (4), the external electric field and the polarization effects are negligible, and in [8]

𝐵 (𝑥) = 𝐵₀𝑥^(𝑛−1)/2. (5) The boundary conditions corresponding to the nonlinear stretching of a sheet are

𝑢 (𝑥, 0) = 𝑐𝑥^𝑛, V(𝑥, 0) = 0,

𝑢 (𝑥, 𝑦) 󳨀→ 0 as𝑦 󳨀→ ∞. (6) Upon making use of the following substitutions:

𝜂 = √𝑐 (𝑛 + 1)

2] 𝑥^(𝑛−1)/2𝑦, 𝑢 = 𝑐𝑥^𝑛𝑓^󸀠(𝜂) , (7) V= −√𝑐](𝑛 + 1)

2 𝑥^(𝑛−1)/2[ 𝑓 (𝜂) +𝑛 − 1

𝑛 + 1𝜂𝑓^󸀠(𝜂)] , (8) Substituting (8) into (3)–(6), the resulting nonlinear differential system can be written in the following form:

𝑓^󸀠󸀠󸀠+ 𝑓𝑓^󸀠󸀠− 𝛽𝑓^󸀠2− 𝑀𝑓^󸀠= 0, (9) 𝑓 (0) = 0, 𝑓^󸀠(0) = 1, 𝑓^󸀠(∞) = 0, (10)

where

𝛽 = 2𝑛

1 + 𝑛, 𝑀 = 2𝜎𝐵²₀

𝜌𝑐 (1 + 𝑛). (11) The parameter 𝛽 is a measure of the pressure gradient, and 𝑀is the magnetic parameter. Positive 𝛽 denotes the favorable negative pressure gradient, and negative𝛽denotes the unfavorable positive pressure gradient; naturally,𝛽 = 0 denotes the flat plate. For the special case of𝛽 = 1, the exact analytical solution of (9) is [11]

𝑓 (𝜂) = 1 −exp(−√1 + 𝑀𝜂)

√1 + 𝑀 . (12)

4. Approximate Solution by the VIM

In order to obtain VIM solution of (9), we construct a correction functional which reads

𝑓_𝑛+1(𝜂)

= 𝑓_𝑛(𝜂) + ∫^𝜂

0 𝜆 (𝜏) [𝜕³𝑓_𝑛(𝜏)

𝜕𝜏³ + ̃𝑓_𝑛(𝜏)𝜕²̃𝑓_𝑛(𝜏)

𝜕𝜏² − 𝛽

× (𝜕̃𝑓_𝑛(𝜏)

𝜕𝜏 )

2

− 𝑀𝜕̃𝑓_𝑛(𝜏)

𝜕𝜏 ] ]

𝑑𝜏, (13) where𝜆(𝜏)is the general Lagrangian multiplier which can be identified optimally via the variational theory. And̃𝑓_𝑛(𝜏)is considered as a restricted variation, that is,𝛿̃𝑓_𝑛(𝜏) = 0. We omit asterisks for simplicity. Its stationary conditions can be obtained as follows:

1 + 𝜆^󸀠󸀠(𝜏)󵄨󵄨󵄨󵄨󵄨𝜏=𝜂= 0, 𝜆^󸀠(𝜏)󵄨󵄨󵄨󵄨󵄨𝜏=𝜂= 0, 𝜆^󸀠󸀠󸀠(𝜏) = 0.

(14) The Lagrange multipliers can be readily identified as the following form:

𝜆 (𝜏) = −1

2(𝜏 − 𝜂)². (15) As a result, we obtain the following variational iteration formula

𝑓_𝑛+1(𝜂)

= 𝑓_𝑛(𝜂) − 1 2∫^𝜂

0 (𝜏 − 𝜂)²[𝜕³𝑓_𝑛(𝜏)

𝜕𝜏³ + ̃𝑓_𝑛(𝜏)𝜕²̃𝑓_𝑛(𝜏)

𝜕𝜏² − 𝛽

× (𝜕̃𝑓_𝑛(𝜏)

𝜕𝜏 )

2

− 𝑀𝜕̃𝑓_𝑛(𝜏)

𝜕𝜏 ] ]

𝑑𝜏.

(16) Now, we assume that an initial approximation

𝑓₀(𝜂) = 𝑎 + 𝑏𝜂 + 𝑐𝜂². (17)

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where 𝑎, 𝑏, and 𝑐 are unknown constants to be further determined.

