The Solution of the One Species Lotka-Volterra Equation Using Variational Iteration Method ABSTRACT INTRODUCTION

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1 Malaysia Joural of Mahemaical Scieces 2(2): 55-6 (28) The Soluio of he Oe Species Loka-Volerra Equaio Usig Variaioal Ieraio Mehod B. Baiha, M.S.M. Noorai, I. Hashim School of Mahemaical Scieces, Uiversii Kebagsaa Malaysia, 436 UKM Bagi Selagor, Malaysia ABSTRACT This paper will prese a umerical compariso bewee he variaioal ieraio mehod (VIM) ad he Adomia decomposiio mehod (ADM) for solvig he oe species Loka-Volerra equaio. The umerical resuls demosrae ha he VIM are quie accurae ad readily implemeed. Keywords: Variaioal ieraio mehod (VIM), Adomia decomposiio mehod (ADM), Loka-Volerra equaios INTRODUCTION The Loka-Volerra equaios model he dyamic behavior of a arbirary umber of compeiors [1]. Though origially formulaed o describe he ime hisory of a biological sysem, hese equaios fid heir applicaio i a umber of egieerig fields such as simulaeous chemical ad oliear corol. I fac, he oe predaor oe prey Loka-Volerra model is oe of he mos popular oes o demosrae a simple oliear corol sysem. The accurae soluio of he Loka-Volerra equaios may become a difficul ask eiher if he equaios are siff (eve wih a small umber of specie, or whe he umber of species is large [2]. The Adomia decomposiio mehod, was proposed by Adomia [3] iiially wih he aims o solve froier physical problem. I has bee applied o a wide class of deermiisic ad sochasic problems, liear ad oliear, i physics, biology ad chemical reacios ec. For oliear models, he mehod has show reliable resuls i supplyig aalyical approximaio ha coverges very rapidly [4].

2 B. Baiha, M.S.M. Noorai, I. Hashim I rece years, a grea deal of aeio has bee devoed o sudy he VIM give by Professor Ji-Hua He for solvig a wide rage of problems whose mahemaical models yield differeial equaio or sysem of differeial equaios [5 13]. The purpose of his paper is o make a umerical compariso bewee VIM ad ADM for solvig he oe species Loka-Volerra equaio. He's variaioal ieraio mehod This mehod, which is a modified geeral Lagrage s muliplier mehod [14], has bee show o solve effecively, easily ad accuraely a large class of oliear problems [5 12]. The mai feaure of he mehod is ha he soluio of a mahemaical problem wih liearizaio assumpio is used as iiial approximaio or rial fucio, he a more highly precise approximaio a some special poi ca be obaied. This approximaio coverges rapidly o a accurae soluio. To illusrae he basic coceps of he VIM, we cosider he followig oliear differeial equaio: Lu + Nu = g(x); (1) where L is a liear operaor, N is a oliear operaor, ad g(x) is a ihomogeeous erm. Accordig o he VIM [5 8], we ca cosruc a correcio fucioal as follows: u + 1 ( = u + λ ( Lu ξ ) + N u~ ξ ) g( ξ )) dξ (2) where is a geeral Lagragia muliplier [14], which ca be ideified opimally via he variaioal heory, he subscrip deoes he h -order approximaio, δ ~ u =. u ~ is cosidered as a resriced variaio [5 8], i.e. Aalysis of oe species Loka-Volerra equaios The oe species compeig for a give fiie source of food: m dn i = N i ( bi + aij N j ), i = d j = 1 1,2,..., m. (3) 56 Malaysia Joural of Mahemaical Scieces

3 The Soluio Of The Oe Species Loka-Volerra Equaio Usig Variaioal Ieraio Mehod where a ad b are cosas. This equaio has a exac soluio: N( = b be b + an() N() b ae for b, (5) N( ) N( = for b =. (6) 1 an( ) To solve equaio (3) by he sadard Adomia decomposiio mehod (ADM) [3] wih iiial codiio N() =.1, usig he iverse operaor L 1 N ( ) = ( ) d we ge: = N( ), N + 1( = ( bn + aa1, ) d,, (7) where he Adomia polyomial: 1 = Nk N k. k= A, (8) To solve equaio (3) by meas of He s variaioal ieraio mehod, we cosruc a correcio fucio: dn ( ~ 2 N+ 1( = N ( + λ ( bn an ( ds, ds (9) where N ~ is cosidered as resriced variaios, which mea δ N ~ =. Is saioary codiios ca be obaied as follows: 1 + λ ( =, (1) λ '( bλ( =. (11) + s= Malaysia Joural of Mahemaical Scieces 57

