A comparative study of a system of Lotka-Voltera type of PDEs through perturbation methods

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1 Computatioal Ecology ad Software, 13, 3(4): Article A comparative study of a system of Lotka-Voltera type of PDEs through perturbatio methods H. A. Wahab 1, M. Shakil 1, T. Kha 1, S. Bhatti, M. Naeem 3 1 Departmet of Mathematics, Hazara Uiversity, Mashera, Pakista Departmet of Mathematics, COMSATS Istitute of Iformatio Techology, Uiversity Road, Abbottabad, Pakista 3 Departmet of Iformatio Techology, Hazara Uiversity, Mashera, Pakista wahabmaths@yahoo.com Received 9 August 13; Accepted 15 September 13; Published olie 1 December 13 Abstract I this paper the Adomia Decompositio Method (ADM) is employed i order to solve liear ad oliear fuctioal equatios ad the results are the compared with those produced by Homotopy Perturbatio Method (HPM) through a system of Lotka Voltera type of PDEs. The result produced by HPM are promisig ad ADM appears as a special case of HPM for Lotka Voltera type of PDEs. Keywords Lotka Voltera PDE; Adomia Decompositio Method; Homotopy Perturbatio Method. Computatioal Ecology ad Software ISSN 71X URL: versio.asp RSS: E mail: ces@iaees.org Editor i Chief: WeJu Zhag Publisher: Iteratioal Academy of Ecology ad Evirometal Scieces 1 Itroductio The Lotka Voltera equatios are a pair of first order o liear differetial equatios these are also kow as the predator prey equatios. The Lotka Voltera type problems were origially itroduced by Lotka i 19 (Lotka, 19) as a model for udumped oscillatig chemical reactios ad after that these were applied by Voltera (Voltera, 196) to predator prey iteractios, cosist of a pair of first order autoomous ordiary differetial equatios. Sice that time the Lotka Voltera model has bee applied to problems i chemical kietics, populatio biology, epidemiology ad eural etworks. These equatios also model the dyamic behavior of a arbitrary umber of competitors. Fidig the solutio of Lotka Voltera equatios may become a difficult task either if the equatios are fractioalized, amely they become a o-local oe. Recetly some ew methods such as Tah fuctio method (Fa, ; Wazwaz, 5) exteded Jacobi elliptic fuctio expasio method (Fu et al., 1) ad the simplest equatio method (Kudryashov, 5) have bee used i literature to fid exact solutios for both partial differetial equatio ad system of partial differetial equatios. However, it is difficult to obtai closed form solutios for oliear problems. I most cases, oly approximated solutios either aalytical oes or umerical oes are fouded. For aalytical solutios of o liear problems Perturbatio method is oe of the

2 Computatioal Ecology ad Software, 13, 3(4): well-kow methods. Perturbatio method is based o the existece of small or large parameters, so-called perturbatio quatities (Hich, 1991; Feradez-Villaverde, 11; Nayfeh, 1993). Applicatio Problem Cosider the followig system of o liear Lotka-Voltera Type of PDEs (Cheriha ad Kig, 5),, w ww w aew pg t x x (1) b f q kw, t x x with iitial approximatios as,, x, x x w x,. () (3) where abe,,, f, g, k, p, q are arbitrary costats such that, ef, gk. 3 Solutios 3.1 Geeral solutio of the problem I the give problem the system of equatios are coupled equatios ad quadratic o liearity terms are also cotaied i it. A exact periodic solutio of this system was preseted i (Alabdullatif et al., 7) as, g c w () t t cos x g t, c (4) 4g g g c () t () t cos x g t, c g c (5) where, 6 ga, 3g s t () t, 3g a (6) 3ga tah ( st) a3 g, a 3g Here e, f, g, k, p, q,, sare arbitrary costats such that, (ag 6 g ) 4g c e f, ba6 g, k cg, p, q p 3 ga. c c 3. Aalysis of the problem by usig the Adomia Decompositio Method The Adomia decompositio method (Adomia, 1994; Bildik ad Bayramoglu, 5) defies a liear operator of the form, t. t The system (1) ca be writte as, w( ww) wa ( ew) pg, t x x (7)