By the iteration formula (16) and the initial approximation (17), we can obtain directly the first-order approximate solution as follows:

𝑓₁(𝜂) = 𝑓₀(𝜂) −1 2∫^𝜂

0 (𝜏 − 𝜂)²

× [𝜕³𝑓₀(𝜏)

𝜕𝜏³ + 𝑓₀(𝜏)𝜕²𝑓₀(𝜏)

𝜕𝜏²

−𝛽(𝜕𝑓₀(𝜏)

𝜕𝜏 )²− 𝑀𝜕𝑓₀(𝜏)

𝜕𝜏 ] 𝑑𝜏

= 𝑎 + 𝑏𝜂 + 𝑐𝜂²− 𝑐²

30𝜂⁵+𝑏𝑀

6 𝜂³+𝑐𝑀 12𝜂⁴𝜂⁵ +𝛽𝑏²

6 𝜂³+𝛽𝑐² 15 −𝑎𝑐

3 𝜂³− 𝑏𝑐

12𝜂⁴+𝑏𝑐𝛽 6 𝜂⁴

= 𝑎 + 𝑏𝜂 + 𝑐𝜂²+𝑏𝑀 + 𝛽𝑏²− 2𝑎𝑐

6 𝜂³

+𝑐𝑀 + 𝑏𝑐 (2𝛽 − 1)

12 𝜂⁴+(2𝛽 − 1) 𝑐² 30 𝜂⁵.

(18) Making use of the initial conditions𝑓(0) = 0, 𝑓^󸀠(0) = 1, we can readily obtain the results as follows:

𝑎 = 0, 𝑏 = 1, 𝑐 = 1

2𝑓^󸀠󸀠(0) , (19) where𝑓^󸀠󸀠(0) = 𝛼will be examined in this work, according the initial condition𝑓^󸀠(∞) = 0.

Then, 𝑓₁(𝜂) = 𝜂 + 1

2𝛼𝜂²+𝑀 + 𝛽

6 𝜂³+𝛼 (𝑀 + 2𝛽 − 1) 24 𝜂⁴ +(2𝛽 − 1) 𝛼²

120 𝜂⁵.

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And the following second-order approximate solution can be obtained

𝑓₂(𝜂) = 𝑓₁(𝜂) −1 2∫^𝜂

0 (𝜏 − 𝜂)²

× [𝜕³𝑓₁(𝜏)

𝜕𝜏³ + 𝑓₁(𝜏)𝜕²𝑓₁(𝜏)

𝜕𝜏²

−𝛽(𝜕𝑓₁(𝜏)

𝜕𝜏 )²− 𝑀𝜕𝑓₁(𝜏)

𝜕𝜏 ] 𝑑𝜏

= 𝜂 + 1

2𝛼𝜂²+𝑀 + 𝛽

6 𝜂³+𝛼 (𝑀 + 2𝛽 − 1) 24 𝜂⁴

+ [(2𝛽 − 1) 𝛼² 120 +𝛽²

60+𝛽𝑀 40 − 1

60 +𝑀² 120−𝑀

60] 𝜂⁵ + (𝛽²𝛼

72 +𝛽𝑀𝛼 72 −𝛽𝛼

60 +𝑀²𝛼 720 −𝑀𝛼

90 + 𝛼 240) 𝜂⁶ + ( 𝛽³

840 +𝛽²𝑀 420 +𝛽²𝛼²

252 − 𝛽²

1260+𝛽𝑀²

840 +𝛽𝑀𝛼² 504

−𝛽𝑀 630 −2𝛽𝛼²

315 − 𝑀²

1260−𝑀𝛼² 630 +11𝛼²

5040) 𝜂⁷ + (𝛽³𝛼

1008+𝛽²𝑀𝛼 672 +𝛽²𝛼³

2016−5𝛽²𝛼

4032+𝛽𝑀²𝛼 2016

−13𝛽𝑀𝛼 8064 −7𝛽𝛼³

1260 + 𝛽𝛼

2688−𝑀²𝛼 2688+ 𝑀𝛼

2688 +11𝛼³

40320) 𝜂⁸ + (𝛽³𝛼²

2592+𝛽²𝑀𝛼²

2592 −37𝛽²𝛼²

60480 +𝛽𝑀²𝛼² 18144

−13𝛽𝑀𝛼²

25920 + 53𝛽𝛼²

181440− 𝑀²𝛼² 24192+𝑀𝛼²

6480

− 𝛼² 24192) 𝜂⁹ + ( 𝛽³𝛼³

12960+𝛽²𝑀𝛼³ 25920 −𝛽²𝛼³

7200−13𝛽𝑀𝛼³ 259200 + 7𝛽𝛼³

86400 +𝑀𝛼³ 64800− 𝛼³

64800) 𝜂¹⁰ + ( 𝛽³𝛼⁴

142560− 𝛽²𝛼⁴

79200+ 7𝛽𝛼⁴

950400− 𝛼⁴

712800) 𝜂¹¹. (21) Therefore, according to (13), we can easily obtain higher- order approximate solution as follows:

