Application of Homotopy Perturbation Method to Biological Population Model

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1 Available a h://vamu.edu/aam Al. Al. Mah. ISSN: Vol. 05, Issue (December 010), (Previously, Vol. 5, Issue 10, ) Alicaios ad Alied Mahemaics: A Ieraioal Joural (AAM) Alicaio of Homooy Perurbaio Mehod o Biological Poulaio Model Pradi Roul Isiue for Numerical Mahemaics Duisburg-Esse Uiversiy D-47058, Duisburg, Germay bauroul@yahoo.com Received: May 18, 010; Acceed: Augus, 010 Absrac I his aricle, a well-kow aalyical aroximaio mehod, so-called he Homooy erurbaio mehod (HPM) is adoed for solvig he oliear arial differeial equaios arisig i he saial diffusio of biological oulaios. The resulig soluios are comared wih hose of he exisig soluios obaied by emloyig he Adomia s decomosiio mehod. The comariso reveals ha our aroximae soluios are i very good agreeme wih he soluios by Adomia s mehod. Moreover, he resuls show ha he roosed mehod is a more reliable, efficie ad coveie oe for solvig he o-liear differeial equaios. Keywords: Homooy erurbaio mehod (HPM); Adomia s decomosiio mehod; biological oulaio model; exac soluio MSC 000 No.: 35K15, 35C05, 65D99, 65M99 1. Iroducio May roblems arisig i scieific fields icludig mahemaical biology, fluid dyamics, visco-elasiciy ad mahemaical hysics ca be successfully modeled by he use of oliear arial differeial equaios. Several aroaches such as he Tha mehod [Evas (005), Fa (000)], he Homooy-Perurbaio mehod [He (1999a, 000a, 005), Jaalizadeh (008), Yildirim (010a, 010b), Koçak (009), Gaji (006)], he Adomia s decomosiio mehod 7

2 AAM: Ier. J., Vol. 05, Issue (December 010) [Previously, Vol. 05, Issue 10, ] 73 [Adomia (1988,1994), Gorguis (006), Momai (005), Shakeri (007)], he Variaioal Ieraio Mehod [(He (1999b, 000b, 006), Mohyud-Di (009), Abdou (005), Xu (009), Noor (008), Shakeri (007)], ad some asymoic mehods [He (006)] have bee used o solve eiher liear or oliear differeial equaios. Amog hese mehods, he variaioal ieraio mehod ad he homooy erurbaio mehod are he mos efficie, coveie ad effecive mehods for fidig he aalyical aroximae soluios of oliear roblems. I his aricle, he oliear degeerae arabolic equaios arise i he sudy of saial diffusio of biological oulaios subjec o some iiial codiios. Cosider he firs-order ime derivaive, oliear biological model i he followig form: u u u x y g u, (1) wih give iiial codiio ux, y,0, where u ad g deoe he oulaio desiy ad oulaio suly due o birhs ad deahs, resecively. k d I his sudy, g is cosidered as g u hu 1 ru, where hdkr,,, are real umbers. I is worh oiig ou ha here are wo examles of cosiuive equaios for g u : if h c, d 1, r 0, his leads o Malhusia law [Guri (1977)], where c is a cosa ad hc, d k 1, r c c, 1 1 Verhuls law [Guri (1977)], c1, c are osiive cosas. The urose of his aer is o exed he homooy erurbaio mehod for comuig he aroximae aalyical soluios of he oliear biological oulaio model ad he see how hese soluios comare wih he available exac soluios imlemeed by Shekeri e al. [Shakeri (007)] adoig a Adomia decomosiio mehod. The homooy erurbaio mehod was origially roosed by He for oliear differeial roblems [He (1999a, 000a)]. I s mai feaure is he codiio of homooy by iroducig a embedme arameer, which akes he value from 0 o 1. If 0, he sysem of equaios (homooy equaios) geerally reduces o a very simlified form, which yields a raher simle soluio. O he oher had, whe 1, i urs ou o be he origial roblem ad rovides he required soluio. The aroximae soluios obaied usig he HPM coverges raidly o he exac soluio wihou ay resricive assumios, liearizaio or rasformaios. I coras o he radiioal erurbaio mehod [Holeme (1995), Nayfeh (000)], he HPM mehod does o eed a small arameer i he sysem. The deailed descriio of HPM is described i Secio. This aricle is orgaized as follows. Secio is devoed o a shor descriio of he aalysis of homooy erurbaio mehod. I Secio 3, we rese he aalyical aroximae soluios obaied by imlemeig he HPM o he oliear biological oulaio model followed by he comariso of resuls bewee he aroximae soluios ad he soluios obaied by adoig a Adomia decomosiio mehod. Fially i Secio 4, we summarize ad discuss he resuls.

3 74 Pradi Roul. Aalysis of Homooy Perurbaio Mehod To illusrae he basis ideas of he homooy-erurbaio mehod, we cosider he followig oliear differeial equaio: 0, Lu Nu g () wih boudary codiios u Bu, 0,, (3) where L is a liear oeraor, N deoes a o-liear oeraor, ad fucio, g is a kow aalyical u is a ukow fucio, is he boudary of he domai ad B is a boudary oeraor. By meas of Homooy erurbaio mehod [He (1999a, 000a)], we ca cosruc a Homooy v, for Equaio () as follows: v, : 0,1 R, which saisfies or 0 H v, 1 Lv Lu Lv Nv g, (4) 0 0 H v, Lv Lu Lu Nv g, (5) where 0,1 is a embeddig arameer, v is a ukow fucio ad u 0 is a iiial aroximaio ha mus saisfies he boudary codiio (3). If 0, he Equaio (5) becomes 0 H v,0 Lv Lu 0 (6) ad whe 1, Equaio (5) akes he origial form of he Equaio (), i.e., H v,1 Lv Nv g 0. (7) To solve he roblem (), i is ecessary o use he erurbaio echique as discussed i Holeme (1995) ad Nayfeh (000). So he combiaio of he erurbaio mehod ad he

4 AAM: Ier. J., Vol. 05, Issue (December 010) [Previously, Vol. 05, Issue 10, ] 75 homooy mehod is kow as he homooy-erurbaio mehod. By alyig he erurbaio echique he soluio of Equaio () ca be exressed as a ower series i vv v v v (8) Leig 1 i Equaio (8), he aroximae soluio of () ca be obaied easily as follows u lim v v v v v 0 v 1 v v 3. (9) The deailed covergece aalysis of he HPM has bee discussed i He (1999a, 000a). The rae of covergece of ower series (8), saed i He (1999a), deeds uo he oliear oeraor of Equaio () which saisfies he followig wo codiios: I. The secod derivaive of oliear oeraor Nv mus be small eough, oherwise he arameer may be large, ha is, aroaches o 1. 1 N II. The orm of L mus be smaller ha 1. v 3. Alicaio of Homooy Perurbaio Mehod I his secio, he homooy erurbaio mehod described i he revious secio for solvig hree differe yes of roblems arisig i biological oulaio models is alied. The comariso is made wih he available aalyical resuls obaied by Shakeri e al. (007) usig he Adomia s decomosiio mehod o assess he accuracy ad he effeciveess of he homooy erurbaio mehod. Examle 1. Le us cosider he followig biological oulaio model: u x, y, u x, y, 8 u x, y, u x, y, 1 u x, y,, x y 9,,0 ex 1. 3 subjec o he iiial codiio ux y x y (10) (11) Accordig o he homooy erurbaio mehod, we ca cosruc he homooy for Equaio (10) which saisfies:

5 76 Pradi Roul 1,,,, u xy u0 xy u xy,, u xy,, u xy,, 8 + uxy,, 1 uxy,,. x y 9 (1) Subsiuig (8) io (1) ad equaig he erms wih ideical owers of, we obai he followig se of liear arial differeial equaios 0 : u0 x, y, 0, 1 u0 x, y, ex x y. 3 (13) u 1 1 xy,, u0 xy,, u0 xy,, 8 : u 0x, y, u0 x, y, 0, x y 9 u1 x, y,0 0, (14) u : x, y, u0 xy,, u1 xy,, u0 xy,, u1 xy,, x y u x, y,0 0, 16,,,,,, 0, 9 + u x y u x y u x y (15) Solvig he above equaios, we obai he followig aroximaios 1 u0 x, y, ex x y 3, (16) 1 u1 x, y, ex xy 3, (17) 1 u x, y, ex x y 3, (18) ad so o, i he same maer he res of he comoes ca be obaied usig he Male ackage. Accordig o he HPM, we ca obai he soluio i a series form as follows ux, y, ex x yex x y ex x y 3 3! 3

6 AAM: Ier. J., Vol. 05, Issue (December 010) [Previously, Vol. 05, Issue 10, ] 77 1 ex x y 1 3! 1 1 ex 3! which has he exac soluio x y, (19) ux, y, ex x yexex x y 3 3. (0) From he above soluio rocess, i ca be see clearly ha, he aroximae soluio coverges very fas o is exac soluio. The soluio i Equaio (0) which obaied by HPM is absoluely same as ha of he soluio ivesigaed by Shkeri e al. [Shakeri (007)] usig he Adomia decomosiio mehod. Furhermore, he mai advaage i usig he HPM for solvig he cosidered model is ha he exac soluios obaied successfully wihou requirig a small arameer i he equaio ad wihou calculaig he comlicaed Adomia s olyomials. Examle. Le us cosider he followig biological oulaio model:,,,, u x y u x y u x, y, hu x, y,, x y (1) wih he iiial codiio ux, y,0 xy. Similarly, by usig he homooy erurbaio mehod, a homooy of (1) ca be obaied as follows 1,,,, u xy u xy 0,,,,,, u xy u xy u xy x y hu x, y,. () Subsiuig (8) io () ad equaig he erms wih ideical owers of, we obai he followig se of liear arial differeial equaios 0 : u0 x, y, 0, u0 x, y, xy, (3) u 1 1 xy,, u0 xy,, u0 xy,, : hu 0 x, y, 0, x y u1 x, y,0 0, (4)

