J. Basic. Appl. Sci. Res., 3(2s) , , TextRoad Publication

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1 , TetRoad Pblication ISSN 9-44 Jornal o Basic and Applied Scientiic Research A Comparison among Homotopy Pertrbation Method and the Decomposition Method with the Variational Iteration Method or Non-Liner Blasis Eqation to Bondary Layer Flow over a Flat Plate K. Gholaminejad *, M. Hajiamiri, A.Majidian M.S stdent at Islamic Azad University Sari Branch, Sari, Iran Lectrer at Mazandaran Institte o technology, Babol, Iran Associate proessor at Islamic Azad University, Sari, Iran ABSTRACT Received: Jne Accepted: Jly In this article, we implement a relatively new nmerical techniqe and we present a comparative stdy among Homotopy pertrbation method, Adomian decomposition method and the variational iterational method. These methods in applied mathematics can be an eective procedre to obtain or approimate soltions. The stdy otlines the signiicant eatres o the three methods. The analysis will be illstrated by investigating the Non-linear Blasis eqation to bondary layer low over a lat plate. This paper is particlarly concerned a nmerical comparison with the Adomian decomposition, Homotopy pertrbation method and the variational iterational method. The nmerical reslts demonstrate that the new methods are qite accrate and readily implemented. KEYWORDS: Non-linear Blasis Eqation, Adomian decomposition method, Homotopy pertrbation Method,The variational iterational Method.INTRODUCTION Partial dierential eqations which arise in real-world physical problems are oten too complicated to be solved eactly. And even i an eact soltion is obtainable, the reqired calclations may be too complicated to be practical, or it might be diclt to interpret the otcome. Very recently, some promising approimate analytical soltions are proposed, sch as Energy Balance method [], Adomian decomposition method [-], variational iteration method [- ] and Homotopy-pertrbation method[-]. HPM is the most eective and convenient on eor both linear and nonlinear eqations. This method does not depend on a small parameter. Using homotopy techniqe in topology, a homotopy is constrcted with an embedding parameter p [,], which is considered as a small parameter. HPM has been shown to eectively, easily and accrately solve a large class o linear and nonlinear problems with components converging rapidly to accrate soltions. HPM was irst proposed by He [4] and was scceslly applied to varios engineering problems [5 7]. Another powerl analytical method, called the variational iteration method(vim), was irst proposed by He [4].VIM has sccesslly been applied to many sitations. VIM is based on the general Lagrange s mltiplier method [8]. The main eatre o the method is that the soltion o a mathematical problem with linearization assmption is sed as initial approimation or trial nction. Then a more highly precise approimation at some special point can be obtained. This approimation converges rapidly to an accrate soltion [9]. The Adomian decomposition method have been shown to solve easily and more accrately a large class o system o partial dierential eqations with approimates that converges rapidly to accrate soltions [,,]. The implementation o the method has shown reliable reslts in that ew terms are needed to obtain either eact soltion or to ind an approimate soltion o a reasonable degree o accracy in real physical models. Moreover, no linearization or pertrbation is reqired in the method.. Nmerical Methods. Fndamentals o the Homotopy Pertrbation Method To illstrate the basic ideas o this method, we consider the ollowing eqation [4]: A() (r) =, r Ω, () * Corresponding athor: K. Gholaminejad, M.S stdent at Islamic Azad University Sari Branch, Sari, Iran Tel: , khosrowgholaminejad@yahoo.com 6

2 Gholaminejad et al., with bondary condition B,, n r () where A is a general dierential operator, B a bondary operator, (r) a known analytical nction and Γ is the bondary o the domain Ω. A can be divided into two parts which are L and N, where L is linear and N is nonlinear. Eq. () can thereore be rewritten as ollows: L()+N() (r)=,r Ω, () Homotopy pertrbation strctre is shown as ollows: H(U, p) = ( - p)[l(v) - L( )] + p[a(v) (r)] =, p [,],r Ω (4) where v(r,p):ω [,] R. (5) In Eq. (4), p [, ] is an embedding parameter and U is the irst approimation that satisies the bondary condition. We can assme that the soltion o Eq. (4) can be written as a power series in p, as ollowing: p p p, (6) and the best approimation or soltion is lim L (7) p The above convergence is discssed in [4].. The variational iterational Method Consider the dierential eqation L + N = g (t) t (8) Where L is a linear operator, N is a non-linear operator and g(t) is a known and Nonlineer analytical nction. Ji Han He has modiied the above method into an iteration method[,]in the ollowing way: t = + ( L ()+N ()- g()) d (9) n n n n where λ is a general Lagrange s mltipler, which can be identiied optimally via the variational is a restricted variation which means δ =. theory, and n It is obvios now that the main steps o He s variational iteration method reqire irst the determination o the Lagrangian mltiplier λ that will be identiied optimally. Having determined the Lagrangian mltiplier, the sccessive approimations n,n, o the soltion will be readily obtained pon sing any selective nction.conseqently, the soltion =lim n,or (n ). () In other words, correction nctional (9) will give several approimations, and thereore the eact soltion is obtained at the limit o the reslting sccessive approimations.. Using Adomian Decomposition Method A system o dierential eqations can be considered as: y = (,y,..., ) y n () y = (,y,..., ) y n y = (,y,..., y ) n n n where each eqation represents the irst derivative o one o the nknown nctions as a mapping depending on the independent variable and n nknown nctions,..., n []. We can present the system (), by sing the i th eqation as: 6

3 where L is the linear operator d / d with the inverse Ly i = i (,y,... y n ) i =,,...,n () L =. d Applying the inverse operator on () we get the ollowing canonical orm, which is sitable or applying Adomian decomposition method. y i = yi + (,y,... y ) d i =,,...,n () i n As sal in Adomian decomposition method the soltion o Eq.() is considered to be as the sm o a series: y i (4) j And the integrand in the Eq.(), as the sm o the ollowing series: where i, j i,, i,,..., i, j i, j (,y,... y ) A,,..., (5) i n i, j i, i, i, j j A are called Adomian polynomials [6]. Sbstitting (4) and (5) into (). we get (6) y + A,,..., d= y + A,,..., d i, j i i, j i, i, i, j i i, j i, i, i, j j j rom which we deine: i, yi i, n i, n i, i, i, n A,,..., d n=,,,... (7).Method o soltion Bondary layer low over a lat plate is governed by the continity and the Navier-Stokes eqations.for a two dimensional, steady state, incompressible low with zero pressre gradient over a lat plate, governing eqations are simpliied to: y y y Sbjected to bondary conditions: y =, =,, = U, y = () y By applying a dimensionless variable (η) deined as: y.5 Re () (Re is the Reynolds nmber and deined as: Re ) The governing eqations o (8) and (9) can be redced to the well-known Blasis eqation where is a nction o variable (η): (8) (9) 6

4 Gholaminejad et al., d d d d () with bondary eqations: d, d d d, () where is related to (velocity) by U, and the prime denotes the derivatives with respect toη. In order to solve Eq.(), sing HPM, we can constrct a homotopy or this eqation: F F F F ( p)( ) p( ) (4) Or F F F p( ) Sppose that the soltion o Eq. (4) to be in the ollowing orm: F = F + pf + pf + (5) Sbstitting (5) into (4), and some algebraic maniplations and rearranging the coeicients o the terms with identical powers o p, we have: p F : - F F F p : p F F F F F : (6) p.. F F F F F F F : First or simplicity we take F =. In the present work we start the iteration by deining as a Taylor series o order two near η= ; so that it cold be reslted in highly accrate soltions near η =, i.e. () F () () (7) By applying F, F and F we derive: =.57 rom [4] and bondary conditions o Eq. () and solving Eq.(7) or 6

5 According to (5) and the assmption p =, we get: (8) 5 ( ) (9) In order to solve this eqation by sing the Adomian decomposition method, we simply take the eqation in an operator orm. Eq.() can be written d d d d () By applying L = =.57 rom [4] and bondary conditions o Eq. () and Using the inverse operator. ddd we get: F F F ddd = () ddd = Some o the symbolically compted components are as ollows F ddd = () And rom (5), we get: F(η) = () In order to solve (9) eqation by sing the variational iteration method, we simply take the eqation in an operator orm. L + N = g (t) t 5 8 And Fn ( ) Fn ( ) Fn ( ) F ( ) n F ( ) n d (4) where λ is a general Lagrange mltiplier [5],and can be identiied optimally via the variational theory[5-7]. 64

6 Gholaminejad et al., ( ) (5) Sbstitting (5) into (4) Fn ( ) Fn ( ) Fn ( ) Fn ( ) ( ) F ( ) n d (6) By applying =.57 rom [4] and bondary conditions o Eq. () we get: From (6) and (7) F ( ).6685 (7) F ( ).6685 ( ).845 d = F (8) d (9) = Approimation soltion via HPM Approimation soltion via VIM Fig..The nmerical reslts or Approimation soltion via ADM F when t and with initial condition o Eq.() by means o HPM,ADM and VIM 65