𝑓 (𝜂) = 𝑟₀+ 𝑟₁𝜂 + 𝑟₂𝜂²+ 𝑟₃𝜂³+ 𝑟₄𝜂⁴+ 𝑟₅𝜂⁵+ ⋅ ⋅ ⋅ , (22) by using mathematical software such as MATLAB.

It is evident that the main problem for solving (21) is to obtain the value of𝑓^󸀠󸀠(0), then we can resort to any numerical integration routine to obtain the solution of the problem. For this purpose, we will employ the Pad´e method to determine this unknown value with high accuracy.

5. Padé Approximation

It is well known that Pad´e approximations [12] have the advantage of manipulating the polynomial approximation into a rational function of polynomials. This manipulation provides us with more information about the mathematical behavior of the solution. Besides that, power series are not

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Table 1: Comparison of the values of 𝑓^󸀠󸀠(0) obtained by the variational iteration method and other methods [1] for various values ofMwhen𝛽 = 1.

𝑀 VIM ADM [1] Exact [1]

1.0 −1.41421 −1.41421 −1.41421

5.0 −2.44948 −2.44948 −2.44948

10.0 −3.31662 −3.31662 −3.31662

50.0 −7.14142 −7.14142 −7.14142

100.0 −10.04987 −10.04987 −10.04987

500.0 −22.38302 −22.38302 −22.38302

Table 2: Comparison of the values of𝑓^󸀠󸀠(0)obtained by the variational iteration method and the modified Adomian decomposition method [1] for various values of𝛽andM.

𝑀 𝛽 = −1.5 𝛽 = 1.5 𝛽 = 5

VIM ADM [1] VIM ADM [1] VIM ADM [1]

1.0 −0.6530 −0.6532 −1.5253 −1.5252 −2.1529 −2.1528 5.0 −2.0852 −2.0852 −2.5162 −2.5161 −2.9414 −2.9414 10 −3.0562 −3.0562 −3.3663 −3.3663 −3.6957 −3.6956 50 −7.0239 −7.0239 −7.1647 −7.1647 −7.3256 −7.3256 100 −9.9667 −9.9666 −10.0776 −10.0776 −10.1816 −10.1816 500−22.3458 −22.3457 −22.3905 −22.3904 −22.4426 −22.4425

useful for large values of𝜂, say𝜂 = ∞. This can be attributed to the possibility that the radius of convergence may not be sufficiently large to contain the boundaries of the domain.

Therefore, the combination of the series solution through the decomposition method or any other series solution method with the Pad´e approximation provides an effective tool for handling boundary value problems on infinite or semi-infinite domains. Furthermore, it is noted that Pad´e approximants can be easily evaluated by using Matlab.

Therefore, we suppose that the solution 𝑓(𝜂) can be expanded as a Taylor series about𝜂 = 0

𝑓 (𝜂) =∑^∞

𝑗=0

𝑓_𝑗𝜂^𝑗. (23)

Pad´e approximant, symbolized by[𝑆/𝑁], is a rational function defined by

[𝑆

𝑁] (𝜂) = ∑^𝑆_𝑗=0𝑝_𝑗𝜂^𝑗

∑^𝑁_𝑗=0𝑞_𝑗𝜂^𝑗. (24) If we selected 𝑆 = 𝑁, then the approximants [𝑁/𝑁]

are called diagonal approximants. More importantly, the diagonal approximants are the most accurate approximants;

therefore, we have to construct only diagonal approximants.

Then,

𝑝₀+ 𝑝₁𝜂 + 𝑝₂𝜂²+ 𝑝₃𝜂³+ ⋅ ⋅ ⋅ + 𝑝_𝑁𝜂^𝑁 𝑞₀+ 𝑞₁𝜂 + 𝑞₂𝜂¹+ 𝑞₃𝜂³+ ⋅ ⋅ ⋅ + 𝑞_𝑁𝜂^𝑁

= 𝑟₀+ 𝑟₁𝜂 + 𝑟₂𝜂²+ 𝑟₃𝜂³+ 𝑟₄𝜂⁴+ ⋅ ⋅ ⋅ .