4 B. Baiha, M.S.M. Noorai, I. Hashim The Lagrage mulipliers, herefore, ca be ideified as λ ( = e ad he followig variaioal ieraio formula ca be obaied by: b( s bs e dn ( 2 N+ 1 ( = N bn an ( ds. b e ds (12) We ca ake he liearized soluio approximaio N (), he we ge: b ( Ce as he iiial N = 2 b b b ac e ( e 1) N1( = Ce +. (13) b The codiio N ( ) =. 1 gives us C =.1. Thus: b b b a. 1 e ( e 1) N1( =. 1e +. (14) b I he same maer, he res of he compoes of he ieraio formulae (12) ca be obaied usig he Maple Package. Numerical resuls ad discussio We ow obai umerical soluios of oe species Loka-Volerra equaios. We compare wih he exac soluio, ADM, VIM. Table 1 shows he case of oe species compariso bewee he 2-ierae of VIM, 3-erm i he series of ADM ad exac soluio for b = 1; a = 3 ad N ( ) = Malaysia Joural of Mahemaical Scieces

5 The Soluio Of The Oe Species Loka-Volerra Equaio Usig Variaioal Ieraio Mehod TABLE 1: Numerical compariso whe b = 1, a = 3, ad N()=.1. Exac soluio ADM, φ 3 2-Ierae VIM CONCLUSION I his paper, we compare bewee he exac soluio, VIM ad ADM applied o oe species Loka-Volerra equaios. Comparisos wih he Adomia decomposiio mehod (ADM) show ha he VIM is a powerful mehod for he soluio of oliear equaios. The advaage of he VIM over he ADM is ha here is o eed for he evaluaio of he Adomia polyomials. REFERENCES Adomia. G., Solvig froier problems of physics: he decomposiio mehod. Boso: Kluwer Academic A.A. Solima, A umerical simulaio ad explici soluios of KdV- Burgers ad Laxs seveh-order KdV equaios. Chaos, Solios ad Fracals, doi:1.116/j.chaos , i press. E.M. Abulwafa, M.A. Abdou ad A.A. Mahmoud, The soluio of oliear coagulaio problem wih mass loss. Chaos, Solios ad Fracals, doi:1.116/j.chaos , i press. Hofbauer, J. ad Sigmud, K., The heory of evoluio ad dyamical sysems, Cambridge Uiversiy press, Lodo Malaysia Joural of Mahemaical Scieces 59

6 B. Baiha, M.S.M. Noorai, I. Hashim J.H. He Variaioal ieraio mehod a kid of o-liear aalyical echique: some examples. I J No-Liear Mech, 34, J.H. He. 2. Variaioal ieraio mehod for auoomous ordiary differeial sysems. Appl Mah Compu, 114(23), J.H. He. 23A simple perurbaio approach o Blasius equaio. Appl Mah Compu., 14(23), J.H. He, Y.Q. Wa ad Q. Guo, 24. A ieraio formulaio for ormalized diode characerisics. I J Circ Theory Appl, 32(6), M.A. Abdou, A.A. Solima, 25. Variaioal ieraio mehod for solvig Burgers ad coupled Burgers equaios. J Compu Appl Mah, 181(2), M. Iokui, H. Sekie ad T. Mura Geeral use of he Lagrage muliplier i oliear mahemaical physics. I: Nema-Nassed S, edior. Variaioal mehod i he mechaics of solids. Pergamo press, M. Moghimi ad F.S.A. Hejazi, Variaioal ieraio mehod for solvig geeralized Burger Fisher ad Burger equaios. Chaos, Solios ad Fracals, doi:1.116/j.chaos , i press. Olek, S., A accurae soluio o he mulispecies Loka-Volerra equaios, Sociey for Idusrial ad Applied Mahemaics, Siam Review, 36(3), S. Momai ad S. Abuasad, 26. Applicaio of He s variaioal ieraio mehod o Helmholz equaio. Chaos, Solios ad Fracals, 27(5), W. Che ad Z. Lu. 24. A algorihm for Adomia decomposiio mehod. Appl. Mah. Compu., 159, Malaysia Joural of Mahemaical Scieces