International Journal of Computer Sciences and Engineering. Research Paper Volume-6, Issue-1 E-ISSN:

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1 Ieriol Jourl of Compuer Scieces d Egieerig Ope ccess Reserch Pper Volume-6, Issue- E-ISSN: pplicios of he boodh Trsform d he Homoopy Perurbio Mehod o he Nolier Oscillors P.K. Ber *, S.K. Ds, P. Ber * Dep. of Physics, Dumkl College, Murshidbd, Wes Begl, Idi Dep. of Mechicl Egieerig, IIT Ropr, Rupgr, Pujb, Idi School of Elecroics Egieerig, VIT Uiversiy, Vellore, Tmil Ndu, Idi * Correspodig uhor: pkbdcb@gmil.com, Tel.: vilble olie : Received: 9/Dec/7, Revised: /Dec/7, cceped: /J/8, Published: /J/8 bsrc I his pper, he differeil equio of moio of he clssicl Helmholz-Duffig oscillor, V der Pol, Duffig oscillor d Duffig-V der Pol oscillor equios hve bee solved lyiclly wih he help of ew iegrl rsform med boodh rsform d homoopy perurbio mehod. By recsig he goverig equios s olier eigevlue problems, we hve obied he excelle pproxime lyicl soluio of he displceme d he relio bewee mpliude d gulr frequecy. We hve lso compred our resuls wih exc umericl resuls grphiclly for few cses. Here, we hve lso demosred he sophisicio d simpliciy of his echique. Keywords boodh Trsform, Homoopy Perurbio Mehod, Helmholz-Duffig Oscillor, V der Pol, Duffig Oscillor, Duffig-V der Pol Oscillor, pproxime lyicl Soluio I. INTRODUCTION My complex problems i ure re due o olier pheome. Nowdys, olier processes re oe of he bigges chlleges i fidig soluios d re o esy o corol, becuse he olier chrcerisic of he sysem bruply chges due o smll chges of vlid prmeers, icludig ime. Thus, he issue becomes more compliced d, hece, eeds ulime soluio. Therefore, he sudy of pproxime soluios of olier differeil equios (NDEs) plys crucil role i udersdig he ierl mechisms of olier pheome. dvced olier echiques re sigific i solvig ihere olier problems, priculrly hose ivolvig differeil equios, dymicl sysems, d reled res. I rece yers, mhemicis, egieers, d physiciss hve mde sigific improvemes i fidig ew mhemicl ools reled o NDEs d dymicl sysems, whose udersdig will rely o oly o lyicl echiques, bu lso o umericl d sympoic mehods. These professiols hve esblished my effecive d powerful mehods o hdle he NDEs. The sudy of give olier problems is of crucil imporce, o oly i ll res of physics, bu lso i egieerig d oher disciplies, sice mos pheome i our world re esseilly olier d re described by NDEs. Moreover, obiig exc soluios for olier oscillory problems hs my difficulies. I is very difficul o solve olier problems d, i geerl, i is ofe more difficul o ge lyicl pproximio soluio h umericl oe for give olier problem. There re my lyicl pproches o solve NDEs. I his ricle, we hve drw he eio owrds he soluio of he differeil equios of he olier oscillors s hey ply impor role i pplied mhemics, physics d egieerig problems. lso i he heory of hrmoics, here re my impor pheome which hve prcicl imporce i demosrig olier effecs. I sciece d egieerig, here exis my olier problems, which do o coi y smll prmeers, especilly hose wih srog olieriy. Thus, i is ecessry o develop d improve some olier lyicl pproximios eve for lrge prmeers. The soluio o he olier problems re difficul o fid d mos of hem re o excly solvble. lhough he umericl soluio o he olier problems is esy, oe desires o fid he lyicl soluio o ge beer isigh of he problem. There re my echiques for solvig olier problems such s he hrmoic blce mehod, Krylov-Bogoliubov-Miropolsky mehod, weighed lierizio mehod, perurbio procedure for limi cycle lysis, modified Lidsed-Poicre mehod, domi decomposiio mehod, rificil prmeer mehod, d Nikiforov-Uvrov mehod [-9]. No oly hese mehods hve complex clculios, bu hey fil o hdle problems wih srog o-lieriy. 8, IJCSE ll Righs Reserved

