Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272, Popl's Rpublic of China Rcivd 1 May 1998 Abstract Th homotopy prturbation tchniqu dos not dpnd upon a small paramtr in th quation. By th homotopy tchniqu in topology, a homotopy is constructd with an imbdding paramtr p 2 ;1Š, which is considrd as a ``small paramtr''. Som xampls ar givn. Th approximations obtaind by th proposd mthod ar uniformly valid not only for small paramtrs, but also for vry larg paramtrs. Ó 1999 Elsvir Scinc S.A. All rights rsrvd. Kywords: Prturbation tchniqus; Homotopy; Nonlinarity 1. Introduction In th last two dcads with th rapid dvlopmnt of nonlinar scinc, thr has appard vrincrasing intrst of scintists and nginrs in th analytical tchniqus for nonlinar problms. Th widly applid tchniqus ar prturbation mthods. But, lik othr nonlinar analytical tchniqus, prturbation mthods hav thir own limitations. At rst, almost all prturbation mthods ar basd on an assumption that a small paramtr must xist in th quation. This so-calld small paramtr assumption gratly rstricts applications of prturbation tchniqus. As is wll known, an ovrwhlming majority of nonlinar problms hav no small paramtrs at all. Scondly, th dtrmination of small paramtrs sms to b a spcial art rquiring spcial tchniqus. An appropriat choic of small paramtrs lads to idal rsults. Howvr, an unsuitabl choic of small paramtrs rsults in bad cts, somtims sriously. Furthrmor, th approximat solutions solvd by th prturbation mthods ar valid, in most cass, only for th small valus of th paramtrs. It is obvious that all ths limitations com from th small paramtr assumption. In this papr, th author will rst propos a nw prturbation tchniqu coupld with th homotopy tchniqu. Th proposd mthod, rquiring no small paramtrs in th quations, can radily liminat th limitations of th traditional prturbation tchniqus. 2. Basic ida of homotopy prturbation mthod To illustrat th basic idas of th nw mthod, w considr th following nonlinar di rntial quation 1 E-mail: glliu@yc.shu.du.cn 45-7825/99/$ ± s front mattr Ó 1999 Elsvir Scinc S.A. All rights rsrvd. PII: S 4 5-7 8 2 5 ( 9 9 ) 18-3
258 J.-H. H / Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 A u f r ˆ ; r 2 X 1 with boundary conditions B u; ou=on ˆ ; r 2 C; 2 whr A is a gnral diffrntial oprator, B is a boundary oprator, f(r) is a known analytic function, C is th boundary of th domain X. Th oprator A can, gnrally spaking, b dividd into two parts L and N, whr L is linar, whil N is nonlinar, Eq. (1), thrfor, can b rwrittn as follows L u N u f r ˆ : 3 By th homotopy tchniqu [1,2], w construct a homotopy v r; p : X ; 1Š! R which satis s H v; p ˆ 1 p L v L u Š p A v f r Š ˆ ; p 2 ; 1Š; r 2 X 4a or H v; p ˆ L v L u pl u p N v f r Š ˆ ; 4b whr p 2 ; 1Š is an mbdding paramtr, u is an initial approximation of Eq. (1), which satis s th boundary conditions. Obviously, from Eq. (4) w hav H v; ˆ L v L u ˆ ; 5 H v; 1 ˆ A v f r ˆ ; 6 th changing procss of p from zro to unity is just that of v(r, p) from u (r) to u(r). In topology, this is calld dformation, and L v L u, A v f r ar calld homotopic. In this papr, th prsnt author will rst us th imbdding paramtr p as a ``small paramtr'', and assum that th solution of Eq. (4) can b writtn as a powr sris in p: v ˆ v pv 1 p 2 v 2 7 Stting p ˆ 1 rsults in