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0 0.5 1 1.5 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

The exact solution

The approximate solution by VIM

Figure 1: Comparison between the approximate solution by the VIM and exact solution for𝛽 = 1and𝑀 = 10.

By using cross multiplication in (25), we find 𝑝₀+ 𝑝₁𝜂 + 𝑝₂𝜂²+ 𝑝₃𝜂³+ ⋅ ⋅ ⋅ + 𝑝_𝑁𝜂^𝑁

= 𝑟₀𝑞₀+ (𝑟₁𝑞₀+ 𝑞₁𝑟₀) 𝜂 + (𝑟₂𝑞₀+ 𝑞₁𝑟₁+ 𝑞₂𝑟₀) 𝜂² + (𝑟₃𝑞₀+ 𝑞₁𝑟₂+ 𝑞₂𝑟₁+ 𝑞₃𝑟₀) 𝜂³+ ⋅ ⋅ ⋅ .

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Using the boundary condition 𝑓^󸀠(∞) = 0, the diagonal approximant[𝑁/𝑁]vanishes if the coefficient of𝜂with the highest power in the numerator vanishes. By putting the coefficients of the highest power of𝜂equal to zero, we can easily obtain the values of 𝑓^󸀠󸀠(0) listed in Tables 1 and 2 andFigure 1, using Matlab. The order of Pad´e approximation [12/12] has sufficient accuracy; on the other hand, if the order of Pad´e approximation increases, the accuracy of the solution increases.

Substituting (21) and the value of𝑓^󸀠󸀠(0)into (8), we can easily obtain the second-order approximate solution of (3)- (4).

6. Conclusion

In this paper, the variational iteration method is used to obtain approximate solutions of magnetohydrodynamics boundary layer equations. The analytical solutions of the governing nonlinear boundary layer problem are obtained.

Without using the Pad´e approximation, the analytical solution that were obtained by the VIM cannot satisfy the boundary condition at infinity𝑓^󸀠(∞) = 0. The combination of the VIM and the Pad´e approximants is shown to be a powerful method for solving two-point boundary value problems consisting of systems of nonlinear differential equations. And the obtained solutions are in good agreement with exact values.

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Acknowledgments

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (CityU 116308) and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 12KJB130002).

References

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[2] J. H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,”International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.

[3] L. Xu, “Variational iteration method for solving integral equations,”Computers & Mathematics with Applications, vol. 54, no.

7-8, pp. 1071–1078, 2007.

[4] S. Abbasbandy and H. R. Ghehsareh, “Solutions of the magnetohydrodynamic flow over a nonlinear stretching sheet and nano boundary layers over stretching surfaces,”International Journal for Numerical Methods in Fluids, vol. 70, pp. 1324–1340, 2012.

[5] M. M. Rashidi, “The modified differential transform method for solving MHD boundary-layer equations,”Computer Physics Communications, vol. 180, no. 11, pp. 2210–2217, 2009.

[6] J.-H. He, “Homotopy perturbation method with an auxiliary term,” Abstract and Applied Analysis, vol. 2012, Article ID 857612, 7 pages, 2012.

[7] L. Xu and J. H. He, “Determination of limit cycle by Hamilto- nian approach for strongly nonlinear oscillators,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 12, pp. 1097–1101, 2010.

[8] T. C. Chaim, “Hydromagnetic flow over a surface stretching with a power-law velocity,”International Journal of Engineering Science, vol. 33, pp. 429–435, 1995.

[9] J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,”Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007.

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[11] K. B. Pavlov, “Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a surface,”Magnitnaya Gidrodinamika, vol. 4, pp. 146–147, 1975.

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2.BasicIdeaoftheVIM 1.Introduction LanXu andEricW.M.Lee VariationalIterationMethodfortheMagnetohydrodynamicFlowoveraNonlinearStretchingSheet ResearchArticle

Research Article

Variational Iteration Method for the Magnetohydrodynamic Flow over a Nonlinear Stretching Sheet

Lan Xu

and Eric W. M. Lee

1. Introduction

2. Basic Idea of the VIM

3. Problem Statement and Governing Equations

4. Approximate Solution by the VIM

5. Padé Approximation

6. Conclusion

Acknowledgments

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis