Research ArticleDual Approximate Solutions of the Unsteady ViscousFlow over a Shrinking Cylinder with Optimal HomotopyAsymptotic Method

Vasile Marinca12 and Remus-Daniel Ene3

1 Department of Mechanics and Vibration Politehnica University of Timisoara 300222 Timisoara Romania2Department of Electromechanics and Vibration Center for Advanced and Fundamental Technical ResearchRomania Academy 300223 Timisoara Romania

3 Department of Mathematics Politehnica University of Timisoara 300006 Timisoara Romania

Correspondence should be addressed to Remus-Daniel Ene eneremusgmailcom

Received 9 January 2014 Revised 10 February 2014 Accepted 17 February 2014 Published 25 March 2014

Academic Editor Waqar Ahmed Khan

Copyright copy 2014 V Marinca and R-D Ene This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The unsteady viscous flow over a continuously shrinking surface with mass suction is investigated using the optimal homotopyasymptotic method (OHAM) The nonlinear differential equation is obtained by means of the similarity transformation The dualsolutions exist for a certain range of mass suction and unsteadiness parameters A very good agreement was found between ourapproximate results andnumerical solutionswhich prove thatOHAMis very efficient in practice ensuring a very rapid convergenceafter only one iteration

1 Introduction

The flow of the Newtonian and non-Newtonian fluids isimportant for engineers and applied mathematicians becauseof its several applications in engineering or industrial pro-cesses In the last few decades these fluids have attractedconsiderable attention from researchers in many branchesof nonlinear dynamical systems in science and technologyThe flow over a stretchingshrinking cylinder is an importantproblem in many engineering processes with applications inindustries such as in plastic and metallurgy industries glass-fiber production and wire drawing The pioneering worksin the area of the flow inside a tube with time dependentdiameter were [1 2] where Uchida and Aoki and Skalak andWang studied the internal flow velocity and pressure dueto tube expansion or contraction Miklavcic and Wang [3]investigated the flow over a shrinking sheet obtaining anexact solution of the Navier-Stokes equations Ishak et al[4] reported that injection reduces the skin friction as wellas the heat transfer rate at the surface while suction actsin the opposite manner Fang et al [5] obtained the exact

solution of the unsteady state Navier-Stokes equations Fanget al [6] studied the viscous flow over a shrinking sheet bya newly proposed second order slip flow model The exactsolution of the full governing Navier-Stokes equation has twobranches in a certain range of the parameters The problemof unsteady viscous flow over a permeable shrinking cylinderwas solved by Zaimi et al [7] numerically using the shootingmethod The effects of suction and unsteadiness parameterson the flow velocity and the skin friction coefficient have beenanalyzed and presented graphically and the same authorsin [8] studied the effects of the unsteadiness parameter andthe Brownian motion parameter on the flow field and heattransfer characteristics Dual solutions are found to exist incertain conditions

Analytical solutions to nonlinear differential equationsplay an important role in the study of the unsteady viscousflow over a shrinking cylinder but it is difficult to find thesesolutions in the presence of strong nonlinearity Many newapproaches have been proposed to find approximate solutionsof nonlinear differential equations Perturbation methodshave been applied to determine approximate solutions to

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 417643 11 pageshttpdxdoiorg1011552014417643

2 Advances in Mathematical Physics

u

w

w r

a(t)

z

U

Figure 1 A schematic model of flow in an expanding cylinder withtime dependent radius

weakly nonlinear problems [9] But the use of perturbationtheory in many problems is invalid for parameters beyond acertain specified range

Homotopy perturbation method is employed to investi-gate steady-state heat conduction with temperature depen-dent thermal conductivity and heat generation in a hollowsphere by Khan et al [10] The same method is applied in thestudy of the effects of temperature distribution andheat trans-fer from solids of arbitrary shapes in [11] Another procedurethe Adomian decomposition method is used to compute theSumudu transformof some typical functions in [12 13] Othermethods have been proposed such as the various modifiedLindstedt-Poincare method [14] some linearizationmethods[15] and the optimal homotopy perturbation method [16]

In this paper we consider the unsteady viscous flow overa shrinking cylinder A version of the optimal homotopyasymptotic method is applied in this study to derive highlyaccurate analytical expressions of solutions Our proceduredoes not depend upon any small or large parameters con-tradistinguishing from other known methods The mainadvantage of this approach is the control of the convergenceof approximate solutions in a very rigorous way A very goodagreement was found between our approximate solutions andnumerical solutions which proves that our procedure is veryefficient and accurate

2 The Governing Equation

In what follows we assume an unsteady laminar boundarylayer flow of a nanofluid over an infinite cylinder or a tubewith a time dependent diameter in shrinking motion asshown in Figure 1

Also we consider the three-dimensional unsteadyNavier-Stokes equations for incompressible fluids without body forcesuch that based on the axisymmetric flow assumption and thefact that there is no azimuthal velocity component we have

nablak = 0 (1)

120597k120597119905

+ k sdot nablak = minus1

120588nabla119901 + ]nabla2k (2)

where k is the velocity vector 120588(119903) is the fluid density 119901 is thepressure and ] is the kinematic viscosityThe diameter of thecylinder is assumed as a function of timewith unsteady radius119886(119905) = 119886

0radic1 minus 120573119905 For a positive value of120573 the cylinder radius

becomes smaller with time that is contracting while for anegative value of 120573 the diameter becomes larger with timethat is expanding In cylindrical polar coordinates 119903 and 119911

are measured in the radial and axial directions respectively(1) and (2) can be written as [5ndash8]

1

119903

120597 (119903119906)

120597119903+

120597119908

120597119911= 0 (3)

120597119906

120597119905+ 119906

120597119906

120597119903+ 119908

120597119906

120597119911

= minus1

120588

119889119901

119889119903+ ](

1205972119906

1205971199032+

1

119903

120597119906

120597119903+

1205972119906

1205971199112minus

119906

1199032)

(4)

120597119908

120597119905+ 119906

120597119908

120597119903+ 119908

120597119908

120597119911

= minus1

120588

119889119901

119889119911+ ](

1205972119908

1205971199032+

1

119903

120597119908

120597119903+

1205972119908

1205971199112)

(5)

If we consider the constant mass transfer velocity119880 (119880 lt

0) and 1198860a positive constant then the boundary conditions

are of the following form

119906 =119880

radic1 minus 120573119905 119908 = minus

4]1199111198862

0(1 minus 120573119905)

at 119903 = 119886 (119905)

119908 = 0 as 119903 997888rarr infin

(6)

By means of the similarity variables [8]

119906 = minus2]

119886 (119905)

119891 (120578)

radic120578 119908 =

4]1199111198862 (119905)

1198911015840(120578) 120578 = (

119903

119886 (119905))

2

(7)

it is clear that 120578 ge 1 and on the other hand (3) is satisfiedautomatically Based on the defined velocity components it isstraightforward to derive from (4) that the pressure gradient120597119901120597119903 is a function of 119905 and 119903 and is independent on 119911 suchthat from (4) we obtain

119901

120588= 119866 (119905 119903) + ](

120597119906

120597119903+

119906

119903) minus

1

21199062+ int

120597119906

120597119905119889119903 (8)

or using (7) the pressure may be written as

119901

120588= 119866 (119905 119903) minus

2]2

1198862 (119905) [1198911015840 (120578) + (1120578) 1198912 (120578) + 2119878119891 (120578)]

(9)

where 119866(119905 119903) is the constant of the integration on 119911 and119878 = 119886

2

01205734] is the unsteadiness parameter for the expanding

(120573 lt 0) or contraction (120573 gt 0) cylinder showing the strengthof expansion or contraction Substituting (7) into (5) andrearranging terms this becomes

120578119891101584010158401015840

(120578) + 11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578) minus 119891

10158402(120578)

minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)] = 0

(10)

Advances in Mathematical Physics 3

with the boundary conditions transformed into the following

119891 (1) = 120574 1198911015840(1) = minus1 119891

1015840(infin) = 0 (11)

where prime denotes differentiation with respect to 120578 and 120574 =

minus11988601198802] gt 0 is the dimensionless suction parameter

3 Basic Ideas of the OHAM

Equation (10) can be written in a more general form

119871 [119891 (120578)] + 119873 [119891 (120578)] = 0 (12)

where 119871 is a linear operator and119873 is a nonlinear operator andthe boundary conditions (11) in the form

119861[119891 (120578) 119889119891 (120578)

119889120578] = 0 (13)

Let 1198910(120578) be an initial approximation of 119891(120578) such as

119871 [1198910(120578)] = 0 119861 [119891

0(120578)

1198891198910(120578)

119889120578] = 0 (14)

We point out that the linear operator 119871 from (12) and (14)is not unique

Let us consider the function 119865(120578 119901 119862119894) in the form

119865 (120578 119901 119862119894) = 1198910(120578) + 119901119891

1(120578 119862119894) (15)

where 119901 isin [0 1] denotes an embedding parameter It followsthat the first-order approximate solution can be written as

119891 (120578 119862119894) = 1198910(120578) + 119891

1(120578 119862119894) (16)

where 1198621 1198622 119862

119904are arbitrary parameters which will be

determined later The boundary conditions are

119861[119891 (120578 119862119894)

119889119891 (120578 119862119894)

119889120578] = 0 (17)

We construct a family of equations [17ndash21]

H [119871 (119865 (120578 119901 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 119901 119862

119894))]

= 119871 [1198910(120578)] + 119901 [119871 (119891

1(120578 119862119894))

minus119867 (120578 119862119894)119873 (119891

0(120578))] = 0

(18)

with the properties

H [119871 (119865 (120578 0 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 0 119862

119894))]

= 119871 [1198910(120578)] = 0

(19)

H [119871 (119865 (120578 1 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 1 119862

119894))]

= 119867 (120578 119862119894) [119871 (119891 (120578 119862

119894)) + 119873 (119891 (120578 119862

119894))] = 0

(20)

where 119867(120578 119862119894) is an arbitrary auxiliary convergence-control

function

From (15) and (16) we get

119865 (120578 0 119862119894) = 1198910(120578) (21)

119865 (120578 1 119862119894) = 119891 (120578 119862

119894) (22)

