AASCIT Couicatios Volue, Issue 6 Septeber, 15 olie ISSN: 375-383 The Differetial Trasfor Method for Solvig Volterra s Populatio Model Khatereh Tabatabaei Departet of Matheatics, Faculty of Sciece, Kafas Uiversity, Kars, Turey Era Guerha Departet of Coputer, Faculty of Egieerig, Kafas Uiversity, Kars, Turey I Keywords Volterra s Populatio Model, Itegro-differetial equatio, Differetial Trasfor Method this article, Differetial trasfor ethod is preseted for solvig Volterra s populatio odel for populatio growth of a species i a closed syste. This odel is a oliear itegro-differetial where the itegral ter represets the effect of toi. This powerful ethod catches the eact solutio. First Volterra s populatio odel has bee coverted to power series by oe-diesioal differetial trasforatio. Thus we obtaied uerical solutio Volterra s populatio odel. Itroductio This ote deals with a atheatical odel of the acculated effect of tois o a populatio livig i a closed syste. Scudo idicates i his review that volterra proposed this odel for a populatio q( ) of idetical idividuals wich ehibits crowdig ad sesitivity to total etabolis : dq t = aq bq cq q( ) d, dt q() = q, (1.1) Where a > is the birth rate coefficiet, b > is the crowdig coefficiet ad c > is the toicity coefficiet. The coefficiet c idicates the essetial behavior of the populatio evolutio before its level falls to zero i the log ter. q is the iitial populatio ad q = q() t deotes the populatio at tie t. This odel is a first-order itegro-ordiary t differetial equatio where the ter cq q( ) d, represets the effect of toi accuulatio o the species. We apply scale tie ad populatio by itroducig the odiesioal variables tc qb λ =, u =, b a to obtai the odiesioal proble: du λ u u u u( ) d dλ = u() = u, (1.) Where u( λ ) is the scaled populatio of idetical idividuals at tie λ ad is a prescribed o- diesioal paraeter. The
63 15; (6): 6-67 odel is characterized by the oliear Volterra itegro-differetial equatio du λ u u u u( ) d dλ = u() =.1, (1.3) c ad is a prescribed paraeter. The o diesioal paraeter is =. ba I recet years, uerous wors have bee focusig o the developet of ore advaced ad efficiet ethods for Volterra's populatio odel such as sigular perturbatio ethod [5]. For eaple hootopy perturbatio ethod [6], quasilierizatio approach ethod [4] ad Adoia decopositio ethod ad Sic-Galeri ethod copared for the solutio of soe atheatical populatio growth odels [3]. I [1], the series solutio ethod ad The decopositio ethod for Volterra's populatio odel is cosidered. I 1986, Zhou [] first itroduced the differetial trasfor ethod (DTM) i solvig liear ad oliear iitial value probles i the electrical circuit aalysis. The differetial trasfor ethod obtais a aalytical solutio i the for of a polyoial. It is differet fro the traditioal high order Taylor s series ethod, which requires sybolic copetitio of the ecessary derivatives of the data fuctios. the results of applyig differetial trasforatio ethod to the Volterra s populatio odel will be preseted. Oe-Diesioal Differetial Trasfor Differetial trasfor of fuctio y( ) is defied as follows: 1 d y( ) Y( ) =,! d (.1) I equatio (.1), y( ) is the origial fuctio ad Y( ) is the trasfored fuctio, which is called the T-fuctio. Differetial iverse trasfor of Y( ) is defied as Fro equatio (.1) ad (.), we obtai y( ) = Y( ), (.) = d y( ) y( ) =,! = d (.3) Equatio (.3) iplies that the cocept of differetial trasfor is derived fro Taylor series epasio, but the ethod does ot evaluate the derivatives sybolically. However, relative derivatives are calculated by a iterative way which are described by the trasfored equatios of the origial fuctios. I this study we use the lower case letter to represet the origial fuctio ad upper case letter represet the trasfored fuctio. Fro the defiitios of equatios (1.) ad (.), it is easily prove that the trasfored fuctios coply with the basic atheatics operatios show i Table 1. I actual applicatios, the fuctio y( ) is epressed by a fiite series ad equatio (.) ca be writte as: = + y( ) = Y( ), (.4) = Equatio (.3) iplies that 1 Y( ) is egligibly sall. I fact, is decided by the covergece of atural frequecy i this study.
