APPLICATION OF HOMOTOPY ANALYSIS TRANSFORM METHOD TO FRACTIONAL BIOLOGICAL POPULATION MODEL

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1 Shiraz Universiy of Technology Fro he SelecedWorks of Habibolla Laifizadeh 13 APPLICATION OF HOMOTOPY ANALYSIS TRANSFORM METHOD TO FRACTIONAL BIOLOGICAL POPULATION MODEL Habibolla Laifizadeh, Shiraz Universiy of Technology Available a: hps://works.bepress.co/habib_laifizadeh/11/

2 Roanian Repors in Physics, Vol. 65, No. 1, P , 13 APPLICATION OF HOMOTOPY ANALYSIS TRANSFORM METHOD TO FRACTIONAL BIOLOGICAL POPULATION MODEL DEVENDRA KUMAR 1, JAGDEV SINGH and SUSHILA 3 1 Jagan Nah Gpa Insie of Engineering and Technology, Deparen of Maheaics, Jaipr-3, Rajashan, India E-ail: devendra.ahs@gail.co Jagan Nah Universiy, Deparen of Maheaics, Village-Rapra, Tehsil-Chaks, Jaipr-3391, Rajashan, India E-ail: jagdevsinghrahore@gail.co 3 Jagan Nah Universiy, Deparen of Physics, Village-Rapra, Tehsil-Chaks, Jaipr-3391, Rajashan, India E-ail: sshila.jag@gail.co Received Febrary 14, 1 Absrac. In his paper, we presen an algorih of he hooopy analysis ransfor ehod (HATM) which is a cobinaion of Laplace ransfor ehod and he hooopy analysis ehod (HAM) o solve generalized biological poplaion odels. The fracional derivaives are described by Capo sense. The proposed ehod presens a procedre of consrcing he se of base fncions and gives he high-order deforaion eqaions in a siple for. The proposed schee provides he solion in he for of a rapidly convergen series. Three exaples are sed o illsrae he preciseness and effeciveness of he proposed ehod. The resls show ha he HATM is very efficien, siple and can be applied o oher nonlinear probles. Key words: Laplace ransfor, hooopy analysis ransfor ehod, biological poplaion odel, Miag-Leffler fncion. 1. INTRODUCTION Fracional differenial eqaions have gained iporance and poplariy dring he pas hree decades or so, ainly de o is deonsraed applicaions in neros seeingly diverse fields of science and engineering. For exaple, he nonlinear oscillaion of earhqake can be odeled wih fracional derivaives and he flid-dynaic raffic odel wih fracional derivaives can eliinae he deficiency arising fro he asspion of conin raffic flow. The fracional differenial eqaions are also sed in odeling of any cheical processes, aheaical biology and any oher probles in physics and engineering [1 16].

