Application of homotopy Analysis Method for Solving non linear Dynamical System

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1 IOSR Journal of Mahemaics (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue 1 Ver. V (Jan. - Feb. 16), PP Applicaion of homoopy Analysis Mehod for Solving non linear Dynamical Sysem G. M. Gharib Mahemaics Deparmen, College of Science and Informaion Technology, Zarqa Universiy, Jordan Absrac: In his paper, he nonlinear dynamical sysems are solved by using he homoopy analysis mehod (HAM).The approximaion soluion of his equaion is calculaed in he form ofa series which is componens are compued easily. The exisence anduniqueness of he soluion and he convergence of he proposed mehodare proved.the resuls obained here demonsrae ha he HAM is an effecive and robus echnique for nonlinear dynamical sysems.nonlinear dynamical sysems are omnipresen in numerous pracical engineering and mahemaics problems. I is hardly o seek he exac soluions in normal circumsances. However, he developmen of analyical mehods can provide an all-embracing undersanding for he sysems Key words: Nonlinear dynamical sysem; Homoopy Analysis Mehod Mahemaics Subjec classificaion: 53C; 58Z5 I. Inroducion Since many physical problems are modeled by nonlinear dynamical sysem, he numerical soluions of such nonlinear dynamical sysem have been highly sudied by many auhors. address: Gharibmusa@gmail.com" This research is funded by he Deanship of Research and Graduae Sudies in Zarqa Universiy/ Jordan" In recen years, numerous works have been focusing on he developmen of more advanced and efficien mehods for inegral equaions and nonlinear dynamical sysem for example [1-6, 15, 16, 18], he HAM is based on homoopy, a fundamenal concep in opology and differenial geomery. Over he las couple of decades, Liao [1-1] described a nonlinear analyical echnique which does no require small parameers and hus can be applied o solve nonlinear problems wihou small or large parameers. This echnique is based on homoopy, which is an imporan par of opology, called he homoopy analysis mehod (HAM). Is main idea is o consruc a class of homoopy in a raher general form by inroducing an auxiliary parameer. This parameer can provide us wih a convenien way o conrol he convergence of approximaion series and adjus convergence rae and region when necessary. A sysemaical descripion of his mehod was presened in [1]. Liao [7] also sudied he convergence properies of he HAM and proved ha as long as an HAM series is convergen, i mus converge o one soluion of he considered problem. Many micro elecro-mechanical sysems (MEMSs) inherenly conain nonlineariies, such as inrinsic and exerior nonlineariies arising from coupling of differen domains [5-8]. Also, here exis mechanical nonlineariies, for example, large deformaions, surface conac, creep phenomena, ime-dependen masses and nonlinear damping effecs, and so forh [9-11]. I seems fair o say ha nonlinear dynamic analysis becomes an increasingly imporan ask in MEMS research and manufacuring. The nonlinear dynamical behaviors of microcanilever-based insrumen in MEMS under various loading condiions have simulaed he curiosiies and ineress of many researchers [11-15]. For example, he oscillaion of an elecrosaically acuaed microcanilever-based device in MEMS was invesigaed hrough a simplified mass-spring-damping model subjeced o nonlinear elecrosaic force [16-18]. Complex nonlinear erms arising from elecrosaic force and from squeeze film damping make i difficul o analyze he sysem direcly using some rouine echniques for nonlinear vibraions. For his reason, Sephan[13] suggesed an approximae reamen by expanding he nonlineariies ino Taylor series and reaining only he firs wo erms. The harmonic balance mehod was hen applied o solve he approximae sysem. Noe ha he aained resuls have no been compared wih any numerical soluions. The purpose of his sudy is o seek highly accurae soluions of he aforemenioned sysem on he basis of he HAM. The paper is organized as follows. In secion, he HAM is briefly presened, in secion 3, his mehod is presened for solving he nonlinear dynamical sysem. Finally, I give some conclusions in are shown insecion 4. Homoopy Analysis Mehod The sudy of nonlinear problems is of crucial imporance in all areas of mahemaicaland physics. Some of he mos ineresing feaures of physical sysems are hidden inheir nonlinear behavior, and can only be DOI: 1.979/ Page

