Research Article Analytical Investigation of...
7
Research Article Analytical Investigation of Magnetohydrodynamic Flow over a Nonlinear Porous Stretching Sheet Fazle Mabood 1 and Nopparat Pochai 2 1 Department of Mathematics, University of Peshawar, Peshawar 25120, Pakistan 2 Department of Mathematics, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, ailand Correspondence should be addressed to Nopparat Pochai; nop [email protected] Received 30 March 2016; Revised 11 May 2016; Accepted 29 May 2016 Academic Editor: Giorgio Kaniadakis Copyright © 2016 F. Mabood and N. Pochai. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigated the magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet with the help of semianalytical method known as optimal homotopy asymptotic method (OHAM). e effects of different parameters on fluid flow are investigated and discussed. e obtained results are compared with numerical Runge-Kutta-Fehlberg fourth-fiſth-order method. It is found that the OHAM solution agrees well with numerical as well as published data for different assigned values of parameters; this thus indicates the feasibility of the proposed method (OHAM). 1. Introduction Nonlinear differential equations are frequently arising from mathematical modeling of many physical phenomena. Sev- eral are solved by means of numerical methods and some are solved using the analytic methods such as perturbation [1, 2]. Researchers and engineers have paid more attention towards the analytical solution of boundary layer equations arising in numerous fluids phenomena [3–5]. e study of boundary layer flow for an incompressible fluid has many important applications in science and engineering, for example, the cooling of metallic plate in a cooling bath, the boundary layer along liquid film condensation process, and polymer industries. In recent years, the analysis of magnetohydrodynamics (MHD) flow of a fluid over a stretching sheet has more popu- larity industrially and consequently becomes a fundamental problem in fluid dynamics [6–11]. McCormack and Crane [12] have initiated the stretching problem. e steady flow over a stretching sheet has numerous aspects, such as MHD flow, Non-Newtonian fluids, porous plate, porous medium, and heat transfer phenomena. Sakiadis [13, 14] is pioneer in this area who has investigated the boundary layer flow with uniform speed over continuously stretching surface. Later on, the work of Sakiadis was investigated and verified experimentally with different aspects by many researchers (see [15–17] and the references therein). Most attention has so far been devoted to the analysis of flow of viscoelastic fluids [18–21] and the joint effect of viscoelasticity and magnetic field has been worked out by Ariel [22]. Khan et al. [23] studied MHD nonlinear porous stretching sheet using homotopy perturbation transform method (HPTM). Moreover, Chiam [24], Dandapat and Gupta [20], and Pavlov [25] have considered the motion of micropolar, power-law fluids and MHD flow over a stretching wall, respectively. e optimal homotopy asymptotic method is a powerful approximate analytical technique that is straightforward to use and does not require the existence of any small or large parameter. Optimal homotopy asymptotic method (OHAM) is employed to construct the series solution of the problem. is method is a consistent analytical tool and it has already been applied to nonlinear differential equations arising in the science and engineering [26–28]. So far as we know there has been no OHAM solution of MHD flow over a nonlinear porous stretching sheet. is paper is organized as follows. First in Section 2, we formulate the problem. In Section 3 we present basic principles of OHAM. e OHAM solution for MHD flow Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2016, Article ID 7821405, 6 pages http://dx.doi.org/10.1155/2016/7821405
Transcript of Research Article Analytical Investigation of...
Research ArticleAnalytical Investigation of Magnetohydrodynamic Flow overa Nonlinear Porous Stretching Sheet
Fazle Mabood1 and Nopparat Pochai2
1Department of Mathematics University of Peshawar Peshawar 25120 Pakistan2Department of Mathematics King Mongkutrsquos Institute of Technology Ladkrabang Bangkok 10520 Thailand
Correspondence should be addressed to Nopparat Pochai nop mathyahoocom
Received 30 March 2016 Revised 11 May 2016 Accepted 29 May 2016
Academic Editor Giorgio Kaniadakis
Copyright copy 2016 F Mabood and N Pochai 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
We investigated the magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet with the helpof semianalytical method known as optimal homotopy asymptotic method (OHAM) The effects of different parameters on fluidflow are investigated and discussed The obtained results are compared with numerical Runge-Kutta-Fehlberg fourth-fifth-ordermethod It is found that the OHAM solution agrees well with numerical as well as published data for different assigned values ofparameters this thus indicates the feasibility of the proposed method (OHAM)
1 Introduction
Nonlinear differential equations are frequently arising frommathematical modeling of many physical phenomena Sev-eral are solved by means of numerical methods and some aresolved using the analytic methods such as perturbation [1 2]
Researchers and engineers have paid more attentiontowards the analytical solution of boundary layer equationsarising in numerous fluids phenomena [3ndash5] The studyof boundary layer flow for an incompressible fluid hasmany important applications in science and engineering forexample the cooling of metallic plate in a cooling bath theboundary layer along liquid film condensation process andpolymer industries
In recent years the analysis of magnetohydrodynamics(MHD) flow of a fluid over a stretching sheet has more popu-larity industrially and consequently becomes a fundamentalproblem in fluid dynamics [6ndash11] McCormack and Crane[12] have initiated the stretching problem The steady flowover a stretching sheet has numerous aspects such as MHDflow Non-Newtonian fluids porous plate porous mediumand heat transfer phenomena Sakiadis [13 14] is pioneerin this area who has investigated the boundary layer flowwith uniform speed over continuously stretching surfaceLater on the work of Sakiadis was investigated and verified
experimentally with different aspects by many researchers(see [15ndash17] and the references therein)
Most attention has so far been devoted to the analysisof flow of viscoelastic fluids [18ndash21] and the joint effect ofviscoelasticity and magnetic field has been worked out byAriel [22] Khan et al [23] studied MHD nonlinear porousstretching sheet using homotopy perturbation transformmethod (HPTM) Moreover Chiam [24] Dandapat andGupta [20] and Pavlov [25] have considered the motion ofmicropolar power-law fluids andMHDflowover a stretchingwall respectively