3 64 Devendra Kar, Jagdev Singh, Sshila Nonlinear probles are iporan for engineers, physiciss and aheaicians naely becase os physical syse are nonlinear in nare. However, he nonlinear eqaions are difficl o solve and lead o ineresing phenoeno, e.g. chaos. The invesigaion of he exac solions of nonlinear evolion eqaions plays an iporan role in he sdy of nonlinear physical phenoena. There are any approaches for seeking exac solions, sch as, Hiroa s ehod, Bäcklnd and Darbox ransforaions, Painlevé expansions. Recenly, any alernaive ehods sed for solving boh nonlinear and linear differenial eqaions of physical ineres. The Adoian decoposiion ehod (ADM) [17 18], he hooopy perrbaion ehod (HPM) [19 3], he hooopy analysis ehod (HAM) [4 3], he variaional ieraion ehod (VIM) [31 38] and oher ehods have been sed o solve linear and nonlinear probles. The Laplace ransfor is oally incapable of handling nonlinear eqaions becase of he difficlies ha are cased by he nonlinear ers. Varios ways have been proposed recenly o deal wih hese nonlineariies sch as he Laplace decoposiion ehod (LDM) [39 43] and he hooopy perrbaion ransfor ehod (HPTM) [44]. Very recenly, he hooopy analysis ehod (HAM) is cobined wih he well-known Laplace ransfor o prodce a highly effecive echniqe called he hooopy analysis ransfor ehod (HATM) [45, 46] for handling any nonlinear probles. The fracional opial conrol probles have been solved by Balean e al. [48]. Golankhaneh e al. have eployed he hooopy perrbaion ehod (HPM) for solving a syse of Schrödinger-Koreweg-de Vries eqaions [49]. In his paper, we consider he nonlinear fracional-order biological poplaion odel in he for: = ( ) + ( ) + f( ), ( x y wih he given iniial condiion xy (,,) = f ( xy, ), () where denoes he poplaion densiy and f represens he poplaion spply de o birh and deahs. This nonlinear fracional biological poplaion odel is obained by replacing he firs ie derivaive er in he corresponding biological poplaion odel by a fracional derivaive of order wih < 1. The derivaives are ndersood in he Capo sense. The general response expression conains a paraeer describing he order of he fracional derivaive ha can be varied o obain varios responses. In he case of = 1 he fracional biological poplaion odel redces o he sandard biological poplaion odel. Soe aspecs of sch a odel have been sdied previosly by oher researchers [51, 5]. In his paper, frher we apply he hooopy analysis ransfor ehod (HATM)

4 3 Applicaion of hooopy analysis ransfor ehod 65 o solve he fracional biological poplaion odels. The objecive of he presen paper is o odify he hooopy analysis ehod (HAM) o solve nonlinear fracional biological poplaion odels. The hooopy analysis ransfor ehod (HATM) is a cobinaion of he hooopy analysis ehod (HAM) and Laplace ransfor ehod. The advanage of his ehod is is capabiliy of cobining wo powerfl ehods for obaining exac and approxiae analyical solions for nonlinear eqaions. The fac ha he HATM solves nonlinear probles wiho sing Adoian s polynoials and He s polynoials is a clear advanage of his echniqe over he Adoian s decoposiion ehod (ADM) and he hooopy perrbaion ransfor ehod (HPTM). The plan of or paper is as follows: Brief definiions of he fracional calcls are given in Secion. The HATM is presened in Secion 3. In Secion 4, hree nerical exaples are solved o illsrae he applicabiliy of he considered ehod. Conclsions are presened in Secion 5.. BASIC DEFINITIONS In his secion, we enion he following basic definiions of fracional calcls. Definiion 1. The Rieann-Lioville fracional inegral operaor of order >, of a fncion f() C µ, µ 1is defined as [5]: 1 Γ ( ) (3) 1 J f( ) = ( τ) f( τ)d τ, ( > ), J f ( ) f( ). = (4) For he Rieann-Lioville fracional inegral we have: Γ( γ+ J γ = +γ. (5) Γγ++ ( Definiion. The fracional derivaive of f ( ) in he Capo sense is defined as [1]: n n D f( ) = J D f( ) = 1 n 1 ( n) ( ) f ( )d, Γ( n ) (6) = τ τ τ for n 1 < n, n N, x>. Definiion 3. The Laplace ransfor of he Capo derivaive is given by Capo [1]; see also Kilbas e al. [13] in he for