Now equating only the coefficients of 1199010 and 1199011 into (18)

we obtain the governing equation of 1198910(120578) given by (14) and

the governing equation on 1198911(120578 119862119894) that is

119871 (1198911(120578 119862119894)) = 119867 (120578 119862

119894)119873 (119891

0(120578))

119861 [1198911(120578 119862119894)

1198891198911(120578 119862119894)

119889120578] = 0 119894 = 1 2 119904

(23)

In general the nonlinear operator from (23) may bewritten as

119873(1198910(120578)) =

119899

sum

119894=1

ℎ119894(120578) 119892119894(120578) (24)

where the functions ℎ119894(120578) and 119892

119894(120578) are known and depend

on the functions 1198910(120578) and also on the nonlinear operator 119899

being a known integer number It is known that the generalsolution of the nonhomogeneous linear equation (23) isequal to the sum of general solution of the correspondinghomogeneous equation and some particular solutions of thenonhomogeneous equation In what follows we do not solve(23) but from the theory of differential equations it is moreconvenient to consider the unknown function119891

1(120578 119862119894) in the

form

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 ℎ119895(120578) 119862

119895) 119892119894(120578) 119895 = 1 2 119904

(25)

or

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 119892119895(120578) 119862

119895) 119891119894(120578) 119895 = 1 2 119904

(26)

119861[1198911(120578 119862119895)

1198891198911(120578 119862119895)

119889120578] = 0 (27)

where within expression of 119867119894(120578 ℎ119895(120578) 119862

119894) from (25) appear

linear combinations of some functions ℎ119895 some of the terms

which are given by corresponding homogeneous equationand a number of unknown parameters 119862

119895 119895 = 1 2 119904 119898

being an arbitrary integer number The same considerationscan be made for (26) where 119891

119894and 119892

119894are interchangeable

4 The Convergence ofthe Approximate Solution (16)

The convergence of the approximate solution 119891(120578 119862119894) given

by (16) depends upon the auxiliary functions119867119894(120578 ℎ119895(120578) 119862

119895)

119895 = 1 119904 which appear in (25)There aremany possibilitiesto choose these functions119867

119894 We try to choose such function

4 Advances in Mathematical Physics

119867119894so that within (25) the terms sum

119898

119894=1119867119894(120578 ℎ119895(120578) 119862

119895)119892119894(120578)

are of the same shape as the terms sum119899

119894=1ℎ119894(119909)119892119894(119909) given by

(24) [14ndash18] The first-order approximate solution 119891(120578 119862119894)

also depends on the parameters 119862119895 119895 = 1 119904 The

values of these parameters can be optimally evaluated viavarious methods the least-square method minimization ofthe square residual error the Galerkin method collocationmethod or the Ritz method and so on In this way it is clearthat the first-order approximate solutions given by (16) arewell determined Because the auxiliary functions 119867

119894are not

unique we have freedom to determine multiple solutionsfor nonlinear differential equations (10) and (11) It shouldbe emphasized that our procedure contains the auxiliaryfunctions 119867

119894(119909 119891119895(120578) 119862

119895) 119894 = 1 119898 119895 = 1 119904 which

provides us with a simple way to adjust and control theconvergence of the approximate solutions

5 Multiple Approximate Solutions of theUnsteady Viscous Flow by OHAM

The linear operator can be chosen in the following forms

119871 (119891 (120578)) = 119891101584010158401015840

(120578) + 11987011989110158401015840(120578) (28)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus 11987021198911015840(120578) (29)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) +2119870

119870120578 + 1 minus 11987011989110158401015840(120578) (30)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus41198702

(119870120578 + 1 minus 119870)21198911015840(120578) (31)

where 119870 gt 0 is an unknown positive parameter and will bedetermined later

The initial approximation 1198910(120578) can be obtained from

(14) with boundary conditions

1198910(1) = 120574 119891

1015840

0(1) = minus1 119891

1015840

0(infin) = 0 (32)

Equation (14) with the linear operators (28) or (29) hasthe solutions

1198910(120578) = 120574 +

119890minus119870(120578minus1)

minus 1

119870

(33)

while (14) with the linear operators (30) or (31) has thesolutions

1198910(120578) = 120574 minus

1

119870+

1

119870 (119870120578 + 1 minus 119870) (34)

The nonlinear operator corresponding to nonlinear dif-ferential equation (10) is defined as

119873(119891 (120578)) = (1 minus 119870120578)11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(35)

for linear operator defined by (28)

The same nonlinear operators for the linear operatordefined by (29) (30) and (31) are respectively

119873(119891 (120578)) = 11989110158401015840(120578) + 119870

21198911015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(36)

119873(119891 (120578)) = (1 minus2119870

119870120578 + 1 minus 119870)11989110158401015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(37)

119873(119891 (120578)) = 11989110158401015840(120578) +

41198702120578

(119870120578 + 1 minus 119870)21198911015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(38)

Substituting (33) into (35) it holds that

119873(1198910(120578))

= [minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1] 119890minus119870(120578minus1)

(39)

Now comparing (24) and (39) one gets

ℎ1(120578) = minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1

1198921(120578) = 119890

minus119870(120578minus1)

ℎ119895(120578) = 119892

119895(120578) = 0 for 119895 ge 2

(40)

The first approximation 1198911(120578 119862119894) can be written in the

form

1198911(120578 119862119894) = 119867

1(120578 119862119894) 119890minus119870(120578minus1)

1198911(1) = 119891

1015840

1(1) = 119891

1015840

1(infin) = 0

(41)

where 1198671(120578 119862119894) are arbitrary functions Of course we have

freedom to choose such functions with conditions obtainedfrom (41)

119887

1198671(1 119862119894) = 119867

1015840

1(1 119862119894) = 0 (42)

Advances in Mathematical Physics 5

For example 1198671are given by

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+ 1198623(120578 minus 1)

4

+1198624(120578 minus 1)

2

119890minus(119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(2119870+120572

2)(120578minus1)

+ 1198626(120578 minus 1)

2

119890minus1198701205723(120578minus1)

+ 1198627(120578 minus 1)

2

119890minus1198701205724(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus1198701205725(120578minus1)

(43)

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

+1198624(120578 minus 1)

5

+ [1198625(120578 minus 1)

2

+1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

] 119890minus119870(120578minus1)

+ [1198628(120578 minus 1)

2

+ 1198629(120578 minus 1)

3

+ 11986210(120578 minus 1)

4

+11986211(120578 minus 1)

5

] 119890minus2119870(120578minus1)

(44)

1198671(120578 119862119894)=1198621(120578 minus 1)+119862

2(120578 minus 1)

2

+ 1198623(120578 minus 1)

3

+ 1198624(120578 minus 1)

4

+[minus1198621(120578 minus 1) + 119862

5(120578 minus 1)

2

+ 1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

+ 1198628(120578 minus 1)

5

+ 1198629(120578 minus 1)

6

]119890minus119870(120578minus1)

+ [11986210(120578 minus 1)

2

+ 11986211(120578 minus 1)

3

] 119890minus2119870(120578minus1)

(45)

Taking into consideration only the expression given by(43) from (33) (41) and (16) we obtain the first-orderapproximate solution of (10) and (11) in the form

119891 (120578 119862119894) = 120574 minus

1

119870+ [

1

119870+ 1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

] 119890minus119870(120578minus1)

+1198624(120578 minus 1)

2

119890minus(2119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(3119870+120572

2)(120578minus1)

+1198626(120578 minus 1)

2

119890minus119870(120572

3+1)(120578minus1)

+1198627(120578 minus 1)

2

119890minus119870(120572

4+1)(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus119870(120572

5+1)(120578minus1)

(46)

where 1198621 1198622 120572

1 1205722 are unknown parameters

Many other approximate solutions can be obtained bymeans of combinations between initial approximations givenby (33) and (34) and the nonlinear operators (36) (37) or(38)

6 Numerical Examples

In order to show the validity and accuracy of the OHAMwe compare previously obtained approximate solutions (46)with numerical integration results obtained by means ofa fourth-order Runge-Kutta method in combination withshootingmethod and theWolframMathematica 60 software

Using the least-square method for determination of theparameters 119862

119894and 120572

119894 we present the following four cases for

the different values of the coefficients 120574 and 119878

61 Case 1 120574 = 1 and 119878 = minus1 We find dual solutions

(a) We have

1198621= 03333335814 119862

2= 00210985545

1198623= 70349253510 sdot 10

minus6 119862

4= minus12078582888

1198625= 11007711694 119862

6= 12527962825

1198627= minus14794094663 119862

8= 00003670989

119870 = 1 1205721= minus10044596362

1205722= minus19960995780 120572

3= 00039364947

1205724= minus00044067908 120572

5= minus00876855705

(47)

The first expression of the first-order approximate solutiongiven by (46) can be written in the form

119891 (120578) = [1 + 03333335814(120578 minus 1)2

+ 00210985545(120578 minus 1)3

+7034925351 sdot 10minus6

(120578 minus 1)4

] 1198901minus120578

+ 12527962825(120578 minus 1)2

119890minus10039364947(120578minus1)

+ 11007711694(120578 minus 1)2

119890minus10039004219(120578minus1)

minus 14794094663(120578 minus 1)2

119890minus09955932091(120578minus1)

minus 12078582888(120578 minus 1)2

119890minus09955403637(120578minus1)

+ 00003670989(120578 minus 1)2

119890minus09123144295(120578minus1)

(48)

(b) We have

1198621= minus21103792246 119862

2= minus01209917376

1198623= 00016464844 119862

4= minus00401496995

1198625= 51986534028 119862

6= 29102631733

1198627= minus31870328789 119862

8= minus30376107137

119870 = 06170257079 1205721= 15253390642

1205722= minus12342665630 120572

3= 10337532632

1205724= 00700064931 120572

5= 10337537195

(49)

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

u

w

w r

a(t)

z

U

Figure 1 A schematic model of flow in an expanding cylinder withtime dependent radius

weakly nonlinear problems [9] But the use of perturbationtheory in many problems is invalid for parameters beyond acertain specified range

Homotopy perturbation method is employed to investi-gate steady-state heat conduction with temperature depen-dent thermal conductivity and heat generation in a hollowsphere by Khan et al [10] The same method is applied in thestudy of the effects of temperature distribution andheat trans-fer from solids of arbitrary shapes in [11] Another procedurethe Adomian decomposition method is used to compute theSumudu transformof some typical functions in [12 13] Othermethods have been proposed such as the various modifiedLindstedt-Poincare method [14] some linearizationmethods[15] and the optimal homotopy perturbation method [16]