The optimal homotopy asymptotic method is a powerfulapproximate analytical technique that is straightforward touse and does not require the existence of any small or largeparameter Optimal homotopy asymptotic method (OHAM)is employed to construct the series solution of the problemThis method is a consistent analytical tool and it has alreadybeen applied to nonlinear differential equations arising in thescience and engineering [26ndash28] So far as we know therehas been no OHAM solution of MHD flow over a nonlinearporous stretching sheet
This paper is organized as follows First in Section 2we formulate the problem In Section 3 we present basicprinciples of OHAM The OHAM solution for MHD flow
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 7821405 6 pageshttpdxdoiorg10115520167821405
2 Advances in Mathematical Physics
problem is given in Section 4 In Section 5 we analyze thecomparison of the solutions using OHAM with numericalmethod (NM) Section 6 is devoted for the concludingremarks
2 Governing Equation
Weconsider theMHDflowof an incompressible viscous fluidover a nonlinear porous stretching sheet at 119910 = 0 Electricallyconducting fluid under the influence of appliedmagnetic field119861(119909) normal to the stretching sheet the induced magneticfield is assumed to be negligible Under such assumption theMHD boundary layer equations are governed by
120597119906
120597119909+120597V120597119910= 0
119906120597119906
120597119909+ V120597119906
120597119910= ]1205972119906
1205971199102minus 1205901198612(119909)
120588119906
(1)
where 119906 and V are the velocity components in the 119909- and119910-directions respectively ] is the kinematic viscosity 120588 isfluid density 120590 is the electrical conductivity and 119861(119909) is themagnetic field strength where 119861(119909) = 119861
0119909(119899minus1)2
The boundary conditions to the nonlinear porous stretch-ing sheet are given below
119906 (119909 0) = 119888119909119899
V (119909 0) = minus1198810
119906 (119909infin) = 0
(2)
where 119888 is the stretching parameter and 1198810is the porosity of
the plate (whereas 1198810gt 0 represents suction and 119881
0lt 0
corresponds to injection)By introducing dimensionless variables for nondimen-
sionalized form of momentum and energy equations
120578 = radic119888 (119899 + 1)
2V119909(119899minus1)2
119910
119906 = 1198881199091198991198911015840(120578)
V = minusradic119888V (119899 + 1)2
119909(119899minus1)2
119891 (120578) +119899 minus 1
119899 + 11205781198911015840(120578)
(3)
Using (3) the governing equations can be reduced to non-linear differential equation where 119891 is a function of thesimilarity variable 120578
119891101584010158401015840+ 11989111989110158401015840minus 12057311989110158402minus119872119891
1015840= 0 (4)
subject to the boundary conditions
119891 997888rarr 119870
1198911015840997888rarr 1
as 120578 997888rarr 0
1198911015840997888rarr 0 as 120578 997888rarr infin
(5)
as
120573 =2119899
119899 + 1
119872 =21205901205730
2
120588119888 (1 + 119899)
119870 =119881119900
radicV119888 (119899 + 1) 2119909(119899minus1)2
(6)
120573 is the nondimensional parameter 119872 is the magneticparameter and119870 is wall mass transfer parameter
3 Basic Principles of OHAM
We review the basic principles of OHAMas developed in [26]in the following steps
(i) Let us consider the following differential equation
119860 [V (119909)] + 119886 (119909) = 0 119909 isin Ω (7)
where Ω is problem domain 119860(V) = 119871(V) + 119873(V) where 119871119873 are linear and nonlinear operators V(119909) is an unknownfunction and 119886(119909) is a known function
(ii) Construct an optimal homotopy equation as
(1 minus 119902) [119871 (120601 (119909 119902)) + 119886 (119909)]
minus 119867 (119902) [119860 (120601 (119909 119902)) + 119886 (119909)] = 0
(8)
where 0 le 119902 le 1 is an embedding parameter and 119867(119902) =sum119898
119896=1119902119896119862119896is auxiliary function on which the convergence
of the solution is greatly dependent The auxiliary function119867(119902) also adjusts the convergence domain and controls theconvergence region
(iii) Expand 120601(119909 119902 119862119895) in Taylorrsquos series about 119902 one has
an approximate solution
120601 (119909 119902 119862119895) = V0(119909) +
infin
sum
119896=1
V119896(119909 119862119895) 119902119896
119895 = 1 2 3
(9)
Many researchers have observed that the convergence of theseries equation (9) depends upon 119862
119895(119895 = 1 2 119898) if it is
convergent then we obtain
V = V0(119909) +
119898
sum
119896=1
V119896(119909 119862119895) (10)
(iv) Substituting (10) into (7) we have the followingresidual
119877 (119909 119862119895) = 119871 (V (119909 119862
119895)) + 119886 (119909) + 119873 (V (119909 119862
119895)) (11)
If119877(119909 119862119895) = 0 then Vwill be the exact solution For nonlinear
problems generally this will not be the case For determining119862119895(119895 = 1 2 119898) Galerkinrsquos Method or the method of
least squares can be used(v) Finally substitute these constants in (10) and one can
get the approximate solution
Advances in Mathematical Physics 3
Table 1 Comparison of OHAM results with numerical method for 119870 = 0 120573 = 5 and119872 = 5
120578119891(120578) 119891
1015840(120578)
OHAM NM error OHAM NM error00 0 0 0 1 1 001 0086665 0086668 00034 0746974 0747032 0007802 0151657 0151663 00039 0562191 0562169 0003903 0200655 0200658 00014 0424101 0424023 0018304 0237614 0237598 00067 0319327 0319201 0039405 0265301 0265277 00090 0237681 0237665 0006706 0285709 0285688 00037 0172819 0172901 0047407 0300262 0300252 00033 0119889 0120035 0121608 0309964 0309969 00016 0075254 0075405 0200209 0315493 0315512 00060 0036055 0036162 0295810 0317278 0317303 00078 0 0 0
4 Series Solution via OHAM
According to the OHAM applying (8) to (4)
(1 minus 119902) (11989110158401015840+ 1198911015840)
minus 119867 (119902) (119891101584010158401015840+ 11989111989110158401015840minus 12057311989110158402minus119872119891
1015840) = 0
(12)
where primes denote differentiation with respect to 120578We consider 119891 and119867(119902) as the following
119891 = 1198910+ 1199021198911+ 11990221198912
119867 (119902) = 1199021198621+ 11990221198622
(13)
Using (13) in (12) and some simplifying and rearranging theterms based on the powers of 119902 we obtain zeroth- first- andsecond-order problems as follows
Zeroth Order Problem Consider
11989110158401015840
0(120578) + 119891
1015840
0(120578) = 0 (14)
with boundary conditions
1198910(0) = 119870
1198911015840
0(0) = 1
(15)
Its solution is
1198910(120578) = 119870 + 1 minus 119890
minus120578 (16)
First Order Problem Consider
11989110158401015840
1+ 1198911015840
1
minus 1198621(119890minus120578minus119872119890minus120578+ (119870 + 1 minus 119890
minus120578) 119890minus120578minus 120573119890minus2120578)
= 0
(17)
with boundary conditions
1198910(0) = 0
1198911015840
0(0) = 0
(18)
It solution is
1198911(120578 1198621) =1
21198621119890minus2120578+ 1198621119872(1 + 120578) 119890
minus120578
+ 1198621119870(1 + 120578) 119890
minus120578+1
21198621120573119890minus2120578
minus 1198621(1 + 120573) 119890
minus120578
+1198621
2(1 minus 2119872 minus 2119870 + 1)
(19)
And this goes onWe obtain the three-term solution using OHAM for 119902 =
1
(120578 1198621 1198622) = 1198910(120578) + 119891