5 66 Devendra Kar, Jagdev Singh, Sshila 4 n 1 r 1 ( r) L [D f ( )] = s L[ f( )] s f ( + ), n 1 < n. (7) r= Definiion 4. The Miag-Leffler is defined as [47]: k z E ( z) = ( C, Re( ) > ). (8) Γ ( k + k = 3. HATM FOR GENERALIZED BIOLOGICAL POPULATION MODEL We consider he generalized biological poplaion odel of he for: ( ) ( ) a b = + + k (1 r ), x y (9) >, x, y, < 1, wih he iniial condiion xy (,,) = f ( xy, ). ( Taking he Laplace ransfor on boh sides of eqaion (9) sbjec o he iniial condiion (, we have 1 1 ( ) ( ) a b L[ xy (,, )] f ( xy, ) L k (1 r ). + + = (1 s s x y We define he nonlinear operaor 1 1 N[ φ ( xyq,, ; )] = L[ φ( xyq,, ; )]] f ( xy, ) L ( ( xyq,, ; )) s s φ x ( (,, ; )) a b + φ xyq + kφ ( xyq,, ; )( 1 rφ ( xyq,, ; )), ( y where q [, 1] and φ ( x, yq, ; ) is a real fncion of x, y, and q. We consrc a hooopy as follows (1 q)l[ φ( x, y, ; q) ( x, y, )] = qh( ) N[ φ( x, y, )], (13) where L denoes he Laplace ransfor, q [, 1] is he ebedding paraeer, H () denoes a nonzero axiliary fncion, ћ is an axiliary paraeer, (,, ) x y is an iniial gess of xy (,, ) and φ ( x, yq, ; ) is a nknown fncion. Obviosly, when he ebedding paraeer q = and q = 1, i holds

6 5 Applicaion of hooopy analysis ransfor ehod 67 φ ( x, y, ;) = ( xy,, ), φ ( xy,, ; = xy (,, ), (14) respecively. Ths, as q increases for o 1, he solion φ ( x, yq, ; ) varies fro he iniial gess (,, ) x y o he solion xy (,, ). Expanding φ ( x, yq, ; ) in Taylor series wih respec o q, we have where φ ( x, yq, ; ) = ( xy,, ) + ( xyq,, ), (15) = 1 1 φ( xyq,, ; ) ( x, y, ) = q. =! q (16) If he axiliary linear operaor, he iniial gess, he axiliary paraeer, and he axiliary fncion are properly chosen, he series (15) converges a q = 1, hen we have xy (,, ) = ( xy,, ) + ( xy,, ), (17) which s be one of he solions of he original nonlinear eqaions. According o he definiion (17), he governing eqaion can be dedced fro he zero-order deforaion (13). Define he vecors = { ( xy,, ), ( xy,, ),..., ( xy,, )}. (18) 1 Differeniaing he zero h -order deforaion eqaion (13) -ies wih respec o q and hen dividing he by! and finally seing q =, we ge he following h -order deforaion eqaion: L[ ( x, y, ) χ ( x, y, )] = qh( ) R ( ). (19) = Applying he inverse Laplace ransfor, we have 1 ( xy,, ) =χ ( xy,, ) + L [ qh ( ) R ( )], () where and N[ φ( x, y, ; q)] R ( = 1 q, = (! q (, 1, χ = 1, > 1. ()

7 68 Devendra Kar, Jagdev Singh, Sshila 6 4. APPLICATIONS In his secion, we se he HATM o solve he generalized biological poplaion odels. Exaple 4.1. Consider he following generalized biological poplaion odel: ( ) ( ) = + + (1 r), (3) x y wih he iniial condiion 1 r xy (,,) = exp ( x+ y). Applying he Laplace ransfor sbjec o he iniial condiion, we have 1 1 r L[ xy (,, )] exp ( x+ y) s 1 ( ) ( ) L (1 r). + + = s x y The nonlinear operaor is 1 1 r N[ φ ( xyq,, ; )] = L[ φ( xyq,, ; )]] exp ( x+ y) s 1 L ( ) ( ) ( xyq,, ; ) ( xyq,, ; ) s φ + φ + x y +φ( xyq,, ; )( 1 rφ( xyq,, ; )), (4) (5) (6) and hs r 1 R ( = L( (1 χ ) exp ( x+ y) L r 1 r s s x r= + r 1 r + 1 r r 1 r. y r= r= (7) The h -order deforaion eqaion is given by

