MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation31829810.1155/2010/318298318298Research ArticleModified Variational Iteration Method for Free-Convective Boundary-Layer Equation Using Padé ApproximationMohyud-DinSyed Tauseef1YildirimAhmet2Anıl SezerSefa2UsmanMuhammad3HeJihuan1HITEC UniversityTaxila CanttPakistanau.edu.pk2Department of MathematicsEge UniversityBornova 35100, IzmirTurkeyege.edu.tr3Department of MathematicsUniversity of Dayton300 College ParkDayton, OH 45469-2316USAudayton.edu201014032010201016102009170120102010Copyright © 2010This 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.

This paper is devoted to the study of a free-convective boundary-layer flow modeled by a system of nonlinear ordinary differential equations. We apply a modified variational iteration method (MVIM) coupled with He's polynomials and Padé approximation to solve free-convective boundary-layer equation. It is observed that the combination of MVIM and the Padé approximation improves the accuracy and enlarges the convergence domain.

1. Introduction

The boundary-layer flows of viscous fluids are of utmost importance for industry and applied sciences. These flows can be modeled by systems of nonlinear ordinary differential equations on an unbounded domain, see  and the references therein. Keeping in view the physical importance of such problems, there is a dire need of extension of some reliable and efficient technique for the solution of such problems. He [1, 2, 515] developed the variational iteration (VIM) and homotopy perturbation (HPM) methods which are very efficient and accurate and are [1, 2, 442] being used very frequently for finding the appropriate solutions of nonlinear problems of physical nature. In a later work, Ghorbani and Nadjfi  introduced He’s polynomials which are calculated for He’s homotopy perturbation method. It is also established  that He’s polynomials are compatible with Adomian’s polynomials but are easier to implement and are more user friendly. Recently, Mohyud-Din, Noor and Noor [4, 3336] made the elegant coupling of He’s polynomials and the correction functional of variational iteration method (VIM) and found the solutions of number of nonlinear singular and nonsingular problems. It is observed that [4, 3336] the modified version of VIM is very efficient in solving nonlinear problems. The basic motivation of this paper is the extension of the modified variational iteration method (MVIM) coupled with Padé approximation to solve a free-convective boundary-layer flow modeled by a system of nonlinear ordinary differential equations. Numerical and figurative illustrations show that it is a promising tool to solve nonlinear problems. It needs to be highlighted that Herisanu and Marinca  suggested an optimal variational iteration algorithm. It needs to be highlighted that He in his latest article “The variational iteration method which should be followed”  presented a very comprehensive and detailed study on various aspects of variational iteration method in connection with partial differential equations, ordinary differential equations, fractional differential equations, fractal-differential equations, and difference-differential equations.

2. Modified Variational Iteration Method (MVIM)

To illustrate the basic concept of the modified variational iteration method (MVIM), we consider the following general differential equation:

Lu+Nu=g(x), where L is a linear operator, N is a nonlinear operator, and g(x) is the forcing term. According to variational iteration method [1, 2, 4, 1023, 28, 3339, 41, 42], we can construct a correction functional as follows:

un+1(x)=un(x)+0xλ(ξ)(Lun(ξ)+Nũn(ξ)-g(ξ))dξ, where λ  is a Lagrange multiplier [1, 2, 1015, 42], which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation; ũn is considered as a restricted variation. That is, δũn=0; (2.2) is called a correction functional. Now, we apply He’s polynomials 

n  =  0p(n)un=u0(x)+p0xλ(ξ)(n  =0p(n)L(un)+n  =  0  p(n)N(ũn))dξ-0xλ(ξ)g(ξ)dξ, which is the coupling of variational iteration method and He’s polynomials and is called the modified variational iteration method (MVIM) [4, 3336]. The comparison of like powers of p gives solutions of various orders.