7 78 Pradi Roul : u x, y, u0 xy,, u1 xy,, u0 xy,, u1 xy,, x y u x, y,0 0, hu1 x, y, 0, (5) Usig he iiial aroximaio aroximaios as follows u0 x, y, xy, ad solvig he above equaios, we obai he u0 x, y, xy, (6) u1 x, y, h xy, (7) u x, y, h xy, (8)! u x, y, h xy. (9)! Iserig he values of u 0,u 1,u, u i Equaio (9), yields he exac soluio of (1) as follows!! u x, y, lim xy h xy h xy h xy 0 h h lim xy xye, (30)! which is he same exac soluio obaied by Shakeri e al. (007) usig he Adomia decomosiio mehod, if we use he arameer h 15. Examle 3. Cosider he followig biological oulaio model:,,,, u x y u x y u x, y, u x, y, x y subjec o he iiial codiio u x, y,0 sixsih y., (31) For solvig (31) by he homooy-erurbaio echique we cosider he followig homooy

8 AAM: Ier. J., Vol. 05, Issue (December 010) [Previously, Vol. 05, Issue 10, ] 79 1,,,, u xy u xy 0,,,,,, u xy u xy u xy x y ux, y,. (3) Subsiuig he value of u from Equaio (8) io (3) ad equaig he erms of he same owers of, i yields ha u 0 0 x, y, : 0, u0 x, y, sixsih y. (33) u 1 1 xy,, u0 xy,, u0 xy,, : u 0 x, y, 0, u1 x, y,0 0, (34) x y x, y, u0 xy,, u1 xy,, u0 xy,, u1 xy,, u : u 1 x, y, 0, x y u x, y,0 0, (35) Usig he iiial aroximaio u0 x, y, sixsih y, ad solvig he above equaios, we obai he aroximaios as follows u0 x, y, sixsih y, (36) u1 x, y, sixsih y, (37) u x, y, sixsih y, (38)! u x, y, sixsih y. (39)! Therefore, he exac soluio of (31) ca be exressed as ux, y, lim si xsih y si xsih y si xsih y si xsih y!! lim si xsih y1 si xsih ye. (40)!!

9 80 Pradi Roul 4. Coclusios The mai goal of his work was o emloy homooy erurbaio mehod for fidig he aroximae aalyical soluio of biological oulaio models. Three examles were reseed i his sudy o illusrae he reliabiliy ad alicabiliy of he mehod. The aalyical soluio i each of he examles obaied i erms of a ifiie series wih easily comuable comoes which coverges very raidly o he exac soluio wihou usig ay resricive assumio, erurbaio or discreizaio of he variables. Furhermore, he aroximae soluios obaied usig HPM are i excelle agreeme wih hose obaied by he decomosiio mehod of Adomia. However, due o is ease i calculaios, he HPM is a more reliable ad owerful mahemaical ool ha ca be alied o oher o-liear arial differeial equaios. Ackowledgemes The auhor wishes o hak he aoymous referees for heir valuable suggesios o imrove he qualiy of he aer. REFERENCES Adomia, G. (1988). A review of he decomosiio mehod i alied mahemaics, J. Mah. Aal. Al. 135, Adomia, G. (1994). Solvig Froier roblems of hysics: The decomosiio mehod, Kluwer, Academic, Dordrech. Abdou, M.A., Solima, A.A. (005). Variaioal ieraio mehod for solvig Burger s ad couled Burger s equaios. J. Comu. Al. Mah. 181, Evas, D. J., Rasla, K.R. (005). The Tha-Fucio mehod for solvig some imora oliear arial differeial equaio. Iera. J. Comuaioal Mahemaic. 8, Fa, E. (000). Tha-Fucio mehod ad is alicaios o oliear equaios. Phys. Le. A 77, Gaji, D. D., Sadighi, A. (006). Alicaio of He s homooy-erurbaio mehod o oliear couled sysems of reacio-diffusio equaios. Ieraioal Joural Noliear Sciece Numeri-cal soluio. 7 (4), Gorguis, A. (006). A comariso bewee Cole-Hof rasformaio ad he decomosiio mehod for solvig Burgers s equaio. Al. Mah. Comu. 173 (1), Guri, M.E. ad MacCamy, R.C. (1977). O he diffusio of biological oulaios, Mah. Biosc. 33, He, J. H. (1999a). Homooy-erurbaio echique, Comu. Mehods Al. Mech. Egrg. 178, He, J. H. (1999b). Variaioal ieraio mehod: a kid of oliear aalyical echique: Some examles, Ieraioal Joural of Noliear Mechaics, 344,

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