For demonstration of the concept of HAM, by considering general non-linear problem (1) Non-linear operator is N and v (t) [ ],

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1 ANNALS of Faculy Engineering Hunedoara Inernaional Journal of Engineering Toe XV [7 Fascicule [May ISSN: [prin; online ISSN: [CD-o; online a free-access ulidisciplinary publicaion of he Faculy of Engineering Hunedoara. Jashad AHMAD. Sundas UBAB EFFICIENT HOMOTOPY ANALYSIS METHOD TO SYSTEM OF DELAY DIFFEENTIAL EQUATIONS. Deparen of Maheaics Faculy of Sciences Uniersiy of Gujra PAKISTAN. Deparen of Maheaics NCBA&E (Gujra Capus) PAKISTAN Absrac: This paper deal wih he applicaion of hooopy analysis ehod o sole he delay differenial equaions syses. This process proides speedy series conergence owards exac soluion of he syses. Seeral Delayed probles of syse are gien in he direcion of explaining he efficiency and alidiy of his echnique. In his paper we are going o effecually work wih he hooopy analysis ehod o find he soluion of delayed differenial syses ha conain nonlineariy. Soluions in he for of nueric lead he approxiaion owards exac for soluion wih higher accuracy. Hooopy analysis ehod will be presened in Secion. Nuerical syses are soled by Hooopy Analysis ehod and heir graphical represenaion is presened in Secion. The conclusion will be gien n Secion 4. Keywords: Delay differenial equaions Hooopy analysis ehod. INTODUCTION Delay differenial equaion odels hae wide and dierse range of applicaions. Huchinson inroduced firs aheaical odel of Delay in Biology for period auraion. Nonlinear delay differenial equaions used o odel nuerous cheical reacions engineering probles econoical and biological syses. Delays show he possessions of ransission ranspor processes and ineria.as hey consider inheriance of hese. In 8 h cenury Laplace & Condorce inroduced delay differenial equaions [. Afer he Second World War deelopen ade in hese equaions which sill coninues. Infinie specru of frequencies occurs in Delay probles. The deailed learning of colleced works discloses ha physical phenoena are nonlinear in naure and grea need o ge heir soluions. Sepan& Insperger hae used he seidiscreizaion ehod o conclude he peranency lobes of DDEs ha odel he dynaics of cuing achine operaions. Firs ie in 99 Liao deeloped Hooopy Analysis ehod and hen any oher people applied his ehod on he applicaion of differen syses [-6.A ariey of aheaical and physical probles hae been soled by HAM [7. Hooopy Analysis Mehod conain a paraeer which is naed as auxiliary paraeer which proides quick conergence of he soluion and due o his paraeer i differ o oher ehods [8.This ehod doesn' depend upon any sall or large paraeers and is alid for os nonlinear odels [9.The ajor adanage of HAM is also ha in his ehod differen base funcions can be choose. iccai equaions Vakhnenko equaion [ Glauer-je equaions [ Hiroa- Sasua KdV equaion [ oion of projecile [ boundary layer probles [4 Bolzann equaions [5 MKdV equaion [6 and any ore equaions were efficienly soled by HAM.. HOMOTOPY ANALYSIS METHOD For deonsraion of he concep of HAM by considering general non-linear proble N [ () Non-linear operaor is N and is unidenified funcion wih independen ariable.. Deforaion equaion of zeroh order Eq () has iniial guess of exac soluion ( ) auxiliary paraeer auxiliary funcion H ( ) & L is operaor which is linear in is naure L g when g () F ascicule [

2 [ φ(; q) q HN[ φ(; q) ( q)l () Ebedding paraeer q aries for o φ is unknown funcion. When q deforaion equaion of zeroh order is φ ( ; ) () (4) When q deforaion equaion of zeroh order becoes N [ φ (;) (5) This is exac soluion. As ariaion in he paraeer q soluion also goes closer o exac soluion. Deforaion deriaies of h order is gien as φ(; q) (6)! q And he soluion in series for is. High Order deforaion equaion q q (7)... (8) Afer differeniaing and diing by! deforaion equaion of h order is χ H (9) and [ [ L if χ if () φ(; q) ( )! q () [. NUMEICAL EXAMPLES This secion conained soluion of delayed syses of non-linear ype soling by hooopy analysis ehod & coparison of resuls wih graphical represenaion. Exaple. Consider a syse of delay differenial equaion of nonlinear ype / e e () / e e wih iniial condiion ( ) () ( ) We can choose freely iniial approxiaion o be e (4) e Deforaion equaion of zeroh order is ( q)l[ φ(; q) qhn[ φ(; q) (5) ( q)l[ φ(; q) qhn[ φ(; q) when q Eq. (5) becoes I gies iniial approxiaion L L [ φ (; q) [ φ (; q) q (6) φ (; q) () (7) φ(; q) () 4 F ascicule