In this paper we consider the unsteady viscous flow overa shrinking cylinder A version of the optimal homotopyasymptotic method is applied in this study to derive highlyaccurate analytical expressions of solutions Our proceduredoes not depend upon any small or large parameters con-tradistinguishing from other known methods The mainadvantage of this approach is the control of the convergenceof approximate solutions in a very rigorous way A very goodagreement was found between our approximate solutions andnumerical solutions which proves that our procedure is veryefficient and accurate

2 The Governing Equation

In what follows we assume an unsteady laminar boundarylayer flow of a nanofluid over an infinite cylinder or a tubewith a time dependent diameter in shrinking motion asshown in Figure 1

Also we consider the three-dimensional unsteadyNavier-Stokes equations for incompressible fluids without body forcesuch that based on the axisymmetric flow assumption and thefact that there is no azimuthal velocity component we have

nablak = 0 (1)

120597k120597119905

+ k sdot nablak = minus1

120588nabla119901 + ]nabla2k (2)

where k is the velocity vector 120588(119903) is the fluid density 119901 is thepressure and ] is the kinematic viscosityThe diameter of thecylinder is assumed as a function of timewith unsteady radius119886(119905) = 119886

0radic1 minus 120573119905 For a positive value of120573 the cylinder radius

becomes smaller with time that is contracting while for anegative value of 120573 the diameter becomes larger with timethat is expanding In cylindrical polar coordinates 119903 and 119911

are measured in the radial and axial directions respectively(1) and (2) can be written as [5ndash8]

1

119903

120597 (119903119906)

120597119903+

120597119908

120597119911= 0 (3)

120597119906

120597119905+ 119906

120597119906

120597119903+ 119908

120597119906

120597119911

= minus1

120588

119889119901

119889119903+ ](

1205972119906

1205971199032+

1

119903

120597119906

120597119903+

1205972119906

1205971199112minus

119906

1199032)

(4)

120597119908

120597119905+ 119906

120597119908

120597119903+ 119908

120597119908

120597119911

= minus1

120588

119889119901

119889119911+ ](

1205972119908

1205971199032+

1

119903

120597119908

120597119903+

1205972119908

1205971199112)

(5)

If we consider the constant mass transfer velocity119880 (119880 lt

0) and 1198860a positive constant then the boundary conditions

are of the following form

119906 =119880

radic1 minus 120573119905 119908 = minus

4]1199111198862

0(1 minus 120573119905)

at 119903 = 119886 (119905)

119908 = 0 as 119903 997888rarr infin

(6)

By means of the similarity variables [8]

119906 = minus2]

119886 (119905)

119891 (120578)

radic120578 119908 =

4]1199111198862 (119905)

1198911015840(120578) 120578 = (

119903

119886 (119905))

2

(7)

it is clear that 120578 ge 1 and on the other hand (3) is satisfiedautomatically Based on the defined velocity components it isstraightforward to derive from (4) that the pressure gradient120597119901120597119903 is a function of 119905 and 119903 and is independent on 119911 suchthat from (4) we obtain

119901

120588= 119866 (119905 119903) + ](

120597119906

120597119903+

119906

119903) minus

1

21199062+ int

120597119906

120597119905119889119903 (8)

or using (7) the pressure may be written as

119901

120588= 119866 (119905 119903) minus

2]2

1198862 (119905) [1198911015840 (120578) + (1120578) 1198912 (120578) + 2119878119891 (120578)]

(9)

where 119866(119905 119903) is the constant of the integration on 119911 and119878 = 119886

2

01205734] is the unsteadiness parameter for the expanding

(120573 lt 0) or contraction (120573 gt 0) cylinder showing the strengthof expansion or contraction Substituting (7) into (5) andrearranging terms this becomes

120578119891101584010158401015840

(120578) + 11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578) minus 119891

10158402(120578)

minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)] = 0

(10)

Advances in Mathematical Physics 3

with the boundary conditions transformed into the following

119891 (1) = 120574 1198911015840(1) = minus1 119891

1015840(infin) = 0 (11)

where prime denotes differentiation with respect to 120578 and 120574 =

minus11988601198802] gt 0 is the dimensionless suction parameter

3 Basic Ideas of the OHAM

Equation (10) can be written in a more general form

119871 [119891 (120578)] + 119873 [119891 (120578)] = 0 (12)

where 119871 is a linear operator and119873 is a nonlinear operator andthe boundary conditions (11) in the form

119861[119891 (120578) 119889119891 (120578)

119889120578] = 0 (13)

Let 1198910(120578) be an initial approximation of 119891(120578) such as

119871 [1198910(120578)] = 0 119861 [119891

0(120578)

1198891198910(120578)

119889120578] = 0 (14)

We point out that the linear operator 119871 from (12) and (14)is not unique

Let us consider the function 119865(120578 119901 119862119894) in the form

119865 (120578 119901 119862119894) = 1198910(120578) + 119901119891

1(120578 119862119894) (15)

where 119901 isin [0 1] denotes an embedding parameter It followsthat the first-order approximate solution can be written as

119891 (120578 119862119894) = 1198910(120578) + 119891

1(120578 119862119894) (16)

where 1198621 1198622 119862

119904are arbitrary parameters which will be

determined later The boundary conditions are

119861[119891 (120578 119862119894)

119889119891 (120578 119862119894)

119889120578] = 0 (17)

We construct a family of equations [17ndash21]

H [119871 (119865 (120578 119901 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 119901 119862

119894))]

= 119871 [1198910(120578)] + 119901 [119871 (119891

1(120578 119862119894))

minus119867 (120578 119862119894)119873 (119891

0(120578))] = 0

(18)

with the properties

H [119871 (119865 (120578 0 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 0 119862

119894))]

= 119871 [1198910(120578)] = 0

(19)

H [119871 (119865 (120578 1 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 1 119862

119894))]

= 119867 (120578 119862119894) [119871 (119891 (120578 119862

119894)) + 119873 (119891 (120578 119862

119894))] = 0

(20)

where 119867(120578 119862119894) is an arbitrary auxiliary convergence-control

function

From (15) and (16) we get

119865 (120578 0 119862119894) = 1198910(120578) (21)

119865 (120578 1 119862119894) = 119891 (120578 119862

119894) (22)

Now equating only the coefficients of 1199010 and 1199011 into (18)

we obtain the governing equation of 1198910(120578) given by (14) and

the governing equation on 1198911(120578 119862119894) that is

119871 (1198911(120578 119862119894)) = 119867 (120578 119862

119894)119873 (119891

0(120578))

119861 [1198911(120578 119862119894)

1198891198911(120578 119862119894)

119889120578] = 0 119894 = 1 2 119904

(23)

In general the nonlinear operator from (23) may bewritten as

119873(1198910(120578)) =

119899

sum

119894=1

ℎ119894(120578) 119892119894(120578) (24)

where the functions ℎ119894(120578) and 119892

119894(120578) are known and depend

on the functions 1198910(120578) and also on the nonlinear operator 119899

being a known integer number It is known that the generalsolution of the nonhomogeneous linear equation (23) isequal to the sum of general solution of the correspondinghomogeneous equation and some particular solutions of thenonhomogeneous equation In what follows we do not solve(23) but from the theory of differential equations it is moreconvenient to consider the unknown function119891

1(120578 119862119894) in the

form

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 ℎ119895(120578) 119862

119895) 119892119894(120578) 119895 = 1 2 119904

(25)

or

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 119892119895(120578) 119862

119895) 119891119894(120578) 119895 = 1 2 119904

(26)

119861[1198911(120578 119862119895)

1198891198911(120578 119862119895)

119889120578] = 0 (27)

where within expression of 119867119894(120578 ℎ119895(120578) 119862

119894) from (25) appear

linear combinations of some functions ℎ119895 some of the terms

which are given by corresponding homogeneous equationand a number of unknown parameters 119862

119895 119895 = 1 2 119904 119898

being an arbitrary integer number The same considerationscan be made for (26) where 119891

119894and 119892

119894are interchangeable

4 The Convergence ofthe Approximate Solution (16)

The convergence of the approximate solution 119891(120578 119862119894) given

by (16) depends upon the auxiliary functions119867119894(120578 ℎ119895(120578) 119862

119895)

119895 = 1 119904 which appear in (25)There aremany possibilitiesto choose these functions119867

119894 We try to choose such function

4 Advances in Mathematical Physics

119867119894so that within (25) the terms sum

119898

119894=1119867119894(120578 ℎ119895(120578) 119862

119895)119892119894(120578)

are of the same shape as the terms sum119899

119894=1ℎ119894(119909)119892119894(119909) given by

(24) [14ndash18] The first-order approximate solution 119891(120578 119862119894)

also depends on the parameters 119862119895 119895 = 1 119904 The

values of these parameters can be optimally evaluated viavarious methods the least-square method minimization ofthe square residual error the Galerkin method collocationmethod or the Ritz method and so on In this way it is clearthat the first-order approximate solutions given by (16) arewell determined Because the auxiliary functions 119867

119894are not

unique we have freedom to determine multiple solutionsfor nonlinear differential equations (10) and (11) It shouldbe emphasized that our procedure contains the auxiliaryfunctions 119867

119894(119909 119891119895(120578) 119862

119895) 119894 = 1 119898 119895 = 1 119904 which

provides us with a simple way to adjust and control theconvergence of the approximate solutions

5 Multiple Approximate Solutions of theUnsteady Viscous Flow by OHAM

The linear operator can be chosen in the following forms

119871 (119891 (120578)) = 119891101584010158401015840

(120578) + 11987011989110158401015840(120578) (28)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus 11987021198911015840(120578) (29)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) +2119870

119870120578 + 1 minus 11987011989110158401015840(120578) (30)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus41198702

(119870120578 + 1 minus 119870)21198911015840(120578) (31)

where 119870 gt 0 is an unknown positive parameter and will bedetermined later

The initial approximation 1198910(120578) can be obtained from

(14) with boundary conditions

1198910(1) = 120574 119891

1015840

0(1) = minus1 119891

1015840

0(infin) = 0 (32)