1(120578 1198621) + 1198912(120578 1198621 1198622) (20)
We use the method of least squares to obtain the unknownconvergent constants 119862
1 1198622in (20) for particular case if
119872 = 1 119870 = 0 and 120573 = 15 then the values of 1198621 1198622are
1198621= minus02233887095 119862
2= 08569476324
5 Results and Discussion
Table 1 shows the comparison of OHAM results with numer-ical (NM) for different values of parameters in Table 2 wecompare the numerical values of 11989110158401015840(0) via OHAM withexisting solution [23 24] It is noteworthy to mention herethat theOHAMgives lowest error than othermethodsThisanalysis shows that OHAM suits for MHD boundary layerflowproblems In Figures 1ndash3we have shown the effects of thedimensionless parameter 120573 the magnetic parameter119872 andthe mass transfer parameter 119870 with various assigned values
4 Advances in Mathematical Physics
Table 2 Comparison of 11989110158401015840(0) via OHAM with other methods for various values of119872 at 119870 = 1 and 120573 = 15
119872 Chiam [24] HPTM [23] Present results error Chiam[24]
error HPTM[23]
errorpresent results
0 minus14902 minus14902 minus14897 02826 02826 024891 minus15253 minus15253 minus152529 00019 00019 000135 minus251616 minus25161 minus251611 00015 00039 0003510 minus336632 minus33658 minus33659 00002 00151 0002150 minus716471 minus716354 minus716366 0 00163 00146100 minus100664 minus100648 minus100653 0 00158 00109
1
08
06
04
02
0
0 1 2 3 4 5 6
120573 = 15M = 02
K
0
05
10
120578
f998400(120578)
Figure 1 Effect of wall mass transfer parameter on dimensionlessvelocity for 120573 = 15119872 = 02
Figure 1 is displayed for the influence of 119870 It is observedthat the dimensionless velocity and associated boundarylayer thickness decrease with an increase in 119870 Figures 2and 3 are given for the velocity profile against 120578 in orderto show the influences of parameters 119872 120573 respectivelyFigure 2 exhibits the effect of magnetic parameter on thedimensionless velocity It is observed that the velocity profileof the fluid is significantly reduced with increasing values of119872 Physically an increase in magnetic parameter 119872 resultsin a strong reduction in dimensionless velocity 1198911015840 This isdue to the fact that magnetic field introduces a retardingbody force which acts transverse to the direction of theappliedmagnetic fieldThis body force known as the Lorentzforce decelerates the boundary layer flow and thickens themomentum boundary layer and hence induces an increase inthe absolute value of the velocity gradient at the surface as
1
08
06
04
02
0
0 1 2 3 4 5 6
M
0
05
20
120578
f998400(120578)
120573 = 05K = 01
Figure 2 Effect of magnetic parameter on dimensionless velocityfor 120573 = 05 119870 = 01
shown in Table 2 Figure 3 is plotted to show the influence of120573 The dimensionless velocity decreases with an increase in120573 and it is also seen that the hydrodynamics boundary layerthickness is higher for small value of 120573
6 Concluding Remarks
In this study we have successfully applied the optimal homo-topy asymptotic method for MHD flow over a nonlinearporous stretching sheet Both numerical and approximateanalytical results are obtained for the problem The resultsare presented in tabular and graphical forms for differentcontrolling parameters It was found that OHAM resultsare closer to numerical results The solution obtained usingOHAM is also consistent with solution obtained using anumerical method for variation in 120573119872 and119870
Advances in Mathematical Physics 5
1
08
06
04
02
0
0 1 2 3 4 5
0
05
20
120578
f998400(120578)
M = 01K = 02
120573
Figure 3 Effect of 120573 parameter on dimensionless velocity for 119870 =02119872 = 01
Competing Interests
There is no conflict of interests with any personorganizationupon the acceptance of this paper
Acknowledgments
This research is supported by the Thailand Research Fundunder the TRG Research Scholar Grant no TRG5780016
References
[1] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons 1979
[2] R H Rand and D Armbruster Perturbation Methods Bifurca-tionTheory and Computer Algebraic Springer Berlin Germany1987
[3] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995
[4] J E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of MechanicsmdashBFluids vol 19 no 1 pp 109ndash122 2000
[5] E Magyari I Pop and B Keller ldquoThe lsquomissingrsquo similarityboundary-layer flowover amoving plane surfacerdquoZeitschrift furAngewandte Mathematik und Physik vol 53 no 5 pp 782ndash7932002
[6] M Zakaria ldquoMagnetohydrodynamic viscoelastic boundarylayer flow past a stretching plate and heat transferrdquo AppliedMathematics and Computation vol 155 no 1 pp 165ndash177 2004
[7] H S Takhar A A Raptis andC P Perdikis ldquoMHDasymmetricflow past a semi-infinite moving platerdquoActaMechanica vol 65no 1 pp 287ndash290 1987
[8] K B Pavlov ldquoOscillatory magnetohydrodynamic flows of vis-coplastic media in plane ductsrdquoMagnetohydrodynamics vol 9no 2 pp 197ndash199 1973
[9] I Pop M Kumari and G Nath ldquoConjugate MHD flow past aflat platerdquo Acta Mechanica vol 106 no 3-4 pp 215ndash220 1994
[10] A J Chamkha ldquoMHD-free convection from a vertical plateembedded in a thermally stratified porous medium with HalleffectsrdquoAppliedMathematicalModelling vol 21 no 10 pp 603ndash609 1997
[11] H S Takhar A J Chamkha andGNath ldquoFlow andmass trans-fer on a stretching sheet with a magnetic field and chemicallyreactive speciesrdquo International Journal of Engineering Sciencevol 38 no 12 pp 1303ndash1314 2000
[12] P D McCormack and L Crane Physics of Fluid DynamicsAcademic Press New York NY USA 1973
[13] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961
[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface II Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 221ndash225 1961
[15] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967
[16] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquo Warme-Und Stoffubertragung vol 29 no 4 pp227ndash234 1994
[17] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999
[18] P D Ariel ldquoComputation of flow of viscoelastic fluids byparameter differentiationrdquo International Journal for NumericalMethods in Fluids vol 15 no 11 pp 1295ndash1312 1992
[19] P D Ariel ldquoOn the second solution of flow of viscoelastic fluidover a stretching sheetrdquo Quarterly of Applied Mathematics vol33 no 4 pp 629ndash632 1995
[20] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989
[21] M Fathizadeh M Madani Y Khan N Faraz A Yildirimand S Tutkun ldquoAn effective modification of the homotopyperturbation method for MHD viscous flow over a stretchingsheetrdquo Journal of King Saud University-Science vol 25 no 2 pp107ndash113 2013
[22] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquoActaMechanica vol 105 no 1ndash4 pp 49ndash561994
[23] Y KhanM A Abdou N Faraz A Yildirim andQWu ldquoNum-erical solution of MHD flow over a nonlinear porous stretchingsheetrdquo Iranian Journal of Chemistry and Chemical Engineeringvol 31 no 3 pp 125ndash132 2012