3 When q nonlinear ers are N φ (;) e N φ (;) e Deforaion equaion of h order is L χ ( L χ By puing and H in eq. (9) and Eq. (9) reduces o where χ Consequenly χ and he soluion is [ ( ) [ () ( ) There exac soluion is () / [ ( ) e / [ ( ) e [ ) H [ ( ) [ H [ ( ) () ( ) () ( χ χ ( ) () ( ) () () ( ) ( ) ( ) () τ ( ) τ ( ) ( ) ( ) L L ( ) ( ) () () () [ ( ) [ () () () ( χ ( χ ( χ ( χ )(e )(e )(e )(e τ τ e e e e / / τ/ τ/ ) ) )dτ )dτ (8) (9) () () if χ () if e e e e e 4e / 8 4e 4e e e 4e e / / 8 4e... 4e /... () (4) e (5) e. 5 F ascicule

4 Exac and ApproxiaeSoluion of u Approxiae Exac 4 Figure : Coparison of Exac and Approxiae Soluion Exaple. Consider non-linear Delayed syse cos ( ) sin ( ) cos iniial condiions are () () (). We can choose freely iniial approxiaion o be e Deforaion equaion of zeros order are ( q)l[ φ(; q) qhn[ φ(; q) ( q)l[ φ(; q) qhn[ φ(; q) ( q)l[ φ(; q) qhn[ φ(; q) when q Eq. (9) becoes [ φ(; q) [ φ(; q) [ φ (; q) L L L The iniial approxiaions are φ(; q) () φ(; q) () φ(; q) () When q nonlinear ers are N φ (;) cos [ ( ) ( ) N φ (;) sin N[ φ (;) cos Deforaion equaion of h order is L χ L χ L χ [ ( ) [ ( ) H [ ( ) [ () H [ () [ H [ () () (6) (7) (8) (9) () () () () 6 F ascicule

5 By puing and H in eq. () and [ ( χ ) cos ( ) ( ) ( ) ( ) [ () ( χ )( sin) ( ) ( ) [ () () ( ) () ( χ )( cos) χ ( ) L [ ( ) Eq. () reduces o χ () L [ ( ) χ () L [ () τ τ ( ) χ ( ) ( ) () () ( χ ) τ cos dτ τ ( ) χ () () () ( χ )( τsin( τ))dτ ( ) χ () τ τ τ χ τ τ τ () ( ) ( ) ( ) () ( ) ( )( cos( ))d where if χ if Consequenly e and he soluion is e cos 5 sin sin! sin cos 4 8 cos!! 4! 5 sin cos sin 4cos! sin 4 sin 6 cos cos! 64 cos cos sin. 4cos 4 e 4 cos! 4! (4) (5) (6) (7) (8) (9) (4) 7 F ascicule

6 Exac Soluion of u u u u ApproxiaeSoluion of 4 4 Figure : Exac Soluion Figure : Approxiae Soluion Exaple. Consider Delay differenial syse ( ) ( ) (.) wih iniial condiions () () (). We can choose freely iniial approxiaion ( ) Deforaion equaion of zeroh order ( q)l[ φ(; q) qhn[ φ(; q) ( q)l[ φ(; q) qhn[ φ(; q) ( q)l[ φ(; q) qhn[ φ(; q) When q Eq. (44) becoes [ φ(; q) [ φ(; q) [ φ (; q) L L L I gies iniial approxiaion φ(; q) () φ(; q) () φ(; q) () when q nonlinear ers are N[ φ (;) ( ) N[ φ (;) ( ) (.) N[ φ (;) Deforaion equaion of h order is L[ χ ( ) H [ ( ) L[ χ () H [ ( ) L[ χ () H [ () By puing and H in eq. (48) and 4 (4) (4) (4) (44) (45) (46) (47) (48) 8 F ascicule