2 The homoopy perurbio mehod (HPM) hs bee foud o be very efficie for solvig o-lier equios wih kow iiil or boudry vlues especilly for sysems wih srog o-lieriy i clssicl d quum mechicl problems [-5]. I his mehod, he soluio is give i ifiie series usully coverges o ccure soluio. boodh iroduced rsform derived from he clssicl Fourier iegrl for solvig ordiry d pril differeil equios esily i he ime () domi [6]. boodh rsform (T) hs bee pplied o differe ypes of problems d is foud o be very simple bu powerful echique [7,8]. I his ricle, we hve pplied boodh rsform bsed homoopy perurbio mehod (THPM) o solve he olier differeil equios o obi he pproxime displceme x d he oscillig frequecy ω wih high ccurcy. This pper is orgized s follows. I secio II, we demosre briefly he formulio of THPM. pplicios of THPM o olier problems hve bee show i secio III. Filly, i secio IV we provide brief discussio d our coclusios. II. FORMULTION OF THPM boodh rsform is ew rsform which is defied for fucio of expoeil order i se, where { ( ) :,, ( ) j x M k k x Me d, ( ) [, )} d x() is deoed by [ x( )] x( ) d defied s [ x( )] x( ) x( ) e d, k k () Some properies of boodh Trsform which re ecessry for our clculios re x '() [ x ''( )] x( ) x() () [cos ], [ si ] ( ) ()! [ ] (4) Le us cosider olier o-homogeeous differeil equio s Lx( ) x( ) Rx( ) Nx( ) g( ) (5) wih he iiil codiio x() x () d x ' (). Here, L is he secod order lier differeil operor ( L / ), R is he lier operor hvig order less h L, N is he olier operor, g () is he o-homogeeous erm d is y prmeer. Now, kig he boodh Trsform o boh sides of (5) we ge [ Lx( )] [ x( )] [ Rx( )] [ Nx( )] [ g( )] (6) Usig he differeil properies of he boodh Trsform s meioed bove d he iiil codiios (6) c be wrie x '() x( ) x() ( ) [ Rx( )] [ Nx( )] [ g( )] Tkig Iverse boodh Trsform o boh sides of (7) leds o x( ) X ( ) [ Rx( )] [ Nx( )] [ g( )] x'() X ( ) x(). ( ) where ccordig o he homoopy perurbio mehod, we c wrie (7) (8) x( ) p x ( ) d he olier erm s Nx( ) p H ( x) where He s polyomil H ( x) c be wrie s d H( x) N p x ( ),,,,! dp (9) p pplyig HPM d subsiuig he vlue of () x d () Nx i (8), we ge 8, IJCSE ll Righs Reserved

3 p u ( ) U ( ) p [ R p u( ) p H () () [ g( )] Here we cosider he boudry codiios =, x(), x'(). Now, for our purpose we rewrie (5) s d x x x x x (6) ( ) d Usig boodh Trsform d pplyig he boudry codiios o (6) we ge Equio () is he couplig bewee he boodh Trsform d he homoopy perurbio mehod usig He s polyomils where p is imbeddig prmeer. Comprig he coefficie of like power of p, we ge from () he followig equios x( ) ( ) [ ] x ( ) [ x ] [ x ] (7) p x X : ( ) ( ) () p : x( ) [ Rx ( )] [ H ( x( ))] [ g( )] p : x( ) [ Rx ( )] [ ( ( ), ( )] H x x The pproxime soluio is () () (4) p x ( ) lim p x ( ) x ( ) x ( ) x ( ) THPM III. PPLICTIONS I order o ssess he ccurcy of he THPM which hs bee preseed i secio II, i is pplied o differe ypes of olier oscillors d he resuls re compred wih he exc resuls. Helmholz-Duffig Oscillor The commo form of he differeil equio of moio of Helmholz-Duffig oscillor is give s d x x ( ) x x d (5) Tkig he iverse boodh Trsform o boh sides of (7) we ge x( ) cos ( ) [ x] ( ) [ x ] [ x ] pplyig he HPM, we c wrie (8) s p p x ( ) cos p ( ) p p x ( ) ( ) p p x () p p x () Equig he coefficie of p d p, we obi from (9) p : x ( ) cos p : x( ) [ x ] x x (8) (9) () () 8, IJCSE ll Righs Reserved

4 fer some mhemicl clculio of iverse boodh Trsform, we ge from () x ( ) si 4 cos cos 6 cos cos cos Here, he erm () si is seculr erm which mus be bse if d oly if gulr frequecy of oscillio is period T 8. Hece, 4 d ime 8. So, we c wrie he pproxime soluio obied from () d () s p () d p x p p x ( ) p d () p p x Equig he coefficie of p d p from (7) we obi p : x ( ) dx p : x( ) x d (7) (8) fer some mhemicl clculio of iverse boodh Trsform we ge from (8) s x () (9) xthpm ( ) cos cos cos cos 6 cos cos Duffig Oscillor () So, he pproxime soluio of (4) up o firs order correcio is obied from (7) d (9) s Duffig Oscillor xthpm ( ) () Le us cosider oher dmpig Duffig oscillor like [] Here we cosider dmpig Duffig oscillor like [9] d x dx x, x(), x'() d (4) d pplyig boodh Trsform o (4), we ge dx x( ) x d (5) Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from (5) dx x( ) x d pplyig HPM o (6) we c wrie (6) d x x x, x(), x'() 5 d () pplyig boodh Trsform o () we ge 5 x( ) x x () Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from () x( ) 5 x x pplyig HPM o () we ge () p p x( ) 5 p p x ( ) p () p p x (4) 8, IJCSE ll Righs Reserved 4

5 Equig he coefficie of p d p from (4) we obi p : x ( ) 5 p : x ( ) x x (5) fer some mhemicl clculio of iverse boodh Trsform we ge from (5) s 5 5 x () (6) So, he soluio of () up o firs order correcio is 5 5 xthpm ( ) (7) Duffig-V der Pol Oscillor Le us cosider Duffig-V der Pol oscillor s d x dx x x, x(), x'() d Now, we rewrie (8) s (8) d d x dx x x x d (9) d pplyig boodh Trsform o (9) we ge x x dx [ x ] d ( ) ( ) [ ] (4) Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from (4) x( ) si ( ) [ x] (4) dx [ x ] d pplyig HPM o (4) we obi p p x ( ) si p ( ) p x () p d p x () p d p x () p Equig he coefficie of p d p from (4) we obi p : x ( ) si p : x( ) [ x ] (4) (4) dx x d (44) fer some mhemicl clculio of iverse boodh Trsform d elimiig he seculr erm, we ge he firs order correcio erm s x ( ) si si si wih gulr frequecy T (45) 8 d ime period / 8. So, we c wrie he pproxime soluio of (8) up o firs order correcio s THPM ( ) si si si x e Duffig-V der Pol Oscillor (46) The clssicl Duffig-V der Pol oscillor ppers i my physicl problems d is govered by he followig olier differeil equio like [] d x dx ( x ) x x, x(), x'() d (47) d Now, we rewrie (47) s d x dx x x x x (48) ( ) d d 8, IJCSE ll Righs Reserved 5

6 pplyig boodh Trsform o (48) we ge x( ) ( ) [ ] x dx dx x d d x [ ] (49) Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from (49) x( ) cos ( ) [ x] dx d dx x [ x ] d (5) x ( ) si si cos cos Wih he mpliude, gulr frequecy d ime period T / 4 4 (54). So, we c wrie he pproxime soluio of (47) up o firs order correcio s xthpm ( ) cos si si cos cos (55) pplyig HPM o (5) we obi p p x ( ) cos p ( ) p x d p x () p () p d p x () p p x () p d () p x p d Equig he coefficie of p d p from (5) we obi p : x ( ) cos (5) (5) p : x( ) [ x] dx d (5) dx x x d fer some mhemicl clculio of iverse boodh Trsform d elimiig he seculr erm, we ge he firs order correcio erm s Figure. Time() vs displceme(x) curves obied from umericl (RK) d THMP wih =, μ =. d α =. We hve ploed he displceme x() from umericl soluio for =, µ =., α =. d compred he sme obied from Ruge-Ku (RK) clculios. I is foud h he displceme obied from RK d THPM re mchig very closely. Whe we ge equio of moio of olier oscillor s d x dx x x, x(), x'() d (56) d d we obi he pproxime soluio from (55) s xthpm ( ) cos si si (57) 8, IJCSE ll Righs Reserved 6