[24] T C Chiam ldquoHydromagnetic flow over a surface stretchingwith a power-law velocityrdquo International Journal of EngineeringScience vol 33 no 3 pp 429ndash435 1995
[25] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by deformation of a plane surfacerdquoMagne-tohydrodynamics vol 10 no 4 pp 507ndash510 1974
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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2 Advances in Mathematical Physics
problem is given in Section 4 In Section 5 we analyze thecomparison of the solutions using OHAM with numericalmethod (NM) Section 6 is devoted for the concludingremarks
2 Governing Equation
Weconsider theMHDflowof an incompressible viscous fluidover a nonlinear porous stretching sheet at 119910 = 0 Electricallyconducting fluid under the influence of appliedmagnetic field119861(119909) normal to the stretching sheet the induced magneticfield is assumed to be negligible Under such assumption theMHD boundary layer equations are governed by
120597119906
120597119909+120597V120597119910= 0
119906120597119906
120597119909+ V120597119906
120597119910= ]1205972119906
1205971199102minus 1205901198612(119909)
120588119906
(1)
where 119906 and V are the velocity components in the 119909- and119910-directions respectively ] is the kinematic viscosity 120588 isfluid density 120590 is the electrical conductivity and 119861(119909) is themagnetic field strength where 119861(119909) = 119861
0119909(119899minus1)2
The boundary conditions to the nonlinear porous stretch-ing sheet are given below
119906 (119909 0) = 119888119909119899
V (119909 0) = minus1198810
119906 (119909infin) = 0
(2)
where 119888 is the stretching parameter and 1198810is the porosity of
the plate (whereas 1198810gt 0 represents suction and 119881
0lt 0
corresponds to injection)By introducing dimensionless variables for nondimen-
sionalized form of momentum and energy equations
120578 = radic119888 (119899 + 1)
2V119909(119899minus1)2
119910
119906 = 1198881199091198991198911015840(120578)
V = minusradic119888V (119899 + 1)2
119909(119899minus1)2
119891 (120578) +119899 minus 1
119899 + 11205781198911015840(120578)
(3)
Using (3) the governing equations can be reduced to non-linear differential equation where 119891 is a function of thesimilarity variable 120578
119891101584010158401015840+ 11989111989110158401015840minus 12057311989110158402minus119872119891
1015840= 0 (4)
subject to the boundary conditions
119891 997888rarr 119870
1198911015840997888rarr 1
as 120578 997888rarr 0
1198911015840997888rarr 0 as 120578 997888rarr infin
(5)
as
120573 =2119899
119899 + 1
119872 =21205901205730
2
120588119888 (1 + 119899)
119870 =119881119900
radicV119888 (119899 + 1) 2119909(119899minus1)2
(6)
120573 is the nondimensional parameter 119872 is the magneticparameter and119870 is wall mass transfer parameter
3 Basic Principles of OHAM
We review the basic principles of OHAMas developed in [26]in the following steps
(i) Let us consider the following differential equation
119860 [V (119909)] + 119886 (119909) = 0 119909 isin Ω (7)
where Ω is problem domain 119860(V) = 119871(V) + 119873(V) where 119871119873 are linear and nonlinear operators V(119909) is an unknownfunction and 119886(119909) is a known function
(ii) Construct an optimal homotopy equation as
(1 minus 119902) [119871 (120601 (119909 119902)) + 119886 (119909)]
minus 119867 (119902) [119860 (120601 (119909 119902)) + 119886 (119909)] = 0
(8)
where 0 le 119902 le 1 is an embedding parameter and 119867(119902) =sum119898
119896=1119902119896119862119896is auxiliary function on which the convergence
of the solution is greatly dependent The auxiliary function119867(119902) also adjusts the convergence domain and controls theconvergence region
(iii) Expand 120601(119909 119902 119862119895) in Taylorrsquos series about 119902 one has
an approximate solution
120601 (119909 119902 119862119895) = V0(119909) +
infin
sum
119896=1
V119896(119909 119862119895) 119902119896
119895 = 1 2 3
(9)
Many researchers have observed that the convergence of theseries equation (9) depends upon 119862
119895(119895 = 1 2 119898) if it is
convergent then we obtain
V = V0(119909) +
119898
sum
119896=1
V119896(119909 119862119895) (10)
(iv) Substituting (10) into (7) we have the followingresidual
119877 (119909 119862119895) = 119871 (V (119909 119862
119895)) + 119886 (119909) + 119873 (V (119909 119862
119895)) (11)
If119877(119909 119862119895) = 0 then Vwill be the exact solution For nonlinear
problems generally this will not be the case For determining119862119895(119895 = 1 2 119898) Galerkinrsquos Method or the method of
least squares can be used(v) Finally substitute these constants in (10) and one can
get the approximate solution
Advances in Mathematical Physics 3
Table 1 Comparison of OHAM results with numerical method for 119870 = 0 120573 = 5 and119872 = 5
120578119891(120578) 119891
1015840(120578)
OHAM NM error OHAM NM error00 0 0 0 1 1 001 0086665 0086668 00034 0746974 0747032 0007802 0151657 0151663 00039 0562191 0562169 0003903 0200655 0200658 00014 0424101 0424023 0018304 0237614 0237598 00067 0319327 0319201 0039405 0265301 0265277 00090 0237681 0237665 0006706 0285709 0285688 00037 0172819 0172901 0047407 0300262 0300252 00033 0119889 0120035 0121608 0309964 0309969 00016 0075254 0075405 0200209 0315493 0315512 00060 0036055 0036162 0295810 0317278 0317303 00078 0 0 0
4 Series Solution via OHAM
According to the OHAM applying (8) to (4)
(1 minus 119902) (11989110158401015840+ 1198911015840)
minus 119867 (119902) (119891101584010158401015840+ 11989111989110158401015840minus 12057311989110158402minus119872119891
1015840) = 0
(12)
where primes denote differentiation with respect to 120578We consider 119891 and119867(119902) as the following
119891 = 1198910+ 1199021198911+ 11990221198912
119867 (119902) = 1199021198621+ 11990221198622
(13)
Using (13) in (12) and some simplifying and rearranging theterms based on the powers of 119902 we obtain zeroth- first- andsecond-order problems as follows
Zeroth Order Problem Consider
11989110158401015840
0(120578) + 119891
1015840
0(120578) = 0 (14)
with boundary conditions
1198910(0) = 119870
1198911015840
0(0) = 1
(15)
Its solution is
1198910(120578) = 119870 + 1 minus 119890
minus120578 (16)
First Order Problem Consider
11989110158401015840
1+ 1198911015840
1
minus 1198621(119890minus120578minus119872119890minus120578+ (119870 + 1 minus 119890
minus120578) 119890minus120578minus 120573119890minus2120578)
= 0
(17)
with boundary conditions
1198910(0) = 0
1198911015840
0(0) = 0
(18)
It solution is
1198911(120578 1198621) =1
21198621119890minus2120578+ 1198621119872(1 + 120578) 119890
minus120578
+ 1198621119870(1 + 120578) 119890
minus120578+1
21198621120573119890minus2120578
minus 1198621(1 + 120573) 119890
minus120578
+1198621
2(1 minus 2119872 minus 2119870 + 1)
(19)
And this goes onWe obtain the three-term solution using OHAM for 119902 =
1
(120578 1198621 1198622) = 1198910(120578) + 119891
1(120578 1198621) + 1198912(120578 1198621 1198622) (20)
We use the method of least squares to obtain the unknownconvergent constants 119862
1 1198622in (20) for particular case if
