Reliable analysis for delay differential equations arising in mathematical biology

Transcription

1 Joural of Kig Saud Uiverity Sciece (1) 4, 9 6 Kig Saud Uiverity Joural of Kig Saud Uiverity Sciece ORIGINAL ARTICLE Reliable aalyi for delay differetial equatio ariig i mathematical biology Ahmet Yıldırım a,hu eyi Koc ak b, *, Serap Tutku a a Ege Uiverity, Departmet of Mathematic, 1 Borova, _Izmir, Turkey b Departmet of Mathematical Sciece, Uiverity of Bath, Bath BA 7AY, UK Received 7 July 11; accepted Augut 11 Available olie 7 Augut 11 KEYWORDS Variatioal iteratio method; Delay differetial equatio; Applicatio i mathematical biology ad egieerig Abtract I thi tudy, delay differetial equatio are ivetigated uig the variatioal iteratio method. Delay differetial equatio (DDE) have a wide rage of applicatio i ciece ad egieerig. They arie whe the rate of chage of a time-depedet proce i it mathematical modelig i ot oly determied by it preet tate but alo by a certai pat tate. Recet tudie i uch divere field a biology, ecoomy, cotrol ad electrodyamic have how that DDE play a importat role i explaiig may differet pheomea. The procedure of preet method i baed o the earch for a olutio i the form of a erie with eaily computed compoet. Some umerical illutratio are give. Thee reult reveal that the propoed method i very effective ad imple to perform. ª 11 Kig Saud Uiverity. Productio ad hotig by Elevier B.V. All right reerved. 1. Itroductio Aalytical method commoly ued to olve oliear equatio are very retricted ad umerical techique ivolvig dicretizatio of the variable o the other had give rie to roudig off error. Recetly itroduced variatioal iteratio method by He (7, 1997a,b, 1998a,b, 1999, ), which * Correpodig author. Tel.: addre: hkocak.ege@gmail.com (H. Koçak) ª 11 Kig Saud Uiverity. Productio ad hotig by Elevier B.V. All right reerved. Peer review uder repoibility of Kig Saud Uiverity. doi:1.116/j.jku Productio ad hotig by Elevier give rapidly coverget ucceive approximatio of the exact olutio if uch a olutio exit, ha prove ucceful i derivig aalytical olutio of liear ad oliear differetial equatio. Thi method i preferable over umerical method a it i free from roudig off error ad either require large computer power/memory. He (7, 1997a,b, 1998a,b, 1999, ) ha applied thi method for obtaiig aalytical olutio of autoomou ordiary differetial equatio, oliear partial differetial equatio with variable coefficiet ad itegro-differetial equatio. The variatioal iteratio method wa uccefully applied to liear ad oliear problem by reearcher (O ziß ad Yıldırım, 7; Yıldırım ad O ziß, 8; Yıldırım, 8a,b; Momai et al., 6; Momai ad Abuaad, 6; Tatari ad Dehgha, 7a,b; Wag ad He, 7; Wazwaz, 7a,b; Xu et al., 7; Abdou ad Solima, a,b; Abdou, 7). I the mathematical decriptio of a phyical proce, oe geerally aume that the behavior of the proce coidered deped oly o the preet tate, a aumptio which i