3 11 Computatioal Ecology ad Software, 13, 3(4): t ( x) x( b f) qkw, Applyig the iverse operator to equatios (7) ad (8) we will have that, wxt x t aw g p Fw 1 (, ) ( ) [ ( )], 1 ( xt, ) ( x) t [ b kwqg( )], (8) (9) (1) where the o liear terms are, Fw ww ew G f ( ) ( x) x, ( ) ( x) x. Accordig to the decompositio method (Bildik ad Bayramoglu, 5) it is assumed that the ukow fuctios w( x, t), ( x, t) are represeted as,, wxt (,) w(,) xt (,) x t (,). x t F( w), G( ) are the o liear ad these o liear terms ca be splited ito a ifiite series of polyomials, as, Fxt (,) A Here the compoets. w x, t, x, t Gxt (,) B ca be fouded by use of recursive relatios ad A ' sb, ' s are the Adomia polyomials of w ' s, ' s respectively. w, For are represeted by the followig recursive relatios. w w( x,) ( x), (11) (,) x (), x The other compoets are as follow, w t [ aw g pa], 1 1 t [ b kw qb ], 1 1 The result ca be geeralized for -terms as, (1) (13) (14) w t aw g p A 1 1 [ ],, 1 1 t [ b kw q B],, By usig the recursive relatios for w x, t, x, t wxt (, ) lim ( xt, ), ( xt, ) lim ( xt, ), x x will be determied as, 1 1 ( x,) t w( xt,), ( xt,) ( xt,). i i i i Case #1 1 Cosider a 3 g, () t 6, 6,,, 1, 1. 3 g b c s t a g p q e f c g k The obtaied solutios

4 Computatioal Ecology ad Software, 13, 3(4): are as follows 1 1 c w 6 6 cos x, 3cs 3cs c 1 1 c 6 6 si x, 3cs 3cs c c 1 t ct 6 g ( 6g cos x ( ( sg B( 6 sg))) s s c cs c c ct cos x ( c(6 sg)) ( sgc( sg)cos x )) ( s w1. c 4g 1 ( 6) g) si x tg( 6 g (( ( 6) g g c s c g s c c c si x )) t( 6 g( ( 6) g)cos x ) atp) s s c (15) (16) (17) t 1 c ( ( c( ( 6) g) g cos x ) g( 6 g ( ( 6) g) g s c s s c c 4g 1 c cos x c( 6 g (( ( 6) g) g si x ))) s c g s c 1 c 1 (( sg c( 6 sg)) g g si x ( g(4 sg c( 6 sg)) sg cs g. c 4g 1 c( 6 sg)) g si x )) ( 6 g (( ( 6) g) g s c g s c c si x )) b q ) (18) Case # 3g a For a 3 g, () t 3ga tah ( st) a3 g, a 3, p 1, g 1., c 3, k 1.., the followig results are obtaied, w cos 3 x, si 3, x (19) ()

5 114 Computatioal Ecology ad Software, 13, 3(4): x x x x cos cos 1. 3 w1 t..19si si 1. 3 x x x x cos cos t..143si si 1. 3 Now i the ext subsectio, the aalysis of this problem is preseted by usig Homotopy Perturbatio Method (He, 1999; He, ). 3.3 Aalysis of the problem by usig the Homotopy Perturbatio Method Costructed Homotopy for the give system of equatios (1) ad () respectively will be as follows, w y h y ww aw ew p g t t [ t ( x) x ], (1) () (3) t zt h[ zt ( x) x b f qkw], The iitial approximatios are choosed as, w x, t y x, t w x, x, x t z x t x x,,,. Let us assume the solutio for the above system of equatios respectively as follow, w x, t: h w hw h w..., 1 (4) (5) (6) (7) xth h h, ;..., 1 Puttig equatios (7) ad (8) i equatios (3) ad (4) ad comparig the same powers of h, h wt yt w x t w x x :,,,, h t zt x t x x :,,, ( ), h : w y w w w aw ew pg, w x,, 1 1t t xx x 1 h : z b e qkw, x,, 1 1t t xx x 1 (8) (9) (3) (31) (3) h wt wxw1x ww1xx ww 1 xx aw1 eww1 1 :. h t x1x 1xx 1 xx a1 e1 ww1 :. Case # 1 (33) (34) Now cosider the first case i.e. a 3. g g (). 3cs c (35) Ad by takig a3, p1, g1., c3, k 1.., we fid the followig relatios,