119872 = 1 119870 = 0 and 120573 = 15 then the values of 1198621 1198622are
1198621= minus02233887095 119862
2= 08569476324
5 Results and Discussion
Table 1 shows the comparison of OHAM results with numer-ical (NM) for different values of parameters in Table 2 wecompare the numerical values of 11989110158401015840(0) via OHAM withexisting solution [23 24] It is noteworthy to mention herethat theOHAMgives lowest error than othermethodsThisanalysis shows that OHAM suits for MHD boundary layerflowproblems In Figures 1ndash3we have shown the effects of thedimensionless parameter 120573 the magnetic parameter119872 andthe mass transfer parameter 119870 with various assigned values
4 Advances in Mathematical Physics
Table 2 Comparison of 11989110158401015840(0) via OHAM with other methods for various values of119872 at 119870 = 1 and 120573 = 15
119872 Chiam [24] HPTM [23] Present results error Chiam[24]
error HPTM[23]
errorpresent results
0 minus14902 minus14902 minus14897 02826 02826 024891 minus15253 minus15253 minus152529 00019 00019 000135 minus251616 minus25161 minus251611 00015 00039 0003510 minus336632 minus33658 minus33659 00002 00151 0002150 minus716471 minus716354 minus716366 0 00163 00146100 minus100664 minus100648 minus100653 0 00158 00109
1
08
06
04
02
0
0 1 2 3 4 5 6
120573 = 15M = 02
K
0
05
10
120578
f998400(120578)
Figure 1 Effect of wall mass transfer parameter on dimensionlessvelocity for 120573 = 15119872 = 02
Figure 1 is displayed for the influence of 119870 It is observedthat the dimensionless velocity and associated boundarylayer thickness decrease with an increase in 119870 Figures 2and 3 are given for the velocity profile against 120578 in orderto show the influences of parameters 119872 120573 respectivelyFigure 2 exhibits the effect of magnetic parameter on thedimensionless velocity It is observed that the velocity profileof the fluid is significantly reduced with increasing values of119872 Physically an increase in magnetic parameter 119872 resultsin a strong reduction in dimensionless velocity 1198911015840 This isdue to the fact that magnetic field introduces a retardingbody force which acts transverse to the direction of theappliedmagnetic fieldThis body force known as the Lorentzforce decelerates the boundary layer flow and thickens themomentum boundary layer and hence induces an increase inthe absolute value of the velocity gradient at the surface as
1
08
06
04
02
0
0 1 2 3 4 5 6
M
0
05
20
120578
f998400(120578)
120573 = 05K = 01
Figure 2 Effect of magnetic parameter on dimensionless velocityfor 120573 = 05 119870 = 01
shown in Table 2 Figure 3 is plotted to show the influence of120573 The dimensionless velocity decreases with an increase in120573 and it is also seen that the hydrodynamics boundary layerthickness is higher for small value of 120573
6 Concluding Remarks
In this study we have successfully applied the optimal homo-topy asymptotic method for MHD flow over a nonlinearporous stretching sheet Both numerical and approximateanalytical results are obtained for the problem The resultsare presented in tabular and graphical forms for differentcontrolling parameters It was found that OHAM resultsare closer to numerical results The solution obtained usingOHAM is also consistent with solution obtained using anumerical method for variation in 120573119872 and119870
Advances in Mathematical Physics 5
1
08
06
04
02
0
0 1 2 3 4 5
0
05
20
120578
f998400(120578)
M = 01K = 02
120573
Figure 3 Effect of 120573 parameter on dimensionless velocity for 119870 =02119872 = 01
Competing Interests
There is no conflict of interests with any personorganizationupon the acceptance of this paper
Acknowledgments
This research is supported by the Thailand Research Fundunder the TRG Research Scholar Grant no TRG5780016
References
[1] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons 1979
[2] R H Rand and D Armbruster Perturbation Methods Bifurca-tionTheory and Computer Algebraic Springer Berlin Germany1987
[3] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995
[4] J E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of MechanicsmdashBFluids vol 19 no 1 pp 109ndash122 2000
[5] E Magyari I Pop and B Keller ldquoThe lsquomissingrsquo similarityboundary-layer flowover amoving plane surfacerdquoZeitschrift furAngewandte Mathematik und Physik vol 53 no 5 pp 782ndash7932002
[6] M Zakaria ldquoMagnetohydrodynamic viscoelastic boundarylayer flow past a stretching plate and heat transferrdquo AppliedMathematics and Computation vol 155 no 1 pp 165ndash177 2004
[7] H S Takhar A A Raptis andC P Perdikis ldquoMHDasymmetricflow past a semi-infinite moving platerdquoActaMechanica vol 65no 1 pp 287ndash290 1987
[8] K B Pavlov ldquoOscillatory magnetohydrodynamic flows of vis-coplastic media in plane ductsrdquoMagnetohydrodynamics vol 9no 2 pp 197ndash199 1973
[9] I Pop M Kumari and G Nath ldquoConjugate MHD flow past aflat platerdquo Acta Mechanica vol 106 no 3-4 pp 215ndash220 1994
[10] A J Chamkha ldquoMHD-free convection from a vertical plateembedded in a thermally stratified porous medium with HalleffectsrdquoAppliedMathematicalModelling vol 21 no 10 pp 603ndash609 1997
[11] H S Takhar A J Chamkha andGNath ldquoFlow andmass trans-fer on a stretching sheet with a magnetic field and chemicallyreactive speciesrdquo International Journal of Engineering Sciencevol 38 no 12 pp 1303ndash1314 2000
[12] P D McCormack and L Crane Physics of Fluid DynamicsAcademic Press New York NY USA 1973
[13] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961
[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface II Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 221ndash225 1961
[15] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967
[16] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquo Warme-Und Stoffubertragung vol 29 no 4 pp227ndash234 1994
[17] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999
[18] P D Ariel ldquoComputation of flow of viscoelastic fluids byparameter differentiationrdquo International Journal for NumericalMethods in Fluids vol 15 no 11 pp 1295ndash1312 1992
[19] P D Ariel ldquoOn the second solution of flow of viscoelastic fluidover a stretching sheetrdquo Quarterly of Applied Mathematics vol33 no 4 pp 629ndash632 1995
[20] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989
[21] M Fathizadeh M Madani Y Khan N Faraz A Yildirimand S Tutkun ldquoAn effective modification of the homotopyperturbation method for MHD viscous flow over a stretchingsheetrdquo Journal of King Saud University-Science vol 25 no 2 pp107ndash113 2013
[22] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquoActaMechanica vol 105 no 1ndash4 pp 49ndash561994
[23] Y KhanM A Abdou N Faraz A Yildirim andQWu ldquoNum-erical solution of MHD flow over a nonlinear porous stretchingsheetrdquo Iranian Journal of Chemistry and Chemical Engineeringvol 31 no 3 pp 125ndash132 2012
[24] T C Chiam ldquoHydromagnetic flow over a surface stretchingwith a power-law velocityrdquo International Journal of EngineeringScience vol 33 no 3 pp 429ndash435 1995
[25] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by deformation of a plane surfacerdquoMagne-tohydrodynamics vol 10 no 4 pp 507ndash510 1974