3. Mathematical Model

Let us consider the problem of cooling of a low-heat-resistance sheet that moves downwards in a viscous fluid :

ux+vy=0,uux+vuy=v2uy2+gβ(T-T0),uTx+vTy=κ2Ty2, subject to

u=0,v=0aty=0,u0,TT0asy, where u and v are the velocity components in the x- and y-directions, respectively.  ψT  is the temperature, T0 is the temperature of the surrounding fluid, ν is the kinematic viscosity, κ is the thermal diffusivity, g is the acceleration due to gravity, and β is the coefficient of thermal expansion. Using the similarity variables

ψ=[gβ(T1-T0)v2x03]1/4f(η),T=T0+(T1-T0)[x0(x0-x)]3θ(η),η=[gβ(T1-T0)x03v2]1/4y(x0-x), where ψ is the stream function defined by u=ψ/y and v=-ψ/x, f  and θ are the similarity functions dependent on η,T  (0,0)=T1 and θ(0)=1, (3.1) is transformed to

f(η)+θ(η)-(f'(η))2=0,θ(η)-3σf(η)θ(η)=0, subject to the boundary conditions

f(0)=0,f'(0)=0,f'(+)=0,θ(0)=1,θ(+)=0, where the primes denote differentiation with respect to η, and σ is the Prandtl number.

We denote L,M Padé approximants to f(z) by

[LM]=PL(z)QM(z), where PL(z) is polynomial of degree at most L and QM(z)(QM(z)0) is a polynomial of degree at most M. The former power series is

f(z)=k=0ck·zk. And we write the PL(z) and QM(z) as

PL(z)=p0+p1·z+p2·z2+p3·z3++pL·zL,QM(z)=q0+q1·z+q2·z2+q3·z3++qM·zM, so

f(z)-PL(z)QM(z)=O(zL+M+1)asz0, and the coefficients of PL(z) and QM(z) are determined by the equation. From (4.4), we have

f(z)·QM(z)-PL(z)=O(zL+M+1), which system of L+M+1 homogeneous equations with L+M+2 unknown quantities. We impose the normalization condition

QM(0)=1. We can write out (4.5) as

cL+1+cL·q1++cL-M+1·qM=0,cL+2+cL+1·q1++cL-M+2·qM=0,,cL+M+cL+M-1·q1++cL·qM=0,c0=p0,c1+c0·q1=p1,c2+c1·q1+c0·q2=p2,,cL+cL-1·q1++c0·qL=pL.

From (4.7), we can obtain qi(1iM). Once the values of q1,q2,,qM are all known (4.8) gives an explicit formula for the unknown quantities p1,p2,,pL. For the diagonal approximants like [2/2],[3/3],[4/4],[5/5],or  [6/6] have the most accurate approximants by built-in utilities of Maple.

5. Solution Procedure

Consider problems (3.4)–(3.5) formulated in Section 3 and is related to the free-convective boundary-layer flow.

The correction functional is given by

fn+1(η)=fn(η)+0xλ1(s)(d3fnds3+θ̃n(η)-(df̃ndη)2)ds,θn+1(η)=θn(η)+0xλ2(s)(d2θnds2-3σ(df̃ndη))θ̃n(η)ds. Making the correction functional stationary, the Lagrange multipliers can easily be identified

λ1(s)=-12!  (s-η)2,λ2(s)=(s-η).   Consequently,

fn+1(η)=fn(η)-0x12!(s-η)2(d3fnds3+θn(η)-(dfndη)2)ds,θn+1(η)=θn(η)+0x(s-  η)(d2θnds2  -3σ(dfndη))θn(η)ds. Applying the modified variational iteration method (MVIM), we get

f0+pf1+=f0(η)-  p0x12!  (s-η)2((d3f0ds3+pd3f1ds3+)+(θ0+pθ1+)-(df0dη+pdf1dη+)2)ds,θ0+pθ1+=θ0(η)+0x(s-η)((d2θ0ds2+pd2θ1ds2+)  -3σ(θ0+pθ1+)-(df0dη+pdf1dη+)2)ds. Comparing the coefficient of like powers of p, we get

p0:f0(η)=(α12)η2,p1:f1(η)=  (α12)η2-  (16)η3+(α224)η4+(α1260)η5,p2:f2(η)=(α12)η2-  (16)η3+(α224)η4+(α1260)η5-(σα1240+α1120)η6+(α1α2630+σα1α2120)η7+(α132016)η8,p3:f3(η)=(α12)η2-  (16)η3+(α224)η4+(α1260)η5-(σα1240+α1120)η6+(α1α2630+σα1α2120+σ1680+1840)η7+(α132016+σα23360+α22016)η8+(-α12σ210080+α12σ8640+σα22130240-11α1230240+α2218144)η9+(-α12α214400-19α12σα2604800-α12σ2α240320)η10,,p0:θ0(η)=1+(α2)η,p1:θ1(η)=1+α2η+(σα1α22)η3+(σα1α24)η4,p2:θ2(η)=1+α2η+(σα1α22)η3+(σα1α24)η4-(α1α210)η5+(α12σ220+α12σ120-α22σ60)η6×(α12σα2168+α12α2σ256)η7,p3:θ3(η)=1+α2η+(σα1α22)η3+(σα1α24)η4-(α1α210)η5+(α12σ220+α12σ120-α22σ60)η6+(α12σα2168+α12α2σ256-σα1280-α1σ235)η7+(-11α1σα23360-41α1σ2α22240)η8+(α12σ2α22360+α12σ6048+σ3α13480-α1α22σ2160)η9+(α13α2σα227560+α12σ2α21120+σ3α2α131680)η10,. The series solution is given by