2 6 A. Yıldırım et al. verified for a large cla of dyamical ytem. However, there exit ituatio where thi aumptio i ot atified ad the ue of a claical model i ytem aalyi ad their deig may lead to poor performace. I uch cae, it i better to coider that the ytem behavior iclude alo iformatio o the former tate. Thee ytem are called time-delay ytem. May of the procee, both atural ad artificial, i biology, medicie, chemitry, phyic, egieerig, ecoomic etc ivolve time delay. I fact, time delay occur o ofte, i almot every ituatio, that to igore them i to igore reality. A imple example i ature i that of reforetatio (Kuag, 199). A cut foret, after replatig, will take at leat year before reachig ay kid of maturity. For certai pecie of tree (redwood for example) it would be much loger. Hece, ay mathematical model of foret harvetig ad regeeratio clearly mut have time delay built ito it. So, while i may applicatio it i aumed that the ytem uder coideratio i govered by a priciple of cauality, that i, the future tate of the ytem i idepedet of the pat ad i determied olely by the preet, oe hould keep i mid that thi i oly a firt approximatio to the true ituatio. A more realitic model mut iclude ome of the pat hitory of the ytem Applicatio of delay differetial equatio I thi ectio, i order to how the great importace of DDE, we preet ome applicatio of them A delayed epidemic model Mathematical biologit A.J. Lotka ivetigated, i a erie of paper from 191 o, a differetial equatio model of malaria epidemic due to Ro (1911). I particular, he examied the effect of icubatio delay. The equatio are for the huma populatio _h t¼ bgm t u½p ht uš=p M þ rh t; ad for the moquito populatio _m t¼ bfh t v½q mt uš=p N þ m t: Here, u =. moth i huma ad v =.6 moth i moquito, p ad q are the total huma ad moquito populatio, treated a cotat quatitie, which i a tadard practice i imple epidemiological model. The fuctio h(t) ad m(t) tad for huma ad moquito populatio carryig the malaria orgaim (the ifected or dieaed populatio), repectively. A fixed proportio of each of thee populatio i aumed to be ifective, with the ifective populatio beig fh ad gm, repectively. The quatitie M ad N are death rate, while r ad are recovery rate. It i aumed that each moquito bite b people i uit time, ad that each pero receive a bite i uit time (Kuag, 199). The delay i from the time of a bite to the time at which the huma or moquito i ifective Delay model i phyiology: Dyamic dieae There are may acute phyiological dieae where the iitial ymptom are maifeted by a alteratio or irregularity i a cotrol ytem that i ormally periodic, or by the oet of a ocillatio i a oocillatory proce. Such phyiological dieae have bee termed dyamical dieae by Gla ad Mackey (1979), who have made a ytematic tudy of everal importat ad iteretig phyiological model with time delay. The followig are three example of thee model: _x t¼ k av mxtx N t h þ x t ; b _p t¼ h h cp t; þ p t _p t¼ b h pt h cp t: þ p t Here, k, a, V m,,, h, b ad c are poitive cotat. The firt equatio i ued to tudy a dyamic dieae ivolvig repiratory diorder, where x(t) deote the arterial CO cocetratio of a mammal, k i the CO productio rate, V m deote the maximum vetilatio rate of CO, ad i the time betwee oxygeatio of blood i the lug ad imulatio of chemoreceptor i the braitem. The ecod ad third equatio are propoed a model of hematopoiei (blood cell productio). I thee two equatio, p(t) deote the deity of mature cell i blood circulatio, ad i the time delay betwee the productio of immature cell i the boe marrow ad their maturatio for releae i the circulatig bloodtream (Kuag, 199) Chemical idutry Coider a firt order, exothermic, irreverible reactio: A