ISSN: 375-383 64 Table 1. The fudaetal operatios of oe-diesioal differetial trasfor ethod. Origial fuctio Trasfored fuctio y( ) = u( ) ± v( ) Y( ) = U( ) ± V( ) y( ) = cw( ) Y( ) = cw( ) y( ) = dw/ d Y( ) = ( + 1) W( + 1) j j y( ) = d w/ d Y( ) = ( + 1)( + ) ( + j) W( + j) y( ) u( ) v( ) = r= Theore 1. if y ( ) = ep ( ) the W ( ) 1 =.! Y( ) = U( r) V( r) Proof: By usig equatio (.1), we have ( ) W ( e ) 1 1 1 = = =. e!!! t= Theore. if u( ) ( + )! y( ) = the Y( ) = U( + ).! Proof: By usig equatio (.1), we have + ( ) ( ) ( ) 1 u 1 u +! Y ( ) = ( ) = = U ( + ).!!! + t= t= Theore 3. if y( ) = the Y( ) = δ( ) = 1 = Proof: By usig equatio (.1), we have 1 ( ) W( ) = =! t= 1 ( )! = = 1 =!! 1 ( ) =! Theore 4. if w( ) = si ( w + α ) the W ( ) Proof: By usig equatio (.1), we have ( ) W( ) = δ = w π. = si( + α )! 1 =. 1 si( w + α) 1 = 1 W 1 = = wcos( w + ) 1! 1! ( ) α 1 π 1 π = wsi(( + α) + w) = wsi( + α), 1! 1!
65 15; (6): 6-67 1 si( w + α) 1 π = W ( ) = = w cos( + + w)! t! I the geeral for we have α 1 π π 1 π 1 π = w si( + α + + w) = w si( + α + w) = w si( + α),!!! 1 si( w + α) w π = W ( ) = = si( + α + w)!! w π = si( + α).! w π! Theore 5. if w( ) = cos( w + α ) the W ( ) = cos( + α ). Proof: By usig equatio (.1), we have I the geeral for we have 1 cos( w + α) 1 = 1 W 1 = = wsi( w + ) 1! t 1! ( ) α 1 π 1 π = wcos(( + α) + w) = wcos( + α), 1! 1! 1 cos( w + α) 1 π = W ( ) = = w si( + α + w)!! t 1 π π 1 π 1 π = w cos + α + + w == w cos( + α + w) = w cos( + α),!!! 1 si( w + α) w π = W ( ) = = cos( + α + w)!! w π w π = cos( + α) = cos( + α).!! Nuerical Eaple I this sectio, we use oe diesioal differetial trasfor ethod for solvig the populatio growth odel characterized by the oliear Volterra itegro-differetial equatio With the iitial coditio du 1 u() t u () t 1 u() t t u( ) d, dt = (3.1) Whe taig the oe diesioal differetial trasfor of (3.1), we ca obtia: u () =.1, (3.) U 1 1 ( h h h 1) 1 ( ) 1 ( ) ( ) 1 ( ) ( 1), ( h 1) U h U s U h s U h s U s + = + s= s= 1 s (3.3)
ISSN: 375-383 66 Fro the iitial coditio (3.), we ca write U () =.1, (3.4) For each h substitutig ito Eq. (3.3) ad by recursive ethod, the values U( h ) ca be evaluated as follows: U() =.1, U(1) =.9, U() = 3.55, U(3) = 6.316666667, U(4) = 5.5375499998, U(5) = 63.74166667, U(6) = 156.1375,..., (3.5) by usig the iverse trasforatio rule for oe diesioal i Eq. (.1), the followig solutio ca be obtaied: h= h 1 3 4 5 6 u() t = U( h) = U() + U(1) t + U() t + U(3) t + U(4) t + U(5) t + U(6) t +..., 3 4 5 6 7 =.1+.9t + 3.55t + 6.316666667t 5.5375499998 t 63.74166667t 156.1375 t + O( t ). (3.6) I a coplete agreeet with the results previously obtaied i the previous sectios. Coclusio I this study, the differetial trasfor ethod for the solutio of the Volterra s populatio odel is successfully epaded. It is observed that the ethod is robust. The ethod gives rapidly covergig series solutios.the uerical results show that differetial trasfor ethod is a accurate ad reliable uerical techique for the solutio of the Volterra s populatio odel. Khatereh Tabatabaei Kh. Tabatabaei received the PhD degree i Matheatics of Atatur Uiversity. Her research iterests are i the areas of applied atheatics icludig the uerical ethods for special odels of differetial equatios ad differetial-algebraic equatios ad itegral equatios. Eail address: Kh_tabatabaey@yahoo.co Era Güerha Lecturer, Kafas Uiversity Faculty of Egieerig, Departet of Coputer, Kars- Turey His research iterests are i the areas of atheatics Egieerig ad coputer sciece ad software prograig ad web desiger. Eail address: Era.guerha@gail.co Refereces [1] A. M. Wazwaz, Aalytical approiatios ad Padé approiats for Volterra s populatio odel, Applied Matheatics ad Coputatio, 1 (1999) 13-5. [] J. K. Zhou, Huazhog Uiversity Press, Wuha, Chia, 1986. [3] K. Al-Khaled, Nuerical Approiatios for Populatio Growth Models. Appl. Math. Coput, 16 (5) 865-873. [4] K. Parad, M. Ghasei ASEMI, S. Rezazadeh, A. Peiravi, A. Ghorbapour, A. Tavaoli golpaygai, quasilierizatio approach for solvig Volterra s populatio odel, Appl. Coput. Math., (1)95-13. [5] R. D. Sall, Populatio growth i a closed syste ad Matheatical Modelig, i: Classroo Notes i Applied Matheatics, SIAM, Philadelphia, PA, (1989) 317-3.
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