2 Applicaion Ofhomoopy Analysis Mehod For Solvingnonlinear Dynamical Sysem sudied wih appropriae mehods designedo ackle nonlinear problems.recenly Liao has developed a mehod for solving nonlinear problems which does no depend on he exisence of a small parameer. This mehod has been used by Liao and ohers o find analyical expressions for several nonlinear problems.suppose he problem o be solved is [ -4] N f = (1) Inroduce a new variable q and consider he equaion H φ ; q, f, ħ, q = 1 q L φ ; q f q ħ H N φ ; q, () where L is a suiably chosen linear operaor, f an iniial approximae soluion and h a parameer o be chosen laer. Subsiuion q in () we ge φ ; = f (). (3) If we le q 1 in (), we ge N φ ; 1 = (4) whose soluion is he required funcion f. Thus as q varies coninuously from o 1, he iniial approximae soluion f evolves o he desired soluion f of he problem Liao gives his reason for calling he mehod as he homoopy analysis mehod. Define f m = 1 m φ (; q) m! q m q= (5) hen If he series converges a q 1, hen The funcions f m are found successively from he equaions L f 1 = ħ H N f (8) when m = 1 and when m. By rearranging his equaion and inroducing he erm χ m χ m = m 1 1 m > 1 we can form he mhorder deformaion equaion: 1 L f m χ m f m 1 = ħ H m 1 N[φ ; q ] (m 1)! q m 1 q= (11) which is rue for m 1. Rearranging φ ; q = φ ; + f() = f () + L f m f m 1 = ħ H 1 m φ (; q) m =1 m! q m q= q m (6) m =1 f m (7) DOI: 1.979/ Page (1) 1 (m 1)! m 1 N[φ ; q ] q m 1 q= (9) L 1 1 ħ H m 1 N[φ ; q ] (m 1)! q m 1 q= (1) where L 1 is he inverse of he linear operaor (i.e. inverse of differeniaion is inegraion). The soluion o f() can be expressed as f() = m = f m (13) which is valid where ever he soluion converges. Nonlinear Dynamical Sysem In his work, he HAM is presened as an alernaive mehod o derive he analyical soluion for nonlinear dynamical sysem. Illusraive example is used o show he validiy and accuracy of he mehod in solving he nonlinear sysem. The resuls obained here demonsrae ha he HAM is an effecive and robus echnique for nonlinear dynamical sysems.nonlinear dynamical sysems are omnipresen in numerouspracical engineering and mahemaics problems. I is hardlyo seek he exac soluions in normal circumsances. However, he developmen of analyical mehods can provide anall-embracing undersanding for he sysems. The HAM [3] is a robus analyical approximaeechnique for solving a class of nonlinear problems. The significance of dynamical sysems is mainly due o is globalbifurcaion, regular and chaoic moions, he inensiveresearch subjecs are hus a he forefron of nonlineardynamics. Recenly, some achievemens and fruiful oucomehave been esablished for dynamical sysems [7-15]. Consider he nonlinear sysem d 3 x d y + dx d 3 d d + dy d = 1, (14)

3 R y,m y m Applicaion Ofhomoopy Analysis Mehod For Solvingnonlinear Dynamical Sysem d 3 y + d x + dx dy + = 1, (15) d 3 d d d wih he iniial condiions x = y =, y = x = 1. (16) The exac soluions for Eqs. (14), (15) subjec o he iniial condiions in Eq. (16) are x() = sin, y() = cos (17) Suppose ha he soluion can be expressed by a se of basefuncions n n =, 1,, 3, }, I choose he iniialapproximaion as x () =, y () = 1. (18) Defining he nonlinear operaor as N φ 1 ; q φ ; q = 3 φ 1 ; q 3 3 φ ; q 3 φ ; q + φ 1 ; q Thus, he zeroh-order deformaion equaion can be wrien in he form + φ 1 ; q + φ 1 ; q + ( φ ; q ). (19) φ + ( ; q ) and hemh order deformaion equaion can be expressed as From he iniial condiions and he iniial approximaion, I have m 1 j = y j y m 1 j 1 χ m, (3) m 1 j = y j y m 1 j 1 χ m, (4) From Eq. (1), i implies ha x m = χ m x m 1 + ħ Then, I obain y m = χ m y m 1 + ħ w w r r R x,m x m 1 R y,m y m 1 ds ds drdw + C 1,m + C,m (5) drdw + C 3,m + C 4,m (6) x 1 = ħ 3 + 5, y = ħ 5 6 (9)R x, x 1 = x 1 y 1 + x x 1 + y y 1 x = ħ ħ , (3) y = ħ ħ , (33) I now successively obain he second-order analyicalapproximaion by HAM as he following x + ħ 3 + ħ ħ + ħ ħ ħ 16 7 ħ 16 8 (34) DOI: 1.979/ Page