Equation (14) with the linear operators (28) or (29) hasthe solutions

1198910(120578) = 120574 +

119890minus119870(120578minus1)

minus 1

119870

(33)

while (14) with the linear operators (30) or (31) has thesolutions

1198910(120578) = 120574 minus

1

119870+

1

119870 (119870120578 + 1 minus 119870) (34)

The nonlinear operator corresponding to nonlinear dif-ferential equation (10) is defined as

119873(119891 (120578)) = (1 minus 119870120578)11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(35)

for linear operator defined by (28)

The same nonlinear operators for the linear operatordefined by (29) (30) and (31) are respectively

119873(119891 (120578)) = 11989110158401015840(120578) + 119870

21198911015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(36)

119873(119891 (120578)) = (1 minus2119870

119870120578 + 1 minus 119870)11989110158401015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(37)

119873(119891 (120578)) = 11989110158401015840(120578) +

41198702120578

(119870120578 + 1 minus 119870)21198911015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(38)

Substituting (33) into (35) it holds that

119873(1198910(120578))

= [minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1] 119890minus119870(120578minus1)

(39)

Now comparing (24) and (39) one gets

ℎ1(120578) = minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1

1198921(120578) = 119890

minus119870(120578minus1)

ℎ119895(120578) = 119892

119895(120578) = 0 for 119895 ge 2

(40)

The first approximation 1198911(120578 119862119894) can be written in the

form

1198911(120578 119862119894) = 119867

1(120578 119862119894) 119890minus119870(120578minus1)

1198911(1) = 119891

1015840

1(1) = 119891

1015840

1(infin) = 0

(41)

where 1198671(120578 119862119894) are arbitrary functions Of course we have

freedom to choose such functions with conditions obtainedfrom (41)

119887

1198671(1 119862119894) = 119867

1015840

1(1 119862119894) = 0 (42)

Advances in Mathematical Physics 5

For example 1198671are given by

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+ 1198623(120578 minus 1)

4

+1198624(120578 minus 1)

2

119890minus(119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(2119870+120572

2)(120578minus1)

+ 1198626(120578 minus 1)

2

119890minus1198701205723(120578minus1)

+ 1198627(120578 minus 1)

2

119890minus1198701205724(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus1198701205725(120578minus1)

(43)

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

+1198624(120578 minus 1)

5

+ [1198625(120578 minus 1)

2

+1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

] 119890minus119870(120578minus1)

+ [1198628(120578 minus 1)

2

+ 1198629(120578 minus 1)

3

+ 11986210(120578 minus 1)

4

+11986211(120578 minus 1)

5

] 119890minus2119870(120578minus1)

(44)

1198671(120578 119862119894)=1198621(120578 minus 1)+119862

2(120578 minus 1)

2

+ 1198623(120578 minus 1)

3

+ 1198624(120578 minus 1)

4

+[minus1198621(120578 minus 1) + 119862

5(120578 minus 1)

2

+ 1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

+ 1198628(120578 minus 1)

5

+ 1198629(120578 minus 1)

6

]119890minus119870(120578minus1)

+ [11986210(120578 minus 1)

2

+ 11986211(120578 minus 1)

3

] 119890minus2119870(120578minus1)

(45)

Taking into consideration only the expression given by(43) from (33) (41) and (16) we obtain the first-orderapproximate solution of (10) and (11) in the form

119891 (120578 119862119894) = 120574 minus

1

119870+ [

1

119870+ 1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

] 119890minus119870(120578minus1)

+1198624(120578 minus 1)

2

119890minus(2119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(3119870+120572

2)(120578minus1)

+1198626(120578 minus 1)

2

119890minus119870(120572

3+1)(120578minus1)

+1198627(120578 minus 1)

2

119890minus119870(120572

4+1)(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus119870(120572

5+1)(120578minus1)

(46)

where 1198621 1198622 120572

1 1205722 are unknown parameters

Many other approximate solutions can be obtained bymeans of combinations between initial approximations givenby (33) and (34) and the nonlinear operators (36) (37) or(38)

6 Numerical Examples

In order to show the validity and accuracy of the OHAMwe compare previously obtained approximate solutions (46)with numerical integration results obtained by means ofa fourth-order Runge-Kutta method in combination withshootingmethod and theWolframMathematica 60 software

Using the least-square method for determination of theparameters 119862

119894and 120572

119894 we present the following four cases for

the different values of the coefficients 120574 and 119878

61 Case 1 120574 = 1 and 119878 = minus1 We find dual solutions

(a) We have

1198621= 03333335814 119862

2= 00210985545

1198623= 70349253510 sdot 10

minus6 119862

4= minus12078582888

1198625= 11007711694 119862

6= 12527962825

1198627= minus14794094663 119862

8= 00003670989

119870 = 1 1205721= minus10044596362

1205722= minus19960995780 120572

3= 00039364947

1205724= minus00044067908 120572

5= minus00876855705

(47)

The first expression of the first-order approximate solutiongiven by (46) can be written in the form

119891 (120578) = [1 + 03333335814(120578 minus 1)2

+ 00210985545(120578 minus 1)3

+7034925351 sdot 10minus6

(120578 minus 1)4

] 1198901minus120578

+ 12527962825(120578 minus 1)2

119890minus10039364947(120578minus1)

+ 11007711694(120578 minus 1)2

119890minus10039004219(120578minus1)

minus 14794094663(120578 minus 1)2

119890minus09955932091(120578minus1)

minus 12078582888(120578 minus 1)2

119890minus09955403637(120578minus1)

+ 00003670989(120578 minus 1)2

119890minus09123144295(120578minus1)

(48)

(b) We have

1198621= minus21103792246 119862

2= minus01209917376

1198623= 00016464844 119862

4= minus00401496995

1198625= 51986534028 119862

6= 29102631733

1198627= minus31870328789 119862

8= minus30376107137

119870 = 06170257079 1205721= 15253390642

1205722= minus12342665630 120572

3= 10337532632

1205724= 00700064931 120572

5= 10337537195

(49)

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

with the boundary conditions transformed into the following

119891 (1) = 120574 1198911015840(1) = minus1 119891

1015840(infin) = 0 (11)

where prime denotes differentiation with respect to 120578 and 120574 =

minus11988601198802] gt 0 is the dimensionless suction parameter

3 Basic Ideas of the OHAM

Equation (10) can be written in a more general form

119871 [119891 (120578)] + 119873 [119891 (120578)] = 0 (12)

where 119871 is a linear operator and119873 is a nonlinear operator andthe boundary conditions (11) in the form

119861[119891 (120578) 119889119891 (120578)

119889120578] = 0 (13)

Let 1198910(120578) be an initial approximation of 119891(120578) such as

119871 [1198910(120578)] = 0 119861 [119891

0(120578)

1198891198910(120578)

119889120578] = 0 (14)

We point out that the linear operator 119871 from (12) and (14)is not unique

Let us consider the function 119865(120578 119901 119862119894) in the form

119865 (120578 119901 119862119894) = 1198910(120578) + 119901119891

1(120578 119862119894) (15)

where 119901 isin [0 1] denotes an embedding parameter It followsthat the first-order approximate solution can be written as

119891 (120578 119862119894) = 1198910(120578) + 119891

1(120578 119862119894) (16)

where 1198621 1198622 119862

119904are arbitrary parameters which will be

determined later The boundary conditions are

119861[119891 (120578 119862119894)

119889119891 (120578 119862119894)

119889120578] = 0 (17)

We construct a family of equations [17ndash21]

H [119871 (119865 (120578 119901 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 119901 119862

119894))]

= 119871 [1198910(120578)] + 119901 [119871 (119891

1(120578 119862119894))

minus119867 (120578 119862119894)119873 (119891

0(120578))] = 0

(18)

with the properties

H [119871 (119865 (120578 0 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 0 119862

119894))]

= 119871 [1198910(120578)] = 0

(19)

H [119871 (119865 (120578 1 119862119894)) 119867 (120578 119862

119894) 119873 (119865 (120578 1 119862

119894))]

= 119867 (120578 119862119894) [119871 (119891 (120578 119862

119894)) + 119873 (119891 (120578 119862

119894))] = 0

(20)

where 119867(120578 119862119894) is an arbitrary auxiliary convergence-control

function

From (15) and (16) we get

119865 (120578 0 119862119894) = 1198910(120578) (21)

119865 (120578 1 119862119894) = 119891 (120578 119862

119894) (22)

Now equating only the coefficients of 1199010 and 1199011 into (18)

we obtain the governing equation of 1198910(120578) given by (14) and

the governing equation on 1198911(120578 119862119894) that is

119871 (1198911(120578 119862119894)) = 119867 (120578 119862

119894)119873 (119891

0(120578))

119861 [1198911(120578 119862119894)

1198891198911(120578 119862119894)

119889120578] = 0 119894 = 1 2 119904

(23)

In general the nonlinear operator from (23) may bewritten as

119873(1198910(120578)) =

119899

sum

119894=1

ℎ119894(120578) 119892119894(120578) (24)

where the functions ℎ119894(120578) and 119892

119894(120578) are known and depend

on the functions 1198910(120578) and also on the nonlinear operator 119899

being a known integer number It is known that the generalsolution of the nonhomogeneous linear equation (23) isequal to the sum of general solution of the correspondinghomogeneous equation and some particular solutions of thenonhomogeneous equation In what follows we do not solve(23) but from the theory of differential equations it is moreconvenient to consider the unknown function119891

1(120578 119862119894) in the

form

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 ℎ119895(120578) 119862

119895) 119892119894(120578) 119895 = 1 2 119904

(25)

or

1198911(120578 119862119895) =

119898

sum

119894=1

119867119894(120578 119892119895(120578) 119862

119895) 119891119894(120578) 119895 = 1 2 119904

(26)

119861[1198911(120578 119862119895)

1198891198911(120578 119862119895)

119889120578] = 0 (27)

where within expression of 119867119894(120578 ℎ119895(120578) 119862

119894) from (25) appear

linear combinations of some functions ℎ119895 some of the terms

which are given by corresponding homogeneous equationand a number of unknown parameters 119862

119895 119895 = 1 2 119904 119898

being an arbitrary integer number The same considerationscan be made for (26) where 119891

119894and 119892

119894are interchangeable

4 The Convergence ofthe Approximate Solution (16)

The convergence of the approximate solution 119891(120578 119862119894) given

by (16) depends upon the auxiliary functions119867119894(120578 ℎ119895(120578) 119862