7 x( ) cos [ ] x x dx dx x d d (6) pplyig HPM o (6) we obi Figure. Time() vs displceme(x) curves obied from umericl (RK) d THMP wih =, μ =. d α = We hve ploed he displceme x() from umericl soluio for =, µ =., α = d compred he sme obied from Ruge-Ku (RK) clculios. I is foud h he displceme obied from RK d THPM re mchig very closely. Clssicl Frciol V der Pol Oscillor Cosider he clssicl frciol V der Pol dmped olier oscillor s [] d x dx x ( x ), x(), x'() (58) d d Now, we rewrie (58) s d x dx dx x x x x (59) d pplyig boodh Trsform o (59) we ge x( ) [ ] x dx x d dx x d d d (6) p p x ( ) cos p p x () p p p x () p d p x () p d p x () d p p x () d Equig he coefficie of p d p from (6) we obi p : x ( ) cos (6) (6) p : x( ) [ x] x (64) dx dx x d d cos The Fourier series for give by hs bee clculed d is cos b cos b cos where b.596, b.99. Wih he help of (6) we ge from (64) Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from (6) 8, IJCSE ll Righs Reserved 7

8 x( ) b cos 4 b cos si si 4 (65) x( ) [ ] x dx x d dx d (69) Tkig iverse boodh Trsform d pplyig he boudry codiios we ge from (69) fer some mhemicl clculio of iverse boodh Trsform d voidig he seculr erms, by puig d b, we obi he 4 mpliude d he gulr frequecy b which is sme s obied by he.8547 ierio procedure [4]. Hece, he pproxime periodic x ( ) cos The exc soluio kes he form pp soluio for he clssicl frciol V der Pol dmped olier oscillor is xex cos ex 4 e where, ex [5]. We ge he pproxime soluio of (58) up o firs order correcio s b xthpm ( ) cos cos cos 8 si si Ryleigh Equio (66) The specil cse of he frciol V der Pol dmped olier oscillor or Ryleigh equio c be represeed by [6] x, x(), x'() d d d x dx dx d Now, we rewrie (67) s d x dx dx x x x d d d pplyig boodh Trsform o (68) we ge (67) (68) x( ) cos [ x] x (7) dx dx d d pplyig HPM o (7) we obi p p x ( ) cos p p x () p p p x () p d p x () p d d p x () d Equig he coefficie of p d p from (7) we obi p : x ( ) cos (7) (7) p : x( ) [ x] x (7) dx dx d d 8, IJCSE ll Righs Reserved 8