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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 3
Table 1 Comparison of OHAM results with numerical method for 119870 = 0 120573 = 5 and119872 = 5
120578119891(120578) 119891
1015840(120578)
OHAM NM error OHAM NM error00 0 0 0 1 1 001 0086665 0086668 00034 0746974 0747032 0007802 0151657 0151663 00039 0562191 0562169 0003903 0200655 0200658 00014 0424101 0424023 0018304 0237614 0237598 00067 0319327 0319201 0039405 0265301 0265277 00090 0237681 0237665 0006706 0285709 0285688 00037 0172819 0172901 0047407 0300262 0300252 00033 0119889 0120035 0121608 0309964 0309969 00016 0075254 0075405 0200209 0315493 0315512 00060 0036055 0036162 0295810 0317278 0317303 00078 0 0 0
4 Series Solution via OHAM
According to the OHAM applying (8) to (4)
(1 minus 119902) (11989110158401015840+ 1198911015840)
minus 119867 (119902) (119891101584010158401015840+ 11989111989110158401015840minus 12057311989110158402minus119872119891
1015840) = 0
(12)
where primes denote differentiation with respect to 120578We consider 119891 and119867(119902) as the following
119891 = 1198910+ 1199021198911+ 11990221198912
119867 (119902) = 1199021198621+ 11990221198622
(13)
Using (13) in (12) and some simplifying and rearranging theterms based on the powers of 119902 we obtain zeroth- first- andsecond-order problems as follows
Zeroth Order Problem Consider
11989110158401015840
0(120578) + 119891
1015840
0(120578) = 0 (14)
with boundary conditions
1198910(0) = 119870
1198911015840
0(0) = 1
(15)
Its solution is
1198910(120578) = 119870 + 1 minus 119890
minus120578 (16)
First Order Problem Consider
11989110158401015840
1+ 1198911015840
1
minus 1198621(119890minus120578minus119872119890minus120578+ (119870 + 1 minus 119890
minus120578) 119890minus120578minus 120573119890minus2120578)
= 0
(17)
with boundary conditions
1198910(0) = 0
1198911015840
0(0) = 0
(18)
It solution is
1198911(120578 1198621) =1
21198621119890minus2120578+ 1198621119872(1 + 120578) 119890
minus120578
+ 1198621119870(1 + 120578) 119890
minus120578+1
21198621120573119890minus2120578
minus 1198621(1 + 120573) 119890
minus120578
+1198621
2(1 minus 2119872 minus 2119870 + 1)
(19)
And this goes onWe obtain the three-term solution using OHAM for 119902 =
1
(120578 1198621 1198622) = 1198910(120578) + 119891
1(120578 1198621) + 1198912(120578 1198621 1198622) (20)
We use the method of least squares to obtain the unknownconvergent constants 119862
1 1198622in (20) for particular case if
119872 = 1 119870 = 0 and 120573 = 15 then the values of 1198621 1198622are
1198621= minus02233887095 119862
2= 08569476324
5 Results and Discussion
Table 1 shows the comparison of OHAM results with numer-ical (NM) for different values of parameters in Table 2 wecompare the numerical values of 11989110158401015840(0) via OHAM withexisting solution [23 24] It is noteworthy to mention herethat theOHAMgives lowest error than othermethodsThisanalysis shows that OHAM suits for MHD boundary layerflowproblems In Figures 1ndash3we have shown the effects of thedimensionless parameter 120573 the magnetic parameter119872 andthe mass transfer parameter 119870 with various assigned values
4 Advances in Mathematical Physics
Table 2 Comparison of 11989110158401015840(0) via OHAM with other methods for various values of119872 at 119870 = 1 and 120573 = 15
119872 Chiam [24] HPTM [23] Present results error Chiam[24]
error HPTM[23]
errorpresent results
0 minus14902 minus14902 minus14897 02826 02826 024891 minus15253 minus15253 minus152529 00019 00019 000135 minus251616 minus25161 minus251611 00015 00039 0003510 minus336632 minus33658 minus33659 00002 00151 0002150 minus716471 minus716354 minus716366 0 00163 00146100 minus100664 minus100648 minus100653 0 00158 00109
1
08
06
04
02
0
0 1 2 3 4 5 6
120573 = 15M = 02
K
0
05
10
120578
f998400(120578)
Figure 1 Effect of wall mass transfer parameter on dimensionlessvelocity for 120573 = 15119872 = 02
Figure 1 is displayed for the influence of 119870 It is observedthat the dimensionless velocity and associated boundarylayer thickness decrease with an increase in 119870 Figures 2and 3 are given for the velocity profile against 120578 in orderto show the influences of parameters 119872 120573 respectivelyFigure 2 exhibits the effect of magnetic parameter on thedimensionless velocity It is observed that the velocity profileof the fluid is significantly reduced with increasing values of119872 Physically an increase in magnetic parameter 119872 resultsin a strong reduction in dimensionless velocity 1198911015840 This isdue to the fact that magnetic field introduces a retardingbody force which acts transverse to the direction of theappliedmagnetic fieldThis body force known as the Lorentzforce decelerates the boundary layer flow and thickens themomentum boundary layer and hence induces an increase inthe absolute value of the velocity gradient at the surface as
1
08
06
04
02
0
0 1 2 3 4 5 6
M
0
05
20
120578
f998400(120578)
120573 = 05K = 01
Figure 2 Effect of magnetic parameter on dimensionless velocityfor 120573 = 05 119870 = 01
shown in Table 2 Figure 3 is plotted to show the influence of120573 The dimensionless velocity decreases with an increase in120573 and it is also seen that the hydrodynamics boundary layerthickness is higher for small value of 120573
6 Concluding Remarks
In this study we have successfully applied the optimal homo-topy asymptotic method for MHD flow over a nonlinearporous stretching sheet Both numerical and approximateanalytical results are obtained for the problem The resultsare presented in tabular and graphical forms for differentcontrolling parameters It was found that OHAM resultsare closer to numerical results The solution obtained usingOHAM is also consistent with solution obtained using anumerical method for variation in 120573119872 and119870
Advances in Mathematical Physics 5
1
08
06
04
02
0
0 1 2 3 4 5
0
05
20
120578
f998400(120578)
M = 01K = 02
120573
Figure 3 Effect of 120573 parameter on dimensionless velocity for 119870 =02119872 = 01
Competing Interests
There is no conflict of interests with any personorganizationupon the acceptance of this paper
Acknowledgments
This research is supported by the Thailand Research Fundunder the TRG Research Scholar Grant no TRG5780016
References
[1] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons 1979
[2] R H Rand and D Armbruster Perturbation Methods Bifurca-tionTheory and Computer Algebraic Springer Berlin Germany1987
[3] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995
[4] J E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of MechanicsmdashBFluids vol 19 no 1 pp 109ndash122 2000
[5] E Magyari I Pop and B Keller ldquoThe lsquomissingrsquo similarityboundary-layer flowover amoving plane surfacerdquoZeitschrift furAngewandte Mathematik und Physik vol 53 no 5 pp 782ndash7932002
[6] M Zakaria ldquoMagnetohydrodynamic viscoelastic boundarylayer flow past a stretching plate and heat transferrdquo AppliedMathematics and Computation vol 155 no 1 pp 165ndash177 2004