6 Computatioal Ecology ad Software, 13, 3(4): c wx, twx, cos x x. 3cs c 3cs c x, tx, si x x. 3cs c 3cs (36) (37) 1 cos cx B 3cs c 3 cs 1 si 1 cx B t 1 cos cx B t 3 s 9 cs 1 1 cos cxb cos cxb w1 a t e t pt 3cs c 3 cs 3cs c 3 cs 1 si cx B g t 3cs c 3 cs 1 si cx B 3cs c 3 cs 1 cos 1 cx B t 1 si cx B t 3 s 9 cs 1 1 si cxb si cxb 1 a t e t pt 3cs c 3 cs 3cs c 3 cs 1 cos cx B g t. 3cs c 3 cs Simplificatio ad compariso shows that these are the same compoets as were calculated by usig Adomia Decompositio Method. Now from equatio (33) ad (34) we have that,

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11 1 Computatioal Ecology ad Software, 13, 3(4): Case # Now cosider the secod case i.e. a 3 g, ad by takig, a 3, p 1, g 1., c 3, k 1., s 4., the for this case we fid that, () Usig these values the followig results are calculated, 3 w x, twx, cos x3, (38)

12 Computatioal Ecology ad Software, 13, 3(4): xt, x,.9.1 si x3. (39) (.1536( si( x))) si( x) t e cos( x) t w1 3.77t.37 si( x) t3(.94.14si( x)) t1.( cos( x)) t. (.1536( cos( x))) cos( x) t e si( x) t t.37 cos( x) t3( cos( x)) t 1.(.94.14si( x)) t. Simplificatio ad compariso shows that these are the same compoets as were calculated by usig Adomia Decompositio Method. Now from equatio (33) ad (34) we have that,

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16 Computatioal Ecology ad Software, 13, 3(4): Cocludig Remarks I this study, approximated solutios for some o liear problems are calculated by Adomia Decompositio Method (ADM) ad Homotopy Perturbatio Method (HPM) ad the the umerical results are compared. It is aalyzed that i Adomia Decompositio Method first Adomia polyomials are calculated which is a bit difficult ad time wastig process ad the fact that HPM solves oliear problems without usig Adomia s polyomials is a clear advatage of this techique over ADM. The comparative study betwee these two methods shows that the results obtaied by usig HPM with a special covex costructed Homotopy is almost equivalet to the results obtaied by usig ADM for these types of o liear problems. Refereces Adomia G Solvig Frotier Problems of Physics-The Decompositio Method. Kluwer Academic, Bosto, MA, USA Alabdullatif M, Abdusalam HA, Fahmy ES. 7. Adomia Decompositio Method for Noliear Reactio Diffusio System of Lotka-Voltera Type. Iteratioal Mathematical Forum () No., Bildik N, Bayramoglu H. 5. The solutio of two dimesioal o liear differetial equatio by Adomia Decompositio Method. Applied Mathematics ad Computatio, 163: Cheriha R, Kig JR. 5. No-liear reactio-diffusio systems with variable diffusivities: Lie symmetries, asatz ad exact solutios. Joural of Mathematical Aalysis ad Applicatios, 38(1): Fa E.. Exteded Tah-Fuctio Method ad its applicatios to oliear equatios. Physics Letters A, 77: 1-18 Feradez-Villaverde J. 11. Perturbatio Methods. Uiversity of Pesylvaia, USA Fu ZT, Liu SK, Zhao Q. 1. New Jacobi elliptic fuctio expasio ad ew periodic solutios of oliear wave equatios. Physics Letters. A, 9(1-): 7-76 He JH Homotopy perturbatio techique. Computer Methods i Applied Mechaics ad Egieerig, 178(3-4): 57-6 He JH.. A couplig method of a homotopy techique ad a perturbatio techique for oliear problems. Iteratioal Joural of No-Liear Mechaics, 35(1): Hich EJ Perturbatio Methods. Cambridge Uiversity Press, UK Kudryashov NA. 5. Simplest equatio method to look for exact solutios of oliear differetial quatios. Chaos, Solitos ad Fractals, 4: Lotka AJ. 19. Udamped oscillatios derived from the las of mass actio. Joural of the America Chemical Society, 4(8): Nayfeh AH Itroductio to Perturbatio Techiques. Wiley Classics Library Editio, USA Voltera V Variatios ad Fluctuatios of the Number of the Idividuals i Aimal Species Livig together. I: Aimal Ecology (Chapma RN, ed). McGraw-Hill, New York, USA Wazwaz AM. 5. The ta h method: solitos ad periodic solutios for the Dodd Bullough Mikhailov ad the Tzitzeica Dodd Bullough equatios. Chaos, Solitos ad Fractals, 5: 55-63