7 Eq. (48) reduces o where I gies and he soluion is [ ( ) ( ) ( ) ( ) [ () () ( ) ( ) () (.) [ () χ χ χ () ( ) () () () L L L [ ( ) [ () [ () χ ( ) ( ) ( ) ( τ ) χ () () ( ) ( τ ) () ( τ.) χ () () () dτ dτ dτ (49) (5) if χ. (5) if!.4 e e / e. (5) (5) (54) (55) 9 F ascicule

8 Exac Soluion of u ApproxiaeSoluion of u Figure 4: Exac soluion Figure 5: Approxiae soluion 4. CONCLUSION In his work we hae worked on delay differenial syses ia Hooopy Analysis Mehod. Three nuerical exaples are soled by using HAM wih good approxiaion. The resuls obained by his ehod proide us quick soluion. The coparison of resuls indicaed ha he ehod is exreely effecie for syse of nonlinear probles. This work illusraes grea prospecie of he echnique for syse of nonlinear phenoenon occurring in differen fields of science and engineering. eferences [. Gorecki H. Fuksa S. Grabowski P. and Koryowski A. Analysis and Synhesis of Tie Delay Syses John Wiley and Sons PWN-Polish ScienificPublishers-Warszawa 989. [. SJ. Liao Beyond perurbaion: inroducion o he Hooopy analysis ehod Boca aon: Chapan & Hall/CC Press. [. SJ. Liao The proposed Hooopy analysis echniques for he soluion of nonlinear probles PhD disseraion Shanghai Jiao Tong Uniersiy 99. [4. SJ. Liao Coparison beween he Hooopy analysis ehod and Hooopy perurbaion ehod Appl. Mah. Copu. 69 (5) [5. SJ. Liao and K. Cheung Hooopy analysis of nonlinear progressie waes in deep waer J. Eng. Mah. 45 () 5-6. [6. SJ. Liao J. Su and AT. Chwang Series soluions for a nonlinear odel of cobined conecie and radiaie cooling of a spherical body In. J.Hea Mass. Transfer. 49 (6) [7. S. J. Liao Beyond Perurbaion: Inroducion o he Hooopy Analysis Mehod Chapan & Hall/CC Bocaaon. [8. Ch. Xiurong Yu. Jiaju Hooopy Analysis Mehod for a Class of Holling Model wih he Funcional eacion he Open Auoaion andconrol Syses Journal 5 () 5-5. [9. Y. Mahoudi E.M. Kazeian The Hooopy Analysis Mehod for Soling he Kuraoo-Tsuzuki Equaion World Applied Sciences Journal () [. Y.Y.Wu S.J.Liao Soling he one loop soluion of he Vakhnenko equaion by eans of he Hooopy analysis ehod Chaos Soli.Frac. (4) 7-74 [. Y.Boureel Explici series soluion for he Glauer-je proble by eans of he Hooopy analysis ehod Inernaional J Non. Lin. Sci. Nuer. Siula. (7) [. S.Abbasbandy The applicaion of Hooopy analysis ehod o sole a generalized Hiroa-Sasua Coupled KdV equaion Physics Le. A. 6(7) [. K.Yabushia M.Yaashia K.Tsuboi An analyic soluion of projecile oion wih he quadraic resisance law using he Hooopy analysis ehod Journal of Phys.A. 4(7) [4. M.Hassani M.M.Tabar H.Neai G.Doairry F.Noori An analyical soluion for boundary layer of a nanouid pas a sreching shee Inernaional J.Ther.Sci 48() -8 [5. C.J.Nassar J.F.reelli.J.Bowan Applicaion of he Hooopy analysis ehod o he Poisson- Bolzann equaion for seiconducor deices Inernaional J. Non. Lin. Sci. Nuer. Siula. 6 () 5-5 [6. W.Zhen Z.Li Z.H.Qing Soliary soluion of discree MKdV equaion by Hooopy analysis ehod Counicaion Theory. Phys. 49(8) 7-78 [7. E. A. Bucher P. Nindujarla and E. Bueler Sabiliy of up and down-illing using chebyshe collocaion ehod Proceedings of he ASME Inernaional Design Engineering Technical Conferences & Copuers and Inforaion in Engineering Conference ol. 6 par A C pp [8. Eans D.J. and aslan K.. The Adoian Decoposiion Mehod for Soling Delay Differenial Equaion. Inernaional Journal of Copuer Maheaics 8 (5) ie 4 F ascicule