119895)

119895 = 1 119904 which appear in (25)There aremany possibilitiesto choose these functions119867

119894 We try to choose such function

4 Advances in Mathematical Physics

119867119894so that within (25) the terms sum

119898

119894=1119867119894(120578 ℎ119895(120578) 119862

119895)119892119894(120578)

are of the same shape as the terms sum119899

119894=1ℎ119894(119909)119892119894(119909) given by

(24) [14ndash18] The first-order approximate solution 119891(120578 119862119894)

also depends on the parameters 119862119895 119895 = 1 119904 The

values of these parameters can be optimally evaluated viavarious methods the least-square method minimization ofthe square residual error the Galerkin method collocationmethod or the Ritz method and so on In this way it is clearthat the first-order approximate solutions given by (16) arewell determined Because the auxiliary functions 119867

119894are not

unique we have freedom to determine multiple solutionsfor nonlinear differential equations (10) and (11) It shouldbe emphasized that our procedure contains the auxiliaryfunctions 119867

119894(119909 119891119895(120578) 119862

119895) 119894 = 1 119898 119895 = 1 119904 which

provides us with a simple way to adjust and control theconvergence of the approximate solutions

5 Multiple Approximate Solutions of theUnsteady Viscous Flow by OHAM

The linear operator can be chosen in the following forms

119871 (119891 (120578)) = 119891101584010158401015840

(120578) + 11987011989110158401015840(120578) (28)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus 11987021198911015840(120578) (29)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) +2119870

119870120578 + 1 minus 11987011989110158401015840(120578) (30)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus41198702

(119870120578 + 1 minus 119870)21198911015840(120578) (31)

where 119870 gt 0 is an unknown positive parameter and will bedetermined later

The initial approximation 1198910(120578) can be obtained from

(14) with boundary conditions

1198910(1) = 120574 119891

1015840

0(1) = minus1 119891

1015840

0(infin) = 0 (32)

Equation (14) with the linear operators (28) or (29) hasthe solutions

1198910(120578) = 120574 +

119890minus119870(120578minus1)

minus 1

119870

(33)

while (14) with the linear operators (30) or (31) has thesolutions

1198910(120578) = 120574 minus

1

119870+

1

119870 (119870120578 + 1 minus 119870) (34)

The nonlinear operator corresponding to nonlinear dif-ferential equation (10) is defined as

119873(119891 (120578)) = (1 minus 119870120578)11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(35)

for linear operator defined by (28)

The same nonlinear operators for the linear operatordefined by (29) (30) and (31) are respectively

119873(119891 (120578)) = 11989110158401015840(120578) + 119870

21198911015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(36)

119873(119891 (120578)) = (1 minus2119870

119870120578 + 1 minus 119870)11989110158401015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(37)

119873(119891 (120578)) = 11989110158401015840(120578) +

41198702120578

(119870120578 + 1 minus 119870)21198911015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(38)

Substituting (33) into (35) it holds that

119873(1198910(120578))

= [minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1] 119890minus119870(120578minus1)

(39)

Now comparing (24) and (39) one gets

ℎ1(120578) = minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1

1198921(120578) = 119890

minus119870(120578minus1)

ℎ119895(120578) = 119892

119895(120578) = 0 for 119895 ge 2

(40)

The first approximation 1198911(120578 119862119894) can be written in the

form

1198911(120578 119862119894) = 119867

1(120578 119862119894) 119890minus119870(120578minus1)

1198911(1) = 119891

1015840

1(1) = 119891

1015840

1(infin) = 0

(41)

where 1198671(120578 119862119894) are arbitrary functions Of course we have

freedom to choose such functions with conditions obtainedfrom (41)

119887

1198671(1 119862119894) = 119867

1015840

1(1 119862119894) = 0 (42)

Advances in Mathematical Physics 5

For example 1198671are given by

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+ 1198623(120578 minus 1)

4

+1198624(120578 minus 1)

2

119890minus(119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(2119870+120572

2)(120578minus1)

+ 1198626(120578 minus 1)

2

119890minus1198701205723(120578minus1)

+ 1198627(120578 minus 1)

2

119890minus1198701205724(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus1198701205725(120578minus1)

(43)

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

+1198624(120578 minus 1)

5

+ [1198625(120578 minus 1)

2

+1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

] 119890minus119870(120578minus1)

+ [1198628(120578 minus 1)

2

+ 1198629(120578 minus 1)

3

+ 11986210(120578 minus 1)

4

+11986211(120578 minus 1)

5

] 119890minus2119870(120578minus1)

(44)

1198671(120578 119862119894)=1198621(120578 minus 1)+119862

2(120578 minus 1)

2

+ 1198623(120578 minus 1)

3

+ 1198624(120578 minus 1)

4

+[minus1198621(120578 minus 1) + 119862

5(120578 minus 1)

2

+ 1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

+ 1198628(120578 minus 1)

5

+ 1198629(120578 minus 1)

6

]119890minus119870(120578minus1)

+ [11986210(120578 minus 1)

2

+ 11986211(120578 minus 1)

3

] 119890minus2119870(120578minus1)

(45)

Taking into consideration only the expression given by(43) from (33) (41) and (16) we obtain the first-orderapproximate solution of (10) and (11) in the form

119891 (120578 119862119894) = 120574 minus

1

119870+ [

1

119870+ 1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

] 119890minus119870(120578minus1)

+1198624(120578 minus 1)

2

119890minus(2119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(3119870+120572

2)(120578minus1)

+1198626(120578 minus 1)

2

119890minus119870(120572

3+1)(120578minus1)

+1198627(120578 minus 1)

2

119890minus119870(120572

4+1)(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus119870(120572

5+1)(120578minus1)

(46)

where 1198621 1198622 120572

1 1205722 are unknown parameters

Many other approximate solutions can be obtained bymeans of combinations between initial approximations givenby (33) and (34) and the nonlinear operators (36) (37) or(38)

6 Numerical Examples

In order to show the validity and accuracy of the OHAMwe compare previously obtained approximate solutions (46)with numerical integration results obtained by means ofa fourth-order Runge-Kutta method in combination withshootingmethod and theWolframMathematica 60 software

Using the least-square method for determination of theparameters 119862

119894and 120572

119894 we present the following four cases for

the different values of the coefficients 120574 and 119878

61 Case 1 120574 = 1 and 119878 = minus1 We find dual solutions

(a) We have

1198621= 03333335814 119862

2= 00210985545

1198623= 70349253510 sdot 10

minus6 119862

4= minus12078582888

1198625= 11007711694 119862

6= 12527962825

1198627= minus14794094663 119862

8= 00003670989

119870 = 1 1205721= minus10044596362

1205722= minus19960995780 120572

3= 00039364947

1205724= minus00044067908 120572

5= minus00876855705

(47)

The first expression of the first-order approximate solutiongiven by (46) can be written in the form

119891 (120578) = [1 + 03333335814(120578 minus 1)2

+ 00210985545(120578 minus 1)3

+7034925351 sdot 10minus6

(120578 minus 1)4

] 1198901minus120578

+ 12527962825(120578 minus 1)2

119890minus10039364947(120578minus1)

+ 11007711694(120578 minus 1)2

119890minus10039004219(120578minus1)

minus 14794094663(120578 minus 1)2

119890minus09955932091(120578minus1)

minus 12078582888(120578 minus 1)2

119890minus09955403637(120578minus1)

+ 00003670989(120578 minus 1)2

119890minus09123144295(120578minus1)

(48)

(b) We have

1198621= minus21103792246 119862

2= minus01209917376

1198623= 00016464844 119862

4= minus00401496995

1198625= 51986534028 119862

6= 29102631733

1198627= minus31870328789 119862

8= minus30376107137

119870 = 06170257079 1205721= 15253390642

1205722= minus12342665630 120572

3= 10337532632

1205724= 00700064931 120572

5= 10337537195

(49)

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

119867119894so that within (25) the terms sum

119898

119894=1119867119894(120578 ℎ119895(120578) 119862

119895)119892119894(120578)

are of the same shape as the terms sum119899

119894=1ℎ119894(119909)119892119894(119909) given by

(24) [14ndash18] The first-order approximate solution 119891(120578 119862119894)

also depends on the parameters 119862119895 119895 = 1 119904 The

values of these parameters can be optimally evaluated viavarious methods the least-square method minimization ofthe square residual error the Galerkin method collocationmethod or the Ritz method and so on In this way it is clearthat the first-order approximate solutions given by (16) arewell determined Because the auxiliary functions 119867

119894are not

unique we have freedom to determine multiple solutionsfor nonlinear differential equations (10) and (11) It shouldbe emphasized that our procedure contains the auxiliaryfunctions 119867

119894(119909 119891119895(120578) 119862

119895) 119894 = 1 119898 119895 = 1 119904 which

provides us with a simple way to adjust and control theconvergence of the approximate solutions

5 Multiple Approximate Solutions of theUnsteady Viscous Flow by OHAM

The linear operator can be chosen in the following forms

119871 (119891 (120578)) = 119891101584010158401015840

(120578) + 11987011989110158401015840(120578) (28)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus 11987021198911015840(120578) (29)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) +2119870

119870120578 + 1 minus 11987011989110158401015840(120578) (30)

119871 (119891 (120578)) = 119891101584010158401015840

(120578) minus41198702

(119870120578 + 1 minus 119870)21198911015840(120578) (31)

where 119870 gt 0 is an unknown positive parameter and will bedetermined later

The initial approximation 1198910(120578) can be obtained from

(14) with boundary conditions

1198910(1) = 120574 119891

1015840

0(1) = minus1 119891

1015840

0(infin) = 0 (32)

Equation (14) with the linear operators (28) or (29) hasthe solutions

1198910(120578) = 120574 +

119890minus119870(120578minus1)

minus 1

119870

(33)

while (14) with the linear operators (30) or (31) has thesolutions

1198910(120578) = 120574 minus

1

119870+

1

119870 (119870120578 + 1 minus 119870) (34)

The nonlinear operator corresponding to nonlinear dif-ferential equation (10) is defined as

119873(119891 (120578)) = (1 minus 119870120578)11989110158401015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(35)

for linear operator defined by (28)

The same nonlinear operators for the linear operatordefined by (29) (30) and (31) are respectively