[7] H S Takhar A A Raptis andC P Perdikis ldquoMHDasymmetricflow past a semi-infinite moving platerdquoActaMechanica vol 65no 1 pp 287ndash290 1987
[8] K B Pavlov ldquoOscillatory magnetohydrodynamic flows of vis-coplastic media in plane ductsrdquoMagnetohydrodynamics vol 9no 2 pp 197ndash199 1973
[9] I Pop M Kumari and G Nath ldquoConjugate MHD flow past aflat platerdquo Acta Mechanica vol 106 no 3-4 pp 215ndash220 1994
[10] A J Chamkha ldquoMHD-free convection from a vertical plateembedded in a thermally stratified porous medium with HalleffectsrdquoAppliedMathematicalModelling vol 21 no 10 pp 603ndash609 1997
[11] H S Takhar A J Chamkha andGNath ldquoFlow andmass trans-fer on a stretching sheet with a magnetic field and chemicallyreactive speciesrdquo International Journal of Engineering Sciencevol 38 no 12 pp 1303ndash1314 2000
[12] P D McCormack and L Crane Physics of Fluid DynamicsAcademic Press New York NY USA 1973
[13] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961
[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface II Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 221ndash225 1961
[15] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967
[16] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquo Warme-Und Stoffubertragung vol 29 no 4 pp227ndash234 1994
[17] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999
[18] P D Ariel ldquoComputation of flow of viscoelastic fluids byparameter differentiationrdquo International Journal for NumericalMethods in Fluids vol 15 no 11 pp 1295ndash1312 1992
[19] P D Ariel ldquoOn the second solution of flow of viscoelastic fluidover a stretching sheetrdquo Quarterly of Applied Mathematics vol33 no 4 pp 629ndash632 1995
[20] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989
[21] M Fathizadeh M Madani Y Khan N Faraz A Yildirimand S Tutkun ldquoAn effective modification of the homotopyperturbation method for MHD viscous flow over a stretchingsheetrdquo Journal of King Saud University-Science vol 25 no 2 pp107ndash113 2013
[22] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquoActaMechanica vol 105 no 1ndash4 pp 49ndash561994
[23] Y KhanM A Abdou N Faraz A Yildirim andQWu ldquoNum-erical solution of MHD flow over a nonlinear porous stretchingsheetrdquo Iranian Journal of Chemistry and Chemical Engineeringvol 31 no 3 pp 125ndash132 2012
[24] T C Chiam ldquoHydromagnetic flow over a surface stretchingwith a power-law velocityrdquo International Journal of EngineeringScience vol 33 no 3 pp 429ndash435 1995
[25] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by deformation of a plane surfacerdquoMagne-tohydrodynamics vol 10 no 4 pp 507ndash510 1974
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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
Table 2 Comparison of 11989110158401015840(0) via OHAM with other methods for various values of119872 at 119870 = 1 and 120573 = 15
119872 Chiam [24] HPTM [23] Present results error Chiam[24]
error HPTM[23]
errorpresent results
0 minus14902 minus14902 minus14897 02826 02826 024891 minus15253 minus15253 minus152529 00019 00019 000135 minus251616 minus25161 minus251611 00015 00039 0003510 minus336632 minus33658 minus33659 00002 00151 0002150 minus716471 minus716354 minus716366 0 00163 00146100 minus100664 minus100648 minus100653 0 00158 00109
1
08
06
04
02
0
0 1 2 3 4 5 6
120573 = 15M = 02
K
0
05
10
120578
f998400(120578)
Figure 1 Effect of wall mass transfer parameter on dimensionlessvelocity for 120573 = 15119872 = 02
Figure 1 is displayed for the influence of 119870 It is observedthat the dimensionless velocity and associated boundarylayer thickness decrease with an increase in 119870 Figures 2and 3 are given for the velocity profile against 120578 in orderto show the influences of parameters 119872 120573 respectivelyFigure 2 exhibits the effect of magnetic parameter on thedimensionless velocity It is observed that the velocity profileof the fluid is significantly reduced with increasing values of119872 Physically an increase in magnetic parameter 119872 resultsin a strong reduction in dimensionless velocity 1198911015840 This isdue to the fact that magnetic field introduces a retardingbody force which acts transverse to the direction of theappliedmagnetic fieldThis body force known as the Lorentzforce decelerates the boundary layer flow and thickens themomentum boundary layer and hence induces an increase inthe absolute value of the velocity gradient at the surface as
1
08
06
04
02
0
0 1 2 3 4 5 6
M
0
05
20
120578
f998400(120578)
120573 = 05K = 01
Figure 2 Effect of magnetic parameter on dimensionless velocityfor 120573 = 05 119870 = 01
shown in Table 2 Figure 3 is plotted to show the influence of120573 The dimensionless velocity decreases with an increase in120573 and it is also seen that the hydrodynamics boundary layerthickness is higher for small value of 120573
6 Concluding Remarks
In this study we have successfully applied the optimal homo-topy asymptotic method for MHD flow over a nonlinearporous stretching sheet Both numerical and approximateanalytical results are obtained for the problem The resultsare presented in tabular and graphical forms for differentcontrolling parameters It was found that OHAM resultsare closer to numerical results The solution obtained usingOHAM is also consistent with solution obtained using anumerical method for variation in 120573119872 and119870
Advances in Mathematical Physics 5
1
08
06
04
02
0
0 1 2 3 4 5
0
05
20
120578
f998400(120578)
M = 01K = 02
120573
Figure 3 Effect of 120573 parameter on dimensionless velocity for 119870 =02119872 = 01
Competing Interests
There is no conflict of interests with any personorganizationupon the acceptance of this paper
Acknowledgments
This research is supported by the Thailand Research Fundunder the TRG Research Scholar Grant no TRG5780016
References
[1] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons 1979
[2] R H Rand and D Armbruster Perturbation Methods Bifurca-tionTheory and Computer Algebraic Springer Berlin Germany1987
[3] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995
[4] J E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of MechanicsmdashBFluids vol 19 no 1 pp 109ndash122 2000
[5] E Magyari I Pop and B Keller ldquoThe lsquomissingrsquo similarityboundary-layer flowover amoving plane surfacerdquoZeitschrift furAngewandte Mathematik und Physik vol 53 no 5 pp 782ndash7932002
[6] M Zakaria ldquoMagnetohydrodynamic viscoelastic boundarylayer flow past a stretching plate and heat transferrdquo AppliedMathematics and Computation vol 155 no 1 pp 165ndash177 2004
[7] H S Takhar A A Raptis andC P Perdikis ldquoMHDasymmetricflow past a semi-infinite moving platerdquoActaMechanica vol 65no 1 pp 287ndash290 1987
[8] K B Pavlov ldquoOscillatory magnetohydrodynamic flows of vis-coplastic media in plane ductsrdquoMagnetohydrodynamics vol 9no 2 pp 197ndash199 1973
[9] I Pop M Kumari and G Nath ldquoConjugate MHD flow past aflat platerdquo Acta Mechanica vol 106 no 3-4 pp 215ndash220 1994
[10] A J Chamkha ldquoMHD-free convection from a vertical plateembedded in a thermally stratified porous medium with HalleffectsrdquoAppliedMathematicalModelling vol 21 no 10 pp 603ndash609 1997