th approximat solution of Eq. (1): u ˆ lim v ˆ v v 1 v 2 8 p!1 Th coupling of th prturbation mthod and th homotopy mthod is calld th homotopy prturbation mthod, which has liminatd limitations of th traditional prturbation mthods. In th othr hand, th proposd tchniqu can tak full advantag of th traditional prturbation tchniqus. Th sris (8) is convrgnt for most cass, howvr, th convrgnt rat dpnds upon th nonlinar oprator A(v) (th following opinions ar suggstd by an unknown rfr): 1. Th scond drivativ of N(v) with rspct to v must b small, bcaus th paramtr p may b rlativly larg, i.. p! 1. 2. Th norm of L 1 on=ov must b smallr than on, in ordr that th sris convrgs. 3. Som simpl xampls 3.1. Exampl 1 At rst, w will considr th Lighthill quation [3], for it is widly studid by th PLK mthod (Poincar± Lightill±Kuo Mthod). Th quation can b writtn as follows: x y dy y ˆ ; y 1 ˆ 1: 9
J.-H. H / Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 259 W can radily construct a homotopy which satis s 1 p Y dy y dy p x Y dy Y ˆ ; p 2 ; 1Š: 1 On may now try to obtain a solution of Eq. (1) in th form Y x ˆ Y x py 1 x p 2 Y 2 x ; 11 whr th Y i x ; i ˆ ; 1; 2;... ar functions yt to b dtrmind. Th substitution of Eq. (11) into Eq. (1) yilds dy Y y dy ˆ ; x Y dy Y ˆ : 12 13 Th initial approximation Y (x) or y (x) can b frly chosn, hr w st Y x ˆ y x ˆ x=; Y 1 ˆ 1= 14 so that th rsidual of Eq. (9) at x ˆ vanishs. Th substitution of Eq. (14) into Eq. (13) yilds x ˆ ; Y 1 1 ˆ 1 1=: 15 Th solution of Eq. (15) may b writtn as follows Y 1 ˆ 1 p x 2 2 2 : 16 If th rst approximation is su cint, thn w obtain y 1 x ˆ Y x Y 1 x ˆ 1 x p x 2 2 2 17 which is th xact solution. 3.2. Exampl 2 In this xampl, a mor complicatd quation [4] is considrd whr th PLK mthod will not b valid, which rads x n y dy nxn 1 y ˆ mx m 1 ; y 1 ˆ b > 1; 18 whr n ˆ 2; 3; 4;..., m ˆ ; 1; 2; 3;... A homotopy can b radily constructd as follows 1 p Y dy y dy p x n Y dy nxn 1 Y mx m 1 ˆ ; p 2 ; 1Š: 19 By th sam manipulation as Exampl 1, w st Y x ˆ y x ˆ x n = so that th rsidual of Eq. (18) at x ˆ is zro. W, thrfor, hav xn Y dy nxn 1 Y mx m 1 ˆ ; Y 1 1 ˆ b 1 2a or nx2n 1 = mx m 1 ˆ ; Y1 1 ˆ b 1 : 2b
26 J.-H. H / Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 Th solution of Eq. (2b) can b asily obtaind r 1 Y 1 x ˆ 2 x2n 2 2b 2 xm b 2 : 21 Th rst-ordr approximation of Eq. (18), thrfor, can b obtaind as follows r y 1 x ˆ Y x Y 1 x ˆ xn 1 2 x2n 2 2b 2 xm b 2 22 which is also th xact solution! It should b pointd out that th approximat solution of Eq. (18) obtaind by th PLK mthod is not uniformly valid. For dtails, plas s th discussion in Rf. [5]. 3.3. Exampl 3 In this xampl, th wll-known Du ng quation [3] will b studid which can b xprssd as follows d 2 u dt u 2 u3 ˆ ; u ˆ A; u ˆ : 23 W construct a homotopy which satis s L v L u pl u pv 3 ˆ ; 24 whr Lu ˆ d 2 u=dt 2 u. By th sam manipulation as bfor, w hav th following linar systms L v L u ˆ ; v ˆ A; v ˆ ; 25 L v 1 L u v 3 ˆ ; v 1 ˆ v 1 ˆ : 26 W st v t ˆ u t ˆ A cosat with an unknown constant a as th initial approximation of Eq. (24). Thrfor from Eq. (26), w hav d 2 v 1 dt v 2 1 A a 2 1 3 cosat 4 A2 A3 cos3at ˆ : 27 4 Th solution of Eq. (27) can b radily obtaind by th so-calld variational itration mthod [6,7] Z t v 1 t ˆ sin s t A a 2 1 3 4 A2 cosas A3 4 cos3as ds ˆ a 2 1 3 A 4 A2 a 2 1 cosat cost A 3 cos3at cost : 4 9a 2 1 28 Th constant a can b idnti d by various mthods such as mthod of wightd rsiduals (last squar mthod, mthod of collocation, Galrkin mthod). In this papr, w will us a vry simpl tchniqu to dtrmin th constant. In ordr to liminat th scular trm which may occur in th nxt itration, w st th cof cint of cost zro: a 2 1 3 A 4 A2 a 2 1 A 3 4 9a 2 1 ˆ 29a or s p a ˆ 1 7A 2 64 14A 2 49 2 A 4 : 18 29b