119873(119891 (120578)) = 11989110158401015840(120578) + 119870

21198911015840(120578) + 119891 (120578) 119891

10158401015840(120578)

minus 11989110158402

(120578) minus 119878 [12057811989110158401015840(120578) minus 119891

1015840(120578)]

(36)

119873(119891 (120578)) = (1 minus2119870

119870120578 + 1 minus 119870)11989110158401015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(37)

119873(119891 (120578)) = 11989110158401015840(120578) +

41198702120578

(119870120578 + 1 minus 119870)21198911015840(120578)

+ 119891 (120578) 11989110158401015840(120578) minus 119891

10158402(120578) minus 119878 [120578119891

10158401015840(120578) minus 119891

1015840(120578)]

(38)

Substituting (33) into (35) it holds that

119873(1198910(120578))

= [minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1] 119890minus119870(120578minus1)

(39)

Now comparing (24) and (39) one gets

ℎ1(120578) = minus119870 (119870 + 119878) 120578 + 119870 (120574 + 1) + 119878 minus 1

1198921(120578) = 119890

minus119870(120578minus1)

ℎ119895(120578) = 119892

119895(120578) = 0 for 119895 ge 2

(40)

The first approximation 1198911(120578 119862119894) can be written in the

form

1198911(120578 119862119894) = 119867

1(120578 119862119894) 119890minus119870(120578minus1)

1198911(1) = 119891

1015840

1(1) = 119891

1015840

1(infin) = 0

(41)

where 1198671(120578 119862119894) are arbitrary functions Of course we have

freedom to choose such functions with conditions obtainedfrom (41)

119887

1198671(1 119862119894) = 119867

1015840

1(1 119862119894) = 0 (42)

Advances in Mathematical Physics 5

For example 1198671are given by

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+ 1198623(120578 minus 1)

4

+1198624(120578 minus 1)

2

119890minus(119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(2119870+120572

2)(120578minus1)

+ 1198626(120578 minus 1)

2

119890minus1198701205723(120578minus1)

+ 1198627(120578 minus 1)

2

119890minus1198701205724(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus1198701205725(120578minus1)

(43)

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

+1198624(120578 minus 1)

5

+ [1198625(120578 minus 1)

2

+1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

] 119890minus119870(120578minus1)

+ [1198628(120578 minus 1)

2

+ 1198629(120578 minus 1)

3

+ 11986210(120578 minus 1)

4

+11986211(120578 minus 1)

5

] 119890minus2119870(120578minus1)

(44)

1198671(120578 119862119894)=1198621(120578 minus 1)+119862

2(120578 minus 1)

2

+ 1198623(120578 minus 1)

3

+ 1198624(120578 minus 1)

4

+[minus1198621(120578 minus 1) + 119862

5(120578 minus 1)

2

+ 1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

+ 1198628(120578 minus 1)

5

+ 1198629(120578 minus 1)

6

]119890minus119870(120578minus1)

+ [11986210(120578 minus 1)

2

+ 11986211(120578 minus 1)

3

] 119890minus2119870(120578minus1)

(45)

Taking into consideration only the expression given by(43) from (33) (41) and (16) we obtain the first-orderapproximate solution of (10) and (11) in the form

119891 (120578 119862119894) = 120574 minus

1

119870+ [

1

119870+ 1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

] 119890minus119870(120578minus1)

+1198624(120578 minus 1)

2

119890minus(2119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(3119870+120572

2)(120578minus1)

+1198626(120578 minus 1)

2

119890minus119870(120572

3+1)(120578minus1)

+1198627(120578 minus 1)

2

119890minus119870(120572

4+1)(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus119870(120572

5+1)(120578minus1)

(46)

where 1198621 1198622 120572

1 1205722 are unknown parameters

Many other approximate solutions can be obtained bymeans of combinations between initial approximations givenby (33) and (34) and the nonlinear operators (36) (37) or(38)

6 Numerical Examples

In order to show the validity and accuracy of the OHAMwe compare previously obtained approximate solutions (46)with numerical integration results obtained by means ofa fourth-order Runge-Kutta method in combination withshootingmethod and theWolframMathematica 60 software

Using the least-square method for determination of theparameters 119862

119894and 120572

119894 we present the following four cases for

the different values of the coefficients 120574 and 119878

61 Case 1 120574 = 1 and 119878 = minus1 We find dual solutions

(a) We have

1198621= 03333335814 119862

2= 00210985545

1198623= 70349253510 sdot 10

minus6 119862

4= minus12078582888

1198625= 11007711694 119862

6= 12527962825

1198627= minus14794094663 119862

8= 00003670989

119870 = 1 1205721= minus10044596362

1205722= minus19960995780 120572

3= 00039364947

1205724= minus00044067908 120572

5= minus00876855705

(47)

The first expression of the first-order approximate solutiongiven by (46) can be written in the form

119891 (120578) = [1 + 03333335814(120578 minus 1)2

+ 00210985545(120578 minus 1)3

+7034925351 sdot 10minus6

(120578 minus 1)4

] 1198901minus120578

+ 12527962825(120578 minus 1)2

119890minus10039364947(120578minus1)

+ 11007711694(120578 minus 1)2

119890minus10039004219(120578minus1)

minus 14794094663(120578 minus 1)2

119890minus09955932091(120578minus1)

minus 12078582888(120578 minus 1)2

119890minus09955403637(120578minus1)

+ 00003670989(120578 minus 1)2

119890minus09123144295(120578minus1)

(48)

(b) We have

1198621= minus21103792246 119862

2= minus01209917376

1198623= 00016464844 119862

4= minus00401496995

1198625= 51986534028 119862

6= 29102631733

1198627= minus31870328789 119862

8= minus30376107137

119870 = 06170257079 1205721= 15253390642

1205722= minus12342665630 120572

3= 10337532632

1205724= 00700064931 120572

5= 10337537195

(49)

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

For example 1198671are given by

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+ 1198623(120578 minus 1)

4

+1198624(120578 minus 1)

2

119890minus(119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(2119870+120572

2)(120578minus1)

+ 1198626(120578 minus 1)

2

119890minus1198701205723(120578minus1)

+ 1198627(120578 minus 1)

2

119890minus1198701205724(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus1198701205725(120578minus1)

(43)

1198671(120578 119862119894)=1198621(120578 minus 1)

2

+1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

+1198624(120578 minus 1)

5

+ [1198625(120578 minus 1)

2

+1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

] 119890minus119870(120578minus1)

+ [1198628(120578 minus 1)

2

+ 1198629(120578 minus 1)

3

+ 11986210(120578 minus 1)

4

+11986211(120578 minus 1)

5

] 119890minus2119870(120578minus1)

(44)

1198671(120578 119862119894)=1198621(120578 minus 1)+119862

2(120578 minus 1)

2

+ 1198623(120578 minus 1)

3

+ 1198624(120578 minus 1)

4

+[minus1198621(120578 minus 1) + 119862

5(120578 minus 1)

2

+ 1198626(120578 minus 1)

3

+1198627(120578 minus 1)

4

+ 1198628(120578 minus 1)

5

+ 1198629(120578 minus 1)

6

]119890minus119870(120578minus1)

+ [11986210(120578 minus 1)

2

+ 11986211(120578 minus 1)

3

] 119890minus2119870(120578minus1)

(45)

Taking into consideration only the expression given by(43) from (33) (41) and (16) we obtain the first-orderapproximate solution of (10) and (11) in the form

119891 (120578 119862119894) = 120574 minus

1

119870+ [

1

119870+ 1198621(120578 minus 1)

2

+ 1198622(120578 minus 1)

3

+1198623(120578 minus 1)

4

] 119890minus119870(120578minus1)

+1198624(120578 minus 1)

2

119890minus(2119870+120572

1)(120578minus1)

+1198625(120578 minus 1)

2

119890minus(3119870+120572

2)(120578minus1)

+1198626(120578 minus 1)

2

119890minus119870(120572

3+1)(120578minus1)

+1198627(120578 minus 1)

2

119890minus119870(120572

4+1)(120578minus1)

+ 1198628(120578 minus 1)

2

119890minus119870(120572

5+1)(120578minus1)

(46)

where 1198621 1198622 120572

1 1205722 are unknown parameters

Many other approximate solutions can be obtained bymeans of combinations between initial approximations givenby (33) and (34) and the nonlinear operators (36) (37) or(38)

6 Numerical Examples

In order to show the validity and accuracy of the OHAMwe compare previously obtained approximate solutions (46)with numerical integration results obtained by means ofa fourth-order Runge-Kutta method in combination withshootingmethod and theWolframMathematica 60 software

Using the least-square method for determination of theparameters 119862

119894and 120572

119894 we present the following four cases for

the different values of the coefficients 120574 and 119878

61 Case 1 120574 = 1 and 119878 = minus1 We find dual solutions

(a) We have

1198621= 03333335814 119862

2= 00210985545

1198623= 70349253510 sdot 10

minus6 119862

4= minus12078582888

1198625= 11007711694 119862

6= 12527962825

1198627= minus14794094663 119862

8= 00003670989

119870 = 1 1205721= minus10044596362

1205722= minus19960995780 120572

3= 00039364947

1205724= minus00044067908 120572

5= minus00876855705

(47)

The first expression of the first-order approximate solutiongiven by (46) can be written in the form

119891 (120578) = [1 + 03333335814(120578 minus 1)2

+ 00210985545(120578 minus 1)3

+7034925351 sdot 10minus6

(120578 minus 1)4

] 1198901minus120578

+ 12527962825(120578 minus 1)2

119890minus10039364947(120578minus1)

+ 11007711694(120578 minus 1)2

119890minus10039004219(120578minus1)

minus 14794094663(120578 minus 1)2

119890minus09955932091(120578minus1)

minus 12078582888(120578 minus 1)2

119890minus09955403637(120578minus1)

+ 00003670989(120578 minus 1)2

119890minus09123144295(120578minus1)

(48)

(b) We have

1198621= minus21103792246 119862

2= minus01209917376

1198623= 00016464844 119862

4= minus00401496995

1198625= 51986534028 119862

6= 29102631733

1198627= minus31870328789 119862

8= minus30376107137

119870 = 06170257079 1205721= 15253390642

1205722= minus12342665630 120572

3= 10337532632

1205724= 00700064931 120572

5= 10337537195

(49)