[11] H S Takhar A J Chamkha andGNath ldquoFlow andmass trans-fer on a stretching sheet with a magnetic field and chemicallyreactive speciesrdquo International Journal of Engineering Sciencevol 38 no 12 pp 1303ndash1314 2000
[12] P D McCormack and L Crane Physics of Fluid DynamicsAcademic Press New York NY USA 1973
[13] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961
[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface II Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 221ndash225 1961
[15] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967
[16] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquo Warme-Und Stoffubertragung vol 29 no 4 pp227ndash234 1994
[17] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999
[18] P D Ariel ldquoComputation of flow of viscoelastic fluids byparameter differentiationrdquo International Journal for NumericalMethods in Fluids vol 15 no 11 pp 1295ndash1312 1992
[19] P D Ariel ldquoOn the second solution of flow of viscoelastic fluidover a stretching sheetrdquo Quarterly of Applied Mathematics vol33 no 4 pp 629ndash632 1995
[20] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989
[21] M Fathizadeh M Madani Y Khan N Faraz A Yildirimand S Tutkun ldquoAn effective modification of the homotopyperturbation method for MHD viscous flow over a stretchingsheetrdquo Journal of King Saud University-Science vol 25 no 2 pp107ndash113 2013
[22] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquoActaMechanica vol 105 no 1ndash4 pp 49ndash561994
[23] Y KhanM A Abdou N Faraz A Yildirim andQWu ldquoNum-erical solution of MHD flow over a nonlinear porous stretchingsheetrdquo Iranian Journal of Chemistry and Chemical Engineeringvol 31 no 3 pp 125ndash132 2012
[24] T C Chiam ldquoHydromagnetic flow over a surface stretchingwith a power-law velocityrdquo International Journal of EngineeringScience vol 33 no 3 pp 429ndash435 1995
[25] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by deformation of a plane surfacerdquoMagne-tohydrodynamics vol 10 no 4 pp 507ndash510 1974
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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 5
1
08
06
04
02
0
0 1 2 3 4 5
0
05
20
120578
f998400(120578)
M = 01K = 02
120573
Figure 3 Effect of 120573 parameter on dimensionless velocity for 119870 =02119872 = 01
Competing Interests
There is no conflict of interests with any personorganizationupon the acceptance of this paper
Acknowledgments
This research is supported by the Thailand Research Fundunder the TRG Research Scholar Grant no TRG5780016
References
[1] A H Nayfeh Introduction to Perturbation Techniques JohnWiley amp Sons 1979
[2] R H Rand and D Armbruster Perturbation Methods Bifurca-tionTheory and Computer Algebraic Springer Berlin Germany1987
[3] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995
[4] J E Magyari and B Keller ldquoExact solutions for self-similarboundary-layer flows induced by permeable stretching wallsrdquoEuropean Journal of MechanicsmdashBFluids vol 19 no 1 pp 109ndash122 2000
[5] E Magyari I Pop and B Keller ldquoThe lsquomissingrsquo similarityboundary-layer flowover amoving plane surfacerdquoZeitschrift furAngewandte Mathematik und Physik vol 53 no 5 pp 782ndash7932002
[6] M Zakaria ldquoMagnetohydrodynamic viscoelastic boundarylayer flow past a stretching plate and heat transferrdquo AppliedMathematics and Computation vol 155 no 1 pp 165ndash177 2004
[7] H S Takhar A A Raptis andC P Perdikis ldquoMHDasymmetricflow past a semi-infinite moving platerdquoActaMechanica vol 65no 1 pp 287ndash290 1987
[8] K B Pavlov ldquoOscillatory magnetohydrodynamic flows of vis-coplastic media in plane ductsrdquoMagnetohydrodynamics vol 9no 2 pp 197ndash199 1973
[9] I Pop M Kumari and G Nath ldquoConjugate MHD flow past aflat platerdquo Acta Mechanica vol 106 no 3-4 pp 215ndash220 1994
[10] A J Chamkha ldquoMHD-free convection from a vertical plateembedded in a thermally stratified porous medium with HalleffectsrdquoAppliedMathematicalModelling vol 21 no 10 pp 603ndash609 1997
[11] H S Takhar A J Chamkha andGNath ldquoFlow andmass trans-fer on a stretching sheet with a magnetic field and chemicallyreactive speciesrdquo International Journal of Engineering Sciencevol 38 no 12 pp 1303ndash1314 2000
[12] P D McCormack and L Crane Physics of Fluid DynamicsAcademic Press New York NY USA 1973
[13] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961
[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface II Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 221ndash225 1961
[15] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967
[16] M E Ali ldquoHeat transfer characteristics of a continuous stretch-ing surfacerdquo Warme-Und Stoffubertragung vol 29 no 4 pp227ndash234 1994
[17] E Magyari and B Keller ldquoHeat and mass transfer in theboundary layers on an exponentially stretching continuoussurfacerdquo Journal of Physics D Applied Physics vol 32 no 5 pp577ndash585 1999
[18] P D Ariel ldquoComputation of flow of viscoelastic fluids byparameter differentiationrdquo International Journal for NumericalMethods in Fluids vol 15 no 11 pp 1295ndash1312 1992
[19] P D Ariel ldquoOn the second solution of flow of viscoelastic fluidover a stretching sheetrdquo Quarterly of Applied Mathematics vol33 no 4 pp 629ndash632 1995
[20] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989
[21] M Fathizadeh M Madani Y Khan N Faraz A Yildirimand S Tutkun ldquoAn effective modification of the homotopyperturbation method for MHD viscous flow over a stretchingsheetrdquo Journal of King Saud University-Science vol 25 no 2 pp107ndash113 2013
[22] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquoActaMechanica vol 105 no 1ndash4 pp 49ndash561994
[23] Y KhanM A Abdou N Faraz A Yildirim andQWu ldquoNum-erical solution of MHD flow over a nonlinear porous stretchingsheetrdquo Iranian Journal of Chemistry and Chemical Engineeringvol 31 no 3 pp 125ndash132 2012
[24] T C Chiam ldquoHydromagnetic flow over a surface stretchingwith a power-law velocityrdquo International Journal of EngineeringScience vol 33 no 3 pp 429ndash435 1995
[25] K B Pavlov ldquoMagnetohydrodynamic flow of an incompressibleviscous fluid caused by deformation of a plane surfacerdquoMagne-tohydrodynamics vol 10 no 4 pp 507ndash510 1974
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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
6 Advances in Mathematical Physics
[26] V Marinca N Herisanu T Dordea and G Madescu ldquoA newanalytical approach to nonlinear vibration of an electricalmachinerdquo Proceedings of the Romanian Academy Series A vol9 no 3 pp 229ndash236 2008
[27] V Marinca and N Herisanu ldquoApplication of Optimal Homo-topy Asymptotic Method for solving nonlinear equations aris-ing in heat transferrdquo International Communications in Heat andMass Transfer vol 35 no 6 pp 710ndash715 2008
[28] V Marinca and N Herisanu ldquoDetermination of periodic solu-tions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journalof Sound and Vibration vol 329 no 9 pp 1450ndash1459 2010
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