8 7 Applicaion of hooopy analysis ransfor ehod 69 L[ ( xy,, ) χ ( xy,, )] = R ( ). (8) 1 1 Applying he inverse Laplace ransfor, we have 1 ( xy,, ) =χ ( xy,, ) + L [ R ( )]. (9) 1 1 Solving he above eqaion (9), for = 1,,3..., we ge 1 r 1( x, y, ) = exp ( x+ y), Γ+ ( 1 r ( x, y, ) = (1 + )exp ( x+ y) + Γ+ ( 1 r + exp ( x+ y), Γ ( + 1 r 3 ( x, y, ) = (1 + ) exp ( x+ y) + Γ+ ( 1 r + (1 + )exp ( x+ y) Γ( r exp ( x+ y), Γ (3 + and so on. Taking = 1, he solion is given by 1 r ( ) xy (,,) = (, xy,) = exp ( x+ y) = = = Γ( + 1 r exp ( x y ) E ( = + ). (3) (3 If we p = 1, we obain he exac solion: 1 r 1 r xy (,, ) = exp ( x+ y) e = exp ( x+ y) +, which is in fll agreeen wih he resls obained by El-Sayed e al. [5] and Arafa e al. [51]. (3)

9 7 Devendra Kar, Jagdev Singh, Sshila 8 Exaple 4.. Consider he following generalized biological poplaion odel: ( ) ( ) = + + k, x y (33) wih he iniial condiion xy (,,) = xy. (34) Applying he Laplace ransfor sbjec o he iniial condiion, we have 1 1 ( ) ( ) L[ (,, )] L x y xy k. + + = s s x y (35) The nonlinear operaor is 1 1 N[ φ ( xyq,, ; )] = L[ φ( xyq,, ; )]] xy L ( ( xyq,, ; )) s s φ + x + ( φ ( xyq,, ; )) + kφ( xyq,, ; ), y and hs r 1 r 1 r= R ( = L[ 1] (1 χ ) xy L r 1 r + s s x r= k y. (36) (37) The h -order deforaion eqaion is given by L[ ( xy,, ) χ ( xy,, )] = R ( ). (38) 1 1 Applying he inverse Laplace ransfor, we have ( xy,, ) =χ ( xy,, ) + L [ R ( )]. (39) Solving he above eqaion (39), for = 1,,3..., we ge

10 9 Applicaion of hooopy analysis ransfor ehod 71 1( x, y, ) = k xy, Γ+ ( x y k xy k xy Γ+ ( Γ(+ (,, ) = (1 + ) +, x y k xy k xy Γ+ ( Γ(+ 3 (,, ) = (1 + ) + (1 + ) k xy, Γ(3+ (4) and so on. Taking = 1, he solion is given by ( k ) ( x, y, ) = ( x, y, ) = xy = xye ( k ). Γ( + = = (4 If we p = 1, we obain he exac solion: xy (,, ) = xye k, (4) which is in fll agreeen wih he resls given by El-Sayed e al. [5] and Arafa e al. [51]. Exaple 4.3. Consider he following generalized biological poplaion odel: ( ) ( ) = + +, x y (43) wih he iniial condiion xy (,,) = sinx sinh y. (44) Applying he Laplace ransfor sbjec o he iniial condiion, we have 1 1 ( ) ( ) L[ (,, )] sin sinh L xy x y. + + = s s x y The nonlinear operaor is (45) 1 1 N[ φ ( xyq,, ; )] = L[ φ( xyq,, ; )]] sinx sinh y L ( ( xyq,, ; )) s s φ + x (46) + ( φ ( xyq,, ; )) +φ( xyq,, ; ), y