f(η)=(α12)η2-(16)η3+(α224)η4+(α1260)η5-(σα1240+α1120)η6+(α1α2630+σα1α2120+σ1680+1840)η7+(α132016+σα23360+α22016)η8+(-α12σ210080+α12σ8640+σα22130240-11α1230240+α2218144)η9+(-α12α214400-19α12σα2604800-α12σ2α240320)η10+,θ(η)=1+α2η+(σα1α22)η3+(σα1α24)η4-(α1α210)η5+(α12σ220+α12σ120-α22σ60)η6+(α12σα2168+α12α2σ256-σα1280-α1σ235)η7+(-11α1σα23360-41α1σ2α22240)η8+(α12σ2α22360+α12σ6048+σ3α13480-α1α22σ2160)η9+(α13α2σα227560+α12σ2α21120+σ3α2α131680)η10+.

It is observed in Figures 1 and 2 that the flow has a boundary-layer structure and the thickness of this boundary-layer decreases with increase in the Prandtl number, σ as expected. This is due to the inhibiting influence of the viscous forces.

Variation of f(η)  using ϕ6[6,6]  for  σ=0.1,ϕ6[5,5]  for  σ=1, and ϕ6[4,4]  for  σ=10.

Variation of f(η)  using ϕ6[6,6]  for  σ=0.1,ϕ6[5,5]  for  σ=1, and ϕ6[4,4]  for  σ=10.

Figure 3 shows the increase of the Prandtl number, σ, that results in the decrease, as expected, of temperature distribution at a particular point of the flow region, that is, there would be a decrease of the thermal boundary-layer thickness with the increase of values of σimplying a slow rate of thermal diffusion. Thus higher Prandtl number σleads to faster cooling of the plane sheet.

Variation of θ(η)using ϖ7[6,6]  for  σ=0.1,ϖ7[5,5]  for  σ=1 and ϕ7[4,4]  for  σ=10.

6. Conclusions

In this study, we employed modified variational iteration method (MVIM) coupled with Padé approximation to solve a system of two nonlinear ordinary differential equations that describes a free-convective boundary-layer in glass-fiber production process. The results show strong effects of the Prandtl number on the velocity and temperature profiles since the two model equations are coupled.

Numerical values of α1=f(0).

σ[4,4][5,5][6,6]α1
0.0011.11355294181.12727604161.12528498541.1231381347
0.011.06317379631.07418956831.06383853511.0633808585
0.10.91280822100.92382262800.92421584930.9240830397
10.69412308610.69295980140.69321951580.6932116298
100.45112407280.45024295440.44767123160.4471165250
1000.26791971510.26814743630.26412956270.2645235434
10000.22040614320.15247832660.15004567550.1512901971
100000.08585871800.08585192490.08447754730.0855408524

Numerical values of α2=θ(0).

σ[4,4][5,5][6,6]α2 of 
0.001−0.0371141028−0.0415417739−0.0436188230−0.0468074648
0.01−0.1274922800−0.1221616907−0.1351353865−0.1357607439
0.1−0.3621215470−0.3505589981−0.3499273453−0.3500596733
1−0.7694165843−0.7695971295−0.7698955992−0.7698611967
10−1.5028543431−1.5007437650−1.4985484075−1.4970992078
100−2.7627624234−2.7637067330−2.7445541894−2.7468855016
1000−5.7787858408−4.9468469883−4.9104728566−4.9349476252
10000−8.8057265644−8.8032691004−8.7384279086−8.8044492660
Acknowledgment

The authors are highly grateful to the referee for his/her very constructive comments.

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