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

The second expression of the first-order approximate solution(48) is

119891 (120578) = minus06206780156

+ [16206780156

minus 21103792246(120578 minus 1)2

minus 01209917376(120578 minus 1)3

+00016464844(120578 minus 1)4

] 119890minus06170257079(120578minus1)

minus 00401496995(120578 minus 1)2

119890minus27593904801(120578minus1)

minus 30376107137(120578 minus 1)2

119890minus12548783286(120578minus1)

+ 29102631733(120578 minus 1)2

119890minus12548780470(120578minus1)

minus 31870328789(120578 minus 1)2

119890minus06602215139(120578minus1)

+ 51986534028(120578 minus 1)2

119890minus06168105608(120578minus1)

(50)

62 Case 2 120574 = 1 and 119878 = minus2 We obtain two dual solutionsrespectively

(a) We have

119891 (120578) = 05913897892

+ [04086102107 minus 10365306560(120578 minus 1)2

minus 00340475469(120578 minus 1)3

minus00208575441(120578 minus 1)4

] 119890minus24473201442(120578minus1)

+ 09265758596(120578 minus 1)2

119890minus51900153052(120578minus1)

minus 09181338819(120578 minus 1)2

119890minus51900088912(120578minus1)

+ 03978132072(120578 minus 1)2

119890minus27650851899(120578minus1)

minus 02705600173(120578 minus 1)2

119890minus24471963928(120578minus1)

+ 09777857487(120578 minus 1)2

119890minus23078302052(120578minus1)

(51)

(b) We have

119891 (120578) = minus22653986945

+ [32653986945

minus 52607190849(120578 minus 1)2

minus 00827319763(120578 minus 1)3

minus00012312519(120578 minus 1)4

] 119890minus03062413179(120578minus1)

minus 00121539864(120578 minus 1)2

119890minus82295440677sdot10

6(120578minus1)

minus 02925877070(120578 minus 1)2

119890minus35914249798(120578minus1)

minus 21949262210(120578 minus 1 )2

119890minus13550840661(120578minus1)

minus 07516910873(120578 minus 1)2

119890minus07893951308(120578minus1)

+ 51689397253(120578 minus 1)2

119890minus02883024664(120578minus1)

(52)

63 Case 3 120574 = 2 and 119878 = minus1 We obtain the correspondingdual solutions respectively

(a) We have

119891 (120578)=15235393146

+ [04764606853

+2146184925584(120578 minus 1)2

minus443762024048(120578minus1)3

+029490275681(120578 minus 1)4

] 119890minus20988090531(120578minus1)

minus 253472639092(120578 minus 1)2

119890minus36896808591(120578minus1)

+ 652289425339(120578 minus 1)2

119890minus35588615870(120578minus1)

minus 103740164628(120578 minus 1)2

119890minus32031469638(120578minus1)

+ 162245606899(120578 minus 1)2

119890minus27947822439(120578minus1)

minus 310683634535(120578 minus 1)2

119890minus24082455720(120578minus1)

(53)

(b) We have

119891 (120578) = minus09034898463

+ [29034898463

minus 104395166917(120578 minus 1)2

+ 00004971369(120578 minus 1)3

minus50246397821 sdot 10minus6

(120578 minus 1)4

] 119890minus03444131210(120578minus1)

minus 01550351871(120578 minus 1)2

119890minus36164143875(120578minus1)

minus 03301513414(120578 minus 1)2

119890minus17295917121(120578minus1)

minus 05465749334(120578 minus 1)2

119890minus09240575546(120578minus1)

minus 01988704213(120578 minus 1)2

119890minus05891517113(120578minus1)

+ 104244457170(120578 minus 1)2

119890minus03444067343(120578minus1)

(54)

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

Table 1 Comparison between the skin friction coefficient11989110158401015840

OHAM(1)

and 11989110158401015840

numerical(1) (error = |11989110158401015840

OHAM(1) minus 11989110158401015840

numerical(1)|)

120574 119878

The first expression of the first-orderapproximate solution

11989110158401015840

OHAM(1) 11989110158401015840

numerical(1) Error1 minus1 10000007544 09999999999 75 sdot 10

minus7

1 minus2 26012206647 26012206637 99 sdot 10minus10

2 minus1 25632048369 25632048269 99 sdot 10minus9

2 minus2 37150911381 37150910381 99 sdot 10minus8

64 Case 4 120574 = 2 and 119878 = minus2 It holds that

(a)

119891 (120578) = 17019613535

+[02980386464

minus 08562787938(120578 minus 1)2

+ 00106882665(120578 minus 1)3

+00310882235(120578 minus 1)4

] 119890minus33552695664(120578minus1)

minus 08757867742(120578 minus 1)2

119890minus63225573014(120578minus1)

+ 09017666577(120578 minus 1)2

119890minus63225571694(120578minus1)

+ 11112669004(120578 minus 1)2

119890minus33547932645(120578minus1)

minus 09948349570(120578 minus 1)2

119890minus27022990501(120578minus1)

+ 08937777528(120578 minus 1)2

119890minus26833589327(120578minus1)

(55)

(b)

119891 (120578) = minus20992008391

+[40992008391

minus 00204738147(120578 minus 1)2

+ 00008263633(120578 minus 1)3

minus95217500205 sdot 10minus6

(120578 minus 1)4

] 119890minus02439499890(120578minus1)

minus 03773706801(120578 minus 1)2

119890minus52745243636(120578minus1)

minus 08947192974(120578 minus 1)2

119890minus27755705539(120578minus1)

minus 31860621314(120578 minus 1)2

119890minus13987895675(120578minus1)

minus 10872627477(120578 minus 1)2

119890minus08222938433(120578minus1)

minus 02399812958(120578 minus 1)2

119890minus04723317954(120578minus1)

(56)

In Table 1 we present a comparison between the skinfriction coefficient 119891

10158401015840

(1) obtained by means of OHAM and

Table 2 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (48) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (48) 119891numerical(120578) Error1 1 1 095 04493290026 04493289954 71 sdot 10

minus9

135 02018965295 02018965301 64 sdot 10minus10

215 00407622028 00407622034 602 sdot 10minus10

295 00082297496 00082297482 14 sdot 10minus9

375 00016615551 00016615578 26 sdot 10minus9

415 00007465838 00007465853 15 sdot 10minus9

9 00003354620 00003354620 14 sdot 10minus11

infin 66613 sdot 10minus16

87647 sdot 10minus16

21 sdot 10minus14

Table 3 Comparison between the derivative 1198911015840

(120578) obtained from(48) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (48) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus04493289667 minus04493289878 21 sdot 10

minus8

135 minus02018965563 minus02018965347 21 sdot 10minus8

215 minus00407621959 minus00407621759 2003 sdot 10minus8

295 minus00082297502 minus00082297476 25 sdot 10minus9

375 minus00016615582 minus00016615571 11 sdot 10minus9

415 minus00007465846 minus00007465884 38 sdot 10minus9

9 minus00003354605 minus00003354620 15 sdot 10minus9

infin 0 0 0

Table 4 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (50) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (50) 119891numerical(120578) Error1 1 1 095 03025681133 03025729017 47 sdot 10

minus6

135 minus01319503386 minus01319549886 46 sdot 10minus6

215 minus04960887571 minus04961000384 11 sdot 10minus5

295 minus05909291088 minus05909208156 82 sdot 10minus6

375 minus06138226907 minus06138228346 14 sdot 10minus7

415 minus06174089320 minus06174180498 91 sdot 10minus6

9 minus06191261345 minus06191351201 89 sdot 10minus6

infin minus06206780156 minus06206780156 45 sdot 10minus14

numerical results The comparisons are found to be in verygood agreement for the first and the second solutions

In Tables 2 3 4 5 6 7 8 9 10 11 12 and 13 we presenta comparison between all approximate solutions 119891(120578) and1198911015840

(120578) and numerical results obtained by the Runge-Kuttamethod in combination with shooting method for differentvalues of variable 120578 and different values of coefficients 120574 and119878

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

Table 5 Comparison between the derivative 1198911015840

(120578) obtained from(50) and numerical results for 120574 = 1 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (50) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus07034979138 minus07036237128 12 sdot 10

minus4

135 minus04016692734 minus04016032225 66 sdot 10minus5

215 minus01094954512 minus01095146046 19 sdot 10minus5

295 minus00270353065 minus00270213000 14 sdot 10minus5

375 minus00063314620 minus00063462255 14 sdot 10minus5

415 minus00030330955 minus00030391496 605 sdot 10minus6

9 minus00014527986 minus00014468075 59 sdot 10minus6

infin 0 0 0

Table 6 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (51) obtained by OHAM andnumerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (51) 119891numerical(120578) Error1 1 1 095 06540357246 06540356881 36 sdot 10

minus8

135 06019106890 06019108087 11 sdot 10minus7

215 05917231482 05917232409 92 sdot 10minus8

295 05914012763 05914011910 85 sdot 10minus8

375 05913902415 05913901904 51 sdot 10minus8

415 05913898860 05913898617 24 sdot 10minus8

9 05913898108 05913897995 11 sdot 10minus8

infin 05913897892 05913897892 69 sdot 10minus14

Table 7 Comparison between the derivative 1198911015840

(120578) obtained from(51) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (51) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01423061993 minus01423072726 107 sdot 10

minus6

135 minus00231135261 minus00231135957 69 sdot 10minus8

215 minus00007095855 minus00007094951 904 sdot 10minus8

295 minus00000238338 minus00000238925 58 sdot 10minus8

375 minus88615 sdot 10minus7

minus84307 sdot 10minus7

43 sdot 10minus8

415 minus18368 sdot 10minus7

minus15887 sdot 10minus7

24 sdot 10minus8

9 minus40119 sdot 10minus8

minus30513 sdot 10minus8

96 sdot 10minus9

infin 0 0 0

It can be observed that the solutions obtained by OHAMare in excellent agreement with numerical results

Figures 2 and 3 present the displacement119891(120578) for differentvalues of unsteadiness 119878 120574 = 1 and 120574 = 2 respectively It isseen that for fixed value of 120574 the displacement 119891(120578) decreasesas 119878 increases for the first solutions The opposite trend isobserved for the second solutions