7 4. Comparison among HPM, VIM and ADM It can be seen rom this stdy, that:. Comparison among HPM, VIM and ADM shows that althogh the reslts o these methods, HPM does not reqire speciic algorithms and comple calclations, sch as ADM or constrction o correction nctionals sing general Lagrange mltipliers, sch as VIM and is mch easier and more convenient than ADM and VIM.. HPM handles linear and nonlinear problems in a simple manner by deorming a diiclt problem into a simple one. Bt in nonlinear problems, we enconter diiclties to calclate the so-called Adomian polynomials, when sing ADM. Also, optimal identiication o Lagrange mltipliers via the variational theory can be diiclt in VIM.. Comparison among HPM, VIM and ADM shows that althogh the reslts o these methods,we have the similar answers or η ; Bt or η 4 there is a noticeable dierence between the answer o ADM and the other methods.(table and igre ) 4. Comparison among HPM, VIM and ADM shows that the answers o HPM and VIM are more similer to Blasis's [8] answer than those one o ADM.(table and igre ) η Table : Comparison among HPM,ADM,VIM and nmerical methods (N.M) or (η) HPM VIM ADM N.M Table : Comparison among HPM,ADM,VIM and Blasis's reslts η HPM VIM ADM BLASIUS

8 Gholaminejad et al., (η) η - - Fig:The comparison o answers obtained by HPM,VIM,ADM ADM VIM HPM (η) η - - ADM VIM HPM Fig:The comparison o answers obtained by HPM,VIM,ADM and Blasis's answers 5. Conclsion In this letter, we have sccesslly developed HPM, ADM and VIM to obtain the eactsoltions o Blasis's eqation. It is apparently seen that these methods are very powerl and eicient techniqes or solving dierent kinds o problems arising in varios ields o science and engineering and present a rapid convergence or the soltions. The soltions obtained show that the reslts o these methods are in agreement bt HPM is an easy and convenient one. REFERENCES. Seyed H.Hashemi Kachapi,D.D.Ganji:Progress in Nonlinear Science;Analytical and Nmerical Methods in Engineering and Applied Science,Xlibris,(). Davood Domairy Ganji,Ehsan Mohseni Langri,Mathematical Methods in Nonlinear Heat Transer,Asian Academic Pblisher Limited,(). D.D.Ganji, Seyed H.Hashemi Kachapi: Progress in Nonlinear Science;Analysis o Nonlinear Eqation in Flids, Asian Academic Pblisher Limited,() 67

9 4. J.H. He, Int. J. Non-Linear Mech. 5 () () A. Rajabi, D.D. Ganji, H. Taherian, Phys. Lett. A 6 (4 5) (7) D.D. Ganji, Phys. Lett. A 55 (4 5) (6) M. Siddiqi, R. Mahmood, Q.K. Ghori, Phys. Lett. A 5 (6) Inokti, M., Sekine, H. and Mra, T. General se o the Lagrange mltiplier in nonlinear mathematical physics. In: Nemat-Nassed S, ed. Variational method in the mechanics o solids (Pergamon Press, 978). 9. He, J.H. Non-pertrbative methods or strongly nonlinear problems (Berlin: dissertation.de- Verlag im Internet GmbH, 6).. J.H. He, M.A. Abdo, Chaos Solitons Fractals 4 (5) (7) 4. S. Saha Ray, Appl. Math. Compt. 75 (6) 46.. J. Biazar, E. Babolian, R. Islam, Soltion o the system o ordinary ierential eqations by Adomian decomposition method, Applied Mathematics and Comptations(in press).. U. M. Ascher, L.R. Petzold, 99, Projected implicit Rnge Ktta methods or dierential-algebraic eqations, SIAM J. Nmer. Anal., 8, pp Howarth, L. : On the Soltion o the Laminar Bondary-Layer Eqations. Proceedings o the Royal Society o London. A. 64: ( 98) 5. J.H. He, X.-H. W, Chaos Solitons Fractals () (6) J.H. He, Int. J. Non-Linear Mech. 4 (999) J.H. He, Int. J. Nonlinear Sci. Nmer. Siml. 6 () (5) Blasis, H.: The Bondary Layers in Flid with Little Friction ( in German ). Zeitschrit r Mathematik nd Physik. 56():-7 (98) ; English translation available as NACATM 56, Febrary(95) 68