4 y Applicaion Ofhomoopy Analysis Mehod For Solvingnonlinear Dynamical Sysem 1 + ħ ħ + ħ ħ ħ 16 7 ħ 16 8 (35) Therefore he approximae soluion is given by aħ = 1 x + 3 y , (36) 16. (37).8.6 x() Fig. 1: Zoom For Comparison beween Exac (17 ) and HAM (36) Soluions a ħ = y() Fig. : Zoom For Comparison beween Exac (17 ) and HAM (37) Soluions a ħ = 1 II. Conclusion In summary,homoopyanalysis approximae mehod is applied o obain analyical approximaion soluion for nonlinear dynamical sysem. The fundamenal idea of he mehod is essenially differen from oher exising analyical mehods. The homoopy analysis approximae mehod provide an ingenious avenue for conrolling he convergences of approximaion series. The exac soluions of his example can be used o verify he accuracy of he mehod. Homoopy analysis mehod has been known as a powerful scheme for solving many funcional equaions such as algebraic equaions, ordinary and parial differenial equaions, inegral equaions and so on. The HAM has been shown o solve effecively, easily and accuraely a large class of nonlinear problems wih he approximaions which are rapidly convergen o he exac soluion, In his work, he HAM has been successfully employed o obain he approximae or analyical soluion of he nonlinear dynamical sysem.the presen echniques can also be furher generalized o invesigae more complicaed nonlinear dynamical sysems ha can only be solved by numerical approaches. References [1]. S. J. Liao, Beyond Perurbaion: Inroducion o he Homoopy Analysis Mehod,Chapman and Hall/CRC Press, Boca Raon, (3). DOI: 1.979/ Page

5 Applicaion Ofhomoopy Analysis Mehod For Solvingnonlinear Dynamical Sysem []. S. J. Liao and A. Campo, Analyic soluions of he emprerauredisrbuion inblasius Viscous flow problems, J. Fluid Mech., 453 (), [3]. S. J. Liao, Noes on he homoopy analysis mehod, some definiions and heorems,commun. in Nonlinear Sci. and Numer. Simula., 14 (9), [4]. S. J. Liao, An explici oally analyic approximaion of Blasiusvisous flowproblems, In. J. Non-Linear Mech., 34 (1999), [5]. S. J. Liao, The proposed homoopy analysis echnique for he soluion of nonlinearproblems, PhD hesis, Shanghai Jiao Tong Universiy, (199). [6]. S. J. Liao, Beyond perurbaion, a review on he basic ideas of he homoopyanalysis mehod and is applicaions AdV Mech., 38 (8), [7]. S. J. Liao, Topology and geomery for physiciss, Academic Press, Florida Press,(1983). [8]. S. J. Liao, An explici, oally analyic approximae soluion for Blasius viscous flow problems, In. J. Non-Linear Mech. 34 (1999), [9]. S. J. Liao, Beyond Perurbaion : Inroducion o homoopy analysis mehod, Chapman and Hall/CRC, New York, 4. [1]. S. J. Liao, A uniformly valid analyic soluion of -D viscous flow over a semi-infinie fla plae, J. Fluid. Mech. 385 (1999) [11]. A. Asaihambi, A finie difference mehod for he soluion of he Falkner-Skan equaion, Appl. Mah. Compu. 156(4), [1]. H. Schliching and K. Gersen, Boundary Layer Theory, 8 h revised ediion, Springer, Berlin. [13]. H. Sephan, Differenial Equaions, Their Soluions Using Symmeries, Cambridge Universiy Press, Cambridge [14]. L. Wang, A new algorihm for solving classical Blasius equaion, Appl. Mah. Compu. 157 (4), 1-9. [15]. C.J.Nassar, J.F.revelli, R.J.Bowman, Applicaion of he homoopy analysismehod o he Poisson-Bolzmann equaion for semiconducor devices,inernaional J. Non. Lin. Sci. Numer. Simula. 16 (11), [16]. J.K.Yang, Sable embedded Solions, Physical Rev.Le. 91(3), [17]. M.Hassani, M.M.Tabar, H.Nemai, G.Domairry, F.Noori, An analyical soluionfor boundary layer flow of a Nano fluid pas a sreching shee, InernaionalJ.Ther.Sci48(11), 1-8 [18]. A.Roozi, E.Alibeiki, S.S. Hosseni, S.M.Sha_of, H.Ebrahimi, Homoopyperurbaion mehod for special nonlinear parial differenial equaions,journal of K.Saud.Univ. 3(11), DOI: 1.979/ Page