Table 8 Comparison between the second expression of the first-order approximate solutions 119891(120578) given by (52) obtained by OHAMand numerical results for 120574 = 1 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (52) 119891numerical(120578) Error1 1 11015840 095 minus04931638904 minus04931976737 33 sdot 10

minus5

135 minus15746868741 minus15746416452 45 sdot 10minus5

215 minus21944651392 minus21944897035 24 sdot 10minus5

295 minus22598561227 minus22598257759 30 sdot 10minus5

375 minus22649614997 minus22650206599 59 sdot 10minus5

415 minus22652663671 minus22653038433 37 sdot 10minus5

9 minus22653903391 minus22653753555 14 sdot 10minus5

infin minus22653986945 minus22653986945 51 sdot 10minus14

Table 9 Comparison between the derivative 1198911015840

(120578) obtained from(52) and numerical results for 120574 = 1 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (52) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus18404107968 minus18405104714 99 sdot 10

minus5

135 minus08900821972 minus08900516447 305 sdot 10minus5

215 minus01082080071 minus01081247967 83 sdot 10minus5

295 minus00091018331 minus00091580377 56 sdot 10minus5

375 minus00006415895 minus00006482600 66 sdot 10minus6

415 minus00002202957 minus00001651469 55 sdot 10minus5

9 minus00001070484 minus00000411696 65 sdot 10minus5

infin 0 0 0

Table 10 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (53) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus1 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (53) 119891numerical(120578) Error1 2 2 095 16330858924 16330861523 25 sdot 10

minus7

135 15550512928 15550531316 18 sdot 10minus6

215 15269871985 15269891353 19 sdot 10minus6

295 15239948037 15239925608 22 sdot 10minus6

375 15236020621 15236049410 28 sdot 10minus6

415 15235615146 15235649024 33 sdot 10minus6

9 15235467934 15235494172 26 sdot 10minus6

infin 15235393146 15235393145 107 sdot 10minus11

Figures 4 and 5 depict the velocity profiles 1198911015840

(120578) for fixedvalue of 120574 and some values of 119878 It is observed that in all casesthe velocity of fluid is damped faster as the magnitude ofthe unsteadiness parameter increases The velocity boundarylayer thickness decreases as 119878 decreases which implies theincrease of the velocity gradient For the first solution thevelocity gradient is positive in contrast with the second

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

Table 11 Comparison between the derivative 1198911015840

(120578) obtained from(53) and numerical results for 120574 = 2 and 119878 = minus1 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (53) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus01815921958 minus01816077514 15 sdot 10

minus5

135 minus00465577528 minus00465561725 15 sdot 10minus6

215 minus00045275785 minus00045260284 15 sdot 10minus6

295 minus00005600263 minus00005588247 12 sdot 10minus6

375 minus00000797560 minus00000778400 19 sdot 10minus6

415 minus00000294916 minus00000299133 42 sdot 10minus7

9 minus00000104089 minus00000116668 12 sdot 10minus6

infin 0 0 0

Table 12 Comparison between the first expression of the first-orderapproximate solutions 119891(120578) given by (55) obtained by OHAM andnumerical results for 120574 = 2 and 119878 = minus2 (error = |119891OHAM(120578) minus

119891numerical(120578)|)

120578 119891OHAM(120578) (55) 119891numerical(120578) Error1 2 2 095 17283791967 17283789517 24 sdot 10

minus7

135 17050648140 17050648331 19 sdot 10minus8

215 17020220942 17020221347 404 sdot 10minus8

295 17019629186 17019628448 73 sdot 10minus8

375 17019614024 17019613948 75 sdot 10minus9

415 17019613622 17019613605 17 sdot 10minus9

9 17019613550 17019613548 21 sdot 10minus10

infin 17019613535 17019613535 44 sdot 10minus13

Table 13 Comparison between the derivative 1198911015840

(120578) obtained from(55) and numerical results for 120574 = 2 and 119878 = minus2 (error = |119891

1015840

OHAM(120578)minus

1198911015840

numerical(120578)|)

120578 1198911015840

OHAM(120578) (55) 1198911015840

numerical(120578) Error1 minus1 minus1 095 minus00740545041 minus00740547752 27 sdot 10

minus7

135 minus00079946060 minus00079954465 84 sdot 10minus7

215 minus00001438623 minus00001440860 22 sdot 10minus7

295 minus34550 sdot 10minus6

minus33915 sdot 10minus6

63 sdot 10minus8

375 minus10521 sdot 10minus7

minus93066 sdot 10minus8

12 sdot 10minus8

415 minus18882 sdot 10minus8

minus15206 sdot 10minus8

36 sdot 10minus9

9 minus33327 sdot 10minus9

minus28550 sdot 10minus9

47 sdot 10minus10

infin 0 0 0

solution These conclusions are in concordance with resultsobtained in [8 9]

From Table 1 it is seen that the magnitude of 11989110158401015840

(1)

increases as the parameters 120574 increase in the case of the firstsolutions given by subcases 61(a) 62(a) 63(a) and 64(a)The opposite trend is observed for the variation of 119878 thatis increasing 119878 is to decrease the magnitude of the skincoefficient 119891

10158401015840

(1) In the case of the second solutions given

First solution

2 4 6 8

1

Second solution

f(120578)

120578

S = minus1

S = minus2

minus1

minus2

minus3

Figure 2 Displacement for different values of 119878 when 120574 = 1

2 4 6 8

1

2

120578

S = minus1S = minus2

minus1

minus2

First solutionSecond solution

f(120578)

Figure 3 Displacement for different values of 119878 when 120574 = 2

2 4 6 8120578

S = minus1

S = minus2

minus08

minus06

minus04

minus02

minus10

minus12

First solutionSecond solution

f998400(120578)

Figure 4 Velocity profile for different values of 119878 and 120574 = 1

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

2 4 6 8120578

S = minus1

S = minus2

minus15

minus10

minus05

minus20

minus25

First solutionSecond solution

f998400(120578)

Figure 5 Velocity profile for different values of 119878 and 120574 = 2

by subcases 61(b) 62(b) 63(b) and 64(b) the variation ofthe skin friction coefficient 119891

10158401015840

(1) is reverse

7 Conclusions

Theproblem of unsteady viscous flowwas solved bymeans ofoptimal homotopy asymptotic method and obtained resultsare compared with numerical results The effects of theparameters 120574 and 119878 have been analyzed and presented graph-ically and in 13 tables This problem admits a lot of solutionsdepending on some convergence-control parameters and incertain conditions (119878 lt 0) every one of these solutionsadmits a dual solution The magnitude of the skin frictioncoefficient decreases with the increasing of the unsteadinessparameter The flow velocity and the skin friction coefficientare influenced by the parameters 120574 and 119878 Our procedureis valid even if the nonlinear differential equation does notcontain small or large parameters In our construction ofthe homotopy appear some distinctive concepts such as theauxiliary convergence-control function 119867

1 the linear oper-

ator 119871 and several optimal convergence-control parameters1198621 1198622 which ensure a fast convergence of the solutions

The examples presented in this work lead to the conclusionthat the obtained results are of the exceptional accuracyusing only one iteration The OHAM provides us with arigorous way to control and adjust the convergence of thesolutions through the auxiliary function119867

1involving several

parameters which are optimally determined

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Uchida and H Aoki ldquoUnsteady flows in a semi-infinitecontracting or expanding piperdquo Journal of Fluid Mechanics vol82 no 2 pp 371ndash387 1977

[2] FM Skalak and C YWang ldquoOn the unsteady squeezing of vis-cous fluid from a tuberdquo Journal of the Australian MathematicalSociety B vol 21 pp 65ndash74 1979

[3] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[4] A Ishak R Nazar and I Pop ldquoUniform suctionblowing effecton flow and heat transfer due to a stretching cylinderrdquo AppliedMathematical Modelling vol 32 no 10 pp 2059ndash2066 2008

[5] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 4 pages 2009

[6] T G Fang S S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[7] W M K A W Zaimi A Ishak and I Pop ldquoUnsteadyviscous flow over a shrinking cylinderrdquo Journal of King SaudUniversitymdashScience vol 25 no 2 pp 143ndash148 2013

[8] K Zaimi A Ishak and I Pop ldquoUnsteady flow due to acontracting cylinder in a nanofluid using Buongiornorsquos modelrdquoInternational Journal of Heat and Mass Transfer vol 68 pp509ndash513 2014

[9] A Nayfeh Problems in Perturbation A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1985

[10] Z H Khan R Gul and W A Khan ldquoEffect of variablethermal conductivity on heat transfer from a hollow spherewith heat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 301ndash309 Jacksonville Fla USA August2008

[11] R Gul Z H Khan and W A Khan ldquoHeat transfer fromsolids with variable thermal conductivity and uniform internalheat generation using homotopy perturbation methodrdquo inProceedings of the ASMEHeat TransferTheory and FundamentalResearch vol 1 pp 311ndash319 Jacksonville Fla USA August 2008

[12] Z H Khan R Gul and W A Khan ldquoApplication of adomiandecomposition method for Sudumu transformrdquo NUST Journalof Engineering Sciences vol 12 no 1 pp 40ndash44 2008

[13] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[14] J H He ldquoModified Lindstedt-Poincare methods for somestrongly non-linear oscillations I Expansion of a constantrdquoInternational Journal of Non-Linear Mechanics vol 37 no 2 pp309ndash314 2002

[15] A Belendez C Pascual C Neipp T Belendez and A Hernan-dez ldquoAn equivalent linearization method for conservative non-linear oscillationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 9 pp 9ndash19 2001

[16] N Herisanu and V Marinca ldquoOptimal homotopy perturbationmethod for a non-conservative dynamical system of a rotatingelectrical machinerdquo Zeitschrift fur Naturforschung A vol 67 pp509ndash516 2012

[17] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

[18] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

[19] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineeringmdashSome Approximate Approaches Springer Heidel-berg Germany 2011

[20] VMarinca andN Herisanu ldquoAn optimal homotopy asymptoticapproach applied to nonlinearMHDJeffery-Hamel flowrdquoMath-ematical Problems in Engineering vol 2011 Article ID 169056 16pages 2011

[21] V Marinca and N Herisanu ldquoOptimal homotopy asymptoticapproach to nonlinear oscillators with discontinuitiesrdquo Scien-tific Research and Essays vol 8 no 4 pp 161ndash167 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of