J.-H. H / Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 261 p For small, Eq. (29b) can b approximatly xprssd as a ˆ 1 3A 2 =4 O 2. Eq. (28), thrfor, can b r-writtn as follows v 1 t ˆ a 2 1 3 A 4 A2 a 2 1 cosat A 3 4 9a 2 1 cos3at 3 with a d nd as Eq. (29b). If, for xampl, th rst-ordr approximation is su cint, thn w hav u 1 t ˆ v t v 1 t ˆ 3A3 4 a 2 1 cosat A 3 4 9a 2 1 cos3at: 31 Th priod of th solution can b xprssd as follows T ˆ 2p a with a dfind as 29b 32 whil th xact priod rads Rf. [3] Z p=2 T x ˆ 4 A 2 p p with k ˆ 1 1 k sin 2 x 2 1 A 2 and th priod obtaind by th prturbation mthod is Rf. [3] 2p T prt ˆ 1 3A 2 =8 : 33 34 It is intrsting to notic that Eq. (34) is valid only for small, whras Eqs. (31) and (32) for a vry larg rgion 6 < 1. Furthrmor th approximations obtaind by th proposd nw mthod ar of high accuracy, vn whn! 1, w hav T x lim!1 T ˆ 2 p 7=9 p Z p=2 p ˆ 2 p 7=9 1:68575 ˆ :946: 1 :5 sin 2 x p Thrfor, for any valu of, it can b asily provd that th maximal rlativ rror is lss than 5.4%. 4. Conclusion In this papr w hav studid fw problms with or without small paramtrs with th homotopy prturbation tchniqu. Th rsults show that: 1. Th proposd mthod dos not rquir small paramtrs in th quations, so th limitations of th traditional prturbation mthods can b liminatd. 2. Th initial approximation can b frly slctd with possibl unknown constants. 3. Th approximations obtaind by this mthod ar valid not only for small paramtrs, but also for vry larg paramtrs. Furthrmor thir rst-ordr approximations ar of xtrm accuracy. Although fw xampls givn in this papr ar nonlinar di rntial quations, it can b applicabl to nonlinar partial di rntial quations. Acknowldgmnts Th work is supportd by National Scinc Foundation of China and Shanghai Education Foundation for Young Scintists. Th author wishs to thank an unknown rfr for his valuabl discussions, and part of his suggstion has bn dirctly citd.
262 J.-H. H / Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 Rfrncs [1] S.J. Liao, An approximat solution tchniqu not dpnding on small paramtrs: a spcial xampl, Int. J. Non-Linar Mchanics 3 (3) (1995) 371±38. [2] S.J. Liao, Boundary lmnt mthod for gnral nonlinar di rntial oprators, Enginring Analysis with Boundary Elmnt 2 (2) (1997) 91±99. [3] A.H. Nayfh, Introduction to Prturbation Tchniqus, Wily, Nw York, 1981. [4] C.C. Lin, Mathmatics Applid to Dtrministic Problms in Natural Scincs, Macmillan, Nw York, 1974. [5] Y.B. Wang t al., An Introduction to Prturbation Tchniqus (in Chins), Shanghai Jiaotong Univrsity Prss, 1986. [6] J.H. H, A nw approach to nonlinar partial di rntial quations, Communications in Nonlinar Scinc and Numrical Simulation 2 (4) (1997) 23±235. [7] J.H. H, Nonlinar oscillation with fractional drivativ and its approximation, Intrnational Confrnc on Vibration Enginring Õ98, 1998, Dalian, China.