11 7 Devendra Kar, Jagdev Singh, Sshila 1 and hs The h 1 R ( = L[ 1] (1 χ ) sinx sinh y s L r 1 r r 1 r s x r= y r= (47) -order deforaion eqaion is given by L[ ( xy,, ) χ ( xy,, )] = R ( ). (48) 1 1 Applying he inverse Laplace ransfor, we have 1 ( xy,, ) =χ ( xy,, ) + L [ R ( )]. (49) 1 1 Solving he above eqaion (49), for = 1,,3..., we ge 1( x, y, ) = sinx sinh y, Γ+ ( ( x, y, ) = (1 + ) sinx sinh y + Γ+ ( + sin x sinh y, Γ(+ 3 ( x, y, ) = (1 + ) sinx sinh y + Γ+ ( (1 + ) sin x sinh y sin x sinh y, Γ(+ Γ(3+ and so on. Taking = 1, he solion is given by ( ) xy (,, ) = ( xy,, ) = sinx sinh y = Γ( + = = = sin x sinh ye ( ). If we p = 1, we obain he exac solion: (5) (5 xy (,, ) = sinx sinh ye, (5) which is in fll agreeen wih he resls given by El-Sayed e al. [5] and Arafa e al. [51].

12 11 Applicaion of hooopy analysis ransfor ehod CONCLUSIONS In his paper, he hooopy analysis ransfor ehod (HATM) has been sccessflly applied o obain he exac solions of he generalized biological poplaion eqaions sbjec o soe iniial condiions. The resls obained sing he schee presened here agree well wih he analyical solions and he nerical resls obained by Adoian s decoposiion ehod (ADM) [5] and hooopy analysis ehod (HAM) [51]. However, El-Sayed e al. [5] have shown ha ADM does no converge in general, in pariclar, when he ehod is applied o linear operaor eqaions. I was also shown ha ADM is eqivalen o Picard ieraion ehod, and herefore i igh diverge. The hooopy analysis ransfor ehod (HATM) is anoher echniqe sed o derive an analyic solion for nonlinear operaors. I provides s wih a siple way o adjs and conrol he convergence region of solion series by choosing proper vales for axiliary paraeer ħ and axiliary fncion H(). The resls reveal ha HATM a very powerfl and efficien echniqe in finding analyical solions for wide classes of nonlinear differenial eqaions. Acknowledgeens. The ahors are graefl o he referee for his invalable sggesions and coens for he iproveen of he paper. REFERENCES 1. G.O. Yong, Definiion of physical consisen daping laws wih fracional derivaives, Z. Angew. Mah. Mech., 75, (1995).. J.H. He, Soe applicaions of nonlinear fracional differenial eqaions and heir approxiaions, Bll. Sci. Technol., 15,, 86 9 (1999). 3. J.H. He, Approxiae analyic solion for seepage flow wih fracional derivaives in poros edia, Cop. Mehods Appl. Mech. Eng., 167, (1998). 4. R. Hilfer (ed.), Applicaions of Fracional Calcls in Physics, World Scienific Pblishing Copany, Singapore-New Jersey-Hong Kong,, pp I. Podlbny, Fracional Differenial Eqaions, Acadeic Press, New York, F. Mainardi, Y. Lchko, G. Pagnini, The fndaenal solion of he space-ie fracional diffsion eqaion, Fracional Calcls and Applied Analysis, 4, (. 7. S.Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, On he solions of ie-fracional reacion-diffsion eqaions, Conicaions in Nonlinear Science and Nerical Silaion, 15,, (. 8. A. Yildiri, He s hooopy perrbaion ehod for solving he space- and ie- fracional elegraph eqaions, Inernaional Jornal of Coper Maheaics, 87, 13, (. 9. L. Debnah, Fracional inegrals and fracional differenial eqaions in flid echanics, Frac. Calc. Appl. Anal., 6, (3). 1. M. Capo, Elasicia e Dissipazione, Zani-Chelli, Bologna, K.S. Miller, B. Ross, An Inrodcion o he fracional Calcls and Fracional Differenial Eqaions, Wiley, New York, K.B. Oldha, J. Spanier, The Fracional Calcls Theory and Applicaions of Differeniaion and Inegraion o Arbirary Order, Acadeic Press, New York, 1974.

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