9 Proceedig i he sme wy s before d voidig he seculr erms, d 4, we ge he 4 pproxime soluio up o he firs order correcio s xthpm ( ) cos cos cos si si (74) I is foud h.5967 d.67. The exc periodic soluio is x ex where cos ex ex 4 ex e ex IV. [5]. CONCLUSION We hve pplied simple perurbio heory THPM o solve he olier differeil equio of moio for olier oscillors. THPM is foud o give lyic soluios wih ll perurbive correcios o boh he displceme d he oscillio frequecy i very simple d srighforwrd mer. Here, we i dded relism d sophisicio of his TPHM by delig wih he differeil equio of moio of olier oscillors o obi lyicl expressio for he frequecy of oscillio d he displceme. I is show h he soluio coverges very fs. Eve firs order correcio is sufficie for geig ccure resuls. This mehod o oly gives very ccure umericl vlues of displceme bu lso gives ide bou he coribuios from differe hrmoics o i. We my coclude h his echique is o oly simple bu lso eleg wy o sudy wide clss of relisic o-excly solvble problems. REFERENCES [].H. Nyfeh, D.T. Mook, Nolier Oscillios, Joh Willey d Sos., New York, 979. [] N.N. Bogoliubov, Y.. Miropolsky, sympoic Mehods i he Theory of Nolier Oscillios Hidus Publishig Compy, Delhi, Chp. I, 96. [] V.P. grwl, H. Dem, Weighed lierizio echique for period pproximio i lrge mpliude Nolier Oscillios, J. Soud Vib., Vol.57, pp.46-47, 985. [4] S.H. Che, Y.K. Cheug, S.L. Lu, O perurbio procedure for limi cycle lysis, I. J. Nolier Mech., Vol.6, pp.5-, 99. [5] Y.K. Cheug, S.H. Che, S.L. Lu, modified Lidsed- Piocre mehod for ceri srog olier oscillios, I. J. No-Lier Mech., Vol.6, pp.67-78, 99. [6] G. domi, review of he decomposiio mehod i pplied mhemics, J. Mh. l. d ppl., Vol.5, pp.5-544, 998. [7] G.L. Lu, New reserch direcio i sigulr perurbio heory, rificil prmeer pproch d iverse-perurbio echique, I he proceedigs of he 997 Niol Coferece o 7h Moder Mhemics d Mechics, pp [8].F. Nikiforov, V.B. Uvrov, Specil fucios of mhemicl physics, Birkhuser, Bsel, 988. [9] P.K. Ber, T. Sil, Exc soluios of Feiberg-Horodecki equio for Time depede hrmoic oscillor, Prm-J. Phys., Vol.8, pp.-9,. [] J.H. He, Homoopy perurbio echique, Comp. Mehods i ppl. Mech. d Egg., Vol.78, pp.57-6, 999. [] J.H. He, couplig mehod of homoopy echique d perurbio echique for olier problems, I. J. No-Lier Mech., Vol., pp.7-4,. [] J. Bizr, M. Eslmi, ew homoopy perurbio mehod for solvig sysems of pril differeil equios, Comp. d Mh. wih ppli., Vol.6, pp.5 4,. []. Yildirim, Homoopy perurbio mehod o obi exc specil soluios wih soliry per for Boussiesq-like B(m,) equios wih fully olier dispersio, J. Mh. Phys., Vol.5, pp.5, 9. [4] M. Gover,.K. Tomer, Compriso of Opiml Homoopy sympoic Mehod wih Homoopy Perurbio Mehod of Twelfh Order Boudry Vlue Problems, Ieriol Jourl of Compuer Scieces d Egieerig, Vol., pp ,. [5] P.K. Ber, T. Sil, Homoopy perurbio mehod i quum mechicl problems, pplied Mh. d Comp., Vol.9, pp. 7 78,. [6] K.S. boodh, The New iegrl Trsform boodh Trsform, Globl Jourl of Pure d pplied Mhemics, Vol.9, pp.5-4,. [7] K. bdelilh, S. Hss, M. Mohd, M. bdelrhim,.s.s. Mueer, pplicio of he ew iegrl rsform i Crypogrphy, Pure d pplied Mhemics Jourl, Vol.5, pp.5-54, 6. [8] P.K. Ber, S.K. Ds, P. Ber, Sudy of Nolier Vibrio of Euler-Beroulli Bems Usig Couplig Bewee The boodh Trsform d The Homoopy Perurbio Mehod, Ieriol Jourl of Compuer Scieces d Egieerig, Vol.5, pp.84-9, 7. [9] B. Bulbul, M. Sezer, Numericl Soluio of Duffig Equio by Usig Improved Tylor Mrix Mehod, ricle ID6964. [] M. Njfi, M. Moghimi, H. Mssh, H. Khormishd, M. Demi, O he pplicio of domi Decomposiio Mehod d Oscillio Equios, I he 9h Ieriol Coferece o pplied Mhemics, Isbul, Turkey, 6. [] Gh. sdi Cordshooli,.R. Vhidi, Siluio of Duffig V der Pol Equio usig Decomposiio Mehod, dv. Sudies Theor. Phys., Vol.5, pp.-9,. [] R.E. Mickes, Ierio mehod soluios for coservive d limi-cycle x^(/) force oscillors, Jourl of Soud d Vibrio, Vol.9, pp , 6. [] I.S. Grdshey, I.S. Ryzhik, Tble of Iegrls, Series d Producs, cdemic Press, New York, 98. [4] H.Li Zhg, Periodic soluios for some srogly olier oscillios by He s eergy blce mehod, Compuer & Mhemics wih pplicios, Vol.58, pp , 9. [5] C.W. Lim, S.K. Li, ccre higher order lyicl pproxime soluios o ocoservive olier oscillors 8, IJCSE ll Righs Reserved 9

10 d pplicios o v der Pol dmped oscillors, Ieriol Jourl of Mechicl Scieces, Vol.48, pp.48-89, 6. [6] J.J. Soker, Nolier Vibrio i Mechicl d Elecricl Sysems, Wiley, New York, 99. uhors Profile Dr. P K Ber ssocie Professor of Physics, Deprme of Physics, Dumkl College hs published more h 8 ppers i repued ieriol jourls. His reserch of ieres re quum mechics d olier dymicl sysems. He hs 7 yers of echig experiece d 7 yers of reserch experiece. Mr. S K Ds pursed Bchelor of Techology i Mechicl Egieerig from Msipl Isiue of Techology, Idi i yer 5. He is currely pursuig M.Tech. i Mechicl Egieerig from Idi Isiue of Techology Ropr, Idi. His mi reserch work focuses o Fluid-Srucure Iercio, Coiuum Mechics d Compuiol Techiques. Ms. P Ber is currely pursuig her bchelor i Elecroics d Commuicio Egieerig from Vellore Isiue of Techology, Tmil Ndu. Her reserch re of ieres icludes o-lier dymicl sysems. 8, IJCSE ll Righs Reserved