B. Sice, i practice, the coverio from A to B i ot complete, oe claical techique ue a recycle tream (which icreae overall coverio, reduce cot of the reactio, etc.). I order to recycle, the output mut be eparated from the iput ad mut flow through ome legth of pipe. Thi proce doe ot take place itataeouly ; it require ome traport time from the output to the iput, ad thu oe may coider a ytem model ivolvig a traport delay. Suppoe ow that the ureacted A ha a recycle flow rate (1 k)q ad i the traport delay. The the material ad eergy balace are decribed by a dyamical ytem icludig delayed tate of the form da t ¼ q V ½kA þ 1 k At AtŠ K e Q T At; dt dt t dt ¼ 1 V ½kT þ 1 kt t TtŠ DH 1 VC q UTt T w ; K e Q T At C q where A(t) i the cocetratio of the compoet A, T(t) i the temperature ad k i the recycle coefficiet, which atifie the coditio k [,1]. The limit ad 1 correpod to o recycle tream ad to a complete recycle, repectively (Dugard ad Verriet, 1997). I the lat year, much reearche have bee focued o the aalyi ad umerical olutio of delay differetial equatio. Recetly Shakeri ad Dehgha (8) ued homotopy perturbatio method uccefully for olvig delay differetial equatio. I thi work we focu o the delay differetial equatio of the followig form which cotai a large cla of the DDE: u ðmþ ðtþ ¼ XJ j¼ k¼ with the iitial coditio l jk ðtþu ðkþ ða jk t þ b jk ÞþgðuÞþfðtÞ; ð1:1þ

3 Reliable aalyi for delay differetial equatio ariig i mathematical biology 61 k¼ c ik u k ðþ ¼k i ; i ¼ ; 1;...; m 1; where gðuþ ¼bu þ P l i¼1 d iu c i, di R, c i N, c i > 1, l N, 6 a jk 6 1, b jk R ad. (m) i u (m) i coidered a the mth derivative of the fuctio u. I order to olve thi equatio, we ue the variatioal iteratio method. Thi paper i orgaized a follow: I Sectio, we decribe the variatioal iteratio method briefly ad apply thi techique to Eq. (1.1). To how the efficiecy of thi method, we give ome example ad umerical reult i Sectio. A cocluio i draw i Sectio 4.. He variatioal iteratio method To clarify the baic idea of He variatioal iteratio method, we coider the followig oliear differetial equatio: LðuÞþNðuÞ fðtþ ¼; t X ð:1þ where L i liear operator, N i a oliear operator ad f(t) i a kow aalytical fuctio. Accordig to the variatioal iteratio method, we ca cotruct a correctio fuctioal a follow: u þ1 ðtþ ¼u ðtþþ kðþðlðu ðþþ þ Nð~u ðþþ fðþþd; ð:þ where k i a geeral Lagrage multiplier, which ca be idetified optimally via the variatioal theory (He, 7), the ubcript deote the th approximatio, ad ~u i coidered a a retricted variatio, i.e., d~u ¼. It i obviou ow that the mai tep of the variatioal iteratio method require firt the determiatio of the Lagragia multiplier k that will be idetified optimally. Havig determied the Lagragia multiplier, the ucceive approximatio u +1, P, of the olutio u will be readily obtaied upo uig ay elective fuctio u. Coequetly, the olutio u ¼ lim u : ð:þ!1 I order to olve Eq. (1.1) by mea of variatioal iteratio method, accordig to (.), we ca cotruct a correctio fuctioal uch that u þ1 ðtþ ¼u ðtþþ kðþ u ðmþ ðþ XJ j¼! gð~u Þ fðþ d; k¼ l jk ðþ~u ðkþ ða jk þ b jk Þ ð:4þ where d~u i coidered a retricted variatio. Makig the correctio fuctioal, Eq. (.4), tatioary, oticig that d~u ¼, du þ1 ðtþ ¼du ðtþþd du þ1 ðtþ ¼du ðtþþd kðþ u ðmþ ðþ XJ j¼ k¼! ~u ðkþ ða jk þ b jk Þ gð~u Þ fðþ d; kðþu ðmþ ðþ d; l jk ðþ Lagrage multiplier ca be eaily idetified a (He, 7): kðþ ¼ð 1Þ m 1 ðm 1Þ! ð tþðm 1Þ : ð:þ A a reult, we obtai the followig iteratio formula of Eq. (1.1): u þ1 ðtþ ¼u ðtþþð 1Þ m u ðmþ ðþ XJ j¼ 1 ð tþðm 1Þ ðm 1Þ! k¼ l jk ðþu ðkþ ða jk þ b jk Þ gðu Þ fðþ! d; ð:6þ The tartig with a iitial approximatio u ad uig above iteratio formula, we ca obtai u for =1,,... which i the th approximatio of the exact olutio. I the ext ectio, we will implemet the above method to obtai umerical reult of give example.. Tet problem I thi ectio, we preet ome example with kow aalytical olutio i order to how the efficiecy ad high accuracy of the method decribed for olvig Eq. (1.1). Example.1. We firt coider Eq. (1.1) with m = 1, J = ad the followig ozero coefficiet l jk, a jk ad b jk (Shakeri ad Dehgha, 8): l ðtþ ¼ t; l 1 ðtþ ¼ t ; l ðtþ ¼ iðtþ; a ¼ 1; a 1 ¼ 1; a ¼ 1 ; b ¼ 1 ; gðuþ ¼u ad fðtþ ¼e t t 4 t þ t t þ te t 7 4 t þ 4 e t t 1 4 t þ 1 iðtþe t t 1 t þ 1 ; with the iitial coditio u() = e. The imple form of the equatio which reult from the above coefficiet i a follow: u ð1þ ðtþ ¼ tu t 1 t uðtþþiðtþu t þ u t þ fðtþ: 4 ð:1þ The exact olutio of thi problem i uðtþ ¼1 t t e t : ð:þ I order to olve thi equatio by mea of the variatioal iteratio method, accordig to (.6), we have u þ1 ðtþ ¼u ðtþ u ð1þ ðþþu 1 þ u ðþ iðþu u fðþ d: ð:þ 4 Begiig with iitial approximatio u (t) =u() = e,by the iteratio formula (.), we obtai the followig ucceive approximatio u ðtþ ¼e ;

4 6 A. Yıldırım et al. Figure 1 (a) Lie: exact olutio (.) of Eq. (.1), poit: ecod-order approximatio olutio u (t), (b) error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t). u 1 ðtþ ¼u ðtþ u ð1þ ðþþu 1 þ ad the ozero coefficiet c ik ad k i i the iitial coditio u ðþ are give a iðþu u fðþ d; c ¼ ; c 1 ¼ 6; c 1 ¼ ; c 11 ¼ 1; 4 4 k ¼ ; k 1 ¼ 1: ¼ 18e 7 4 þ t þ e 6 7 The above coefficiet reult i the followig problem: þ t t co t e 491 u ðþ ðtþ ¼e t u ð1þ t 1 þðt 1ÞuðtÞ u t þ u ðtþþfðtþ; þ t 14 4 t e 8 t 6 ð:4þ with iitial coditio þð18 þ 91t þ 4t þ 4t Þe t 7 4 þ ( ad o o, th approximatio ca be calculated uig (.). Gettig =, we obtai the ecod-order approximatio, uðþþ6u ð1þ ðþ ¼; uðþþu ð1þ ðþ ¼ 1 : ð:þ u (t). The ecod-order approximatio olutio, exact olutio The exact olutio of thi problem i (.) ad error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t) are plotted i Fig. 1. I the ame maer, we obtaied the higher-order approximatio olutio of Eq. (.1) with high accuracy by uig the iteratio formula (.) ad Maple. Example.. We ext coider Eq. (1.1) with m =, J = 1 ad the followig ozero coefficiet l jk, a jk ad b jk (Shakeri ad Dehgha, 8): l ðtþ ¼t 1; l 1 ðtþ ¼e t ; l 1 ðtþ ¼ ; a ¼ 1; a 1 ¼ 1; a 1 ¼ 1 ; b 1 ¼ 1 ; gðuþ ¼u ad fðtþ ¼ 1 t 4 i 1 i t t co 1 t 9 co þ 1 i t 1 co t t þ 1 4 co t e t 1 6 co t co t 1 þ i t 1 9 þ co t þ i t uðtþ ¼ 1 i t þ 1 co t : ð:6þ To olve thi equatio uig the variatioal iteratio method, accordig to (.6), we have u þ1 ðtþ ¼u ðtþþ ð tþ u ðþ ðþ e u ð1þ 1 1Þu ðþþu u ðþ fðþ d: ð:7þ We tart with iitial approximatio u (t)=a + bt. Now we fid the coefficiet a ad b uch that u (t) atifie the iitial coditio (.). The we have the followig ytem ( a þ 6b ¼ ; a þ b ¼ 1 ; which yield a ¼ 1 ; b ¼ 1 6 ; ad therefore u ðtþ ¼ 1 þ 1 6 t. Begiig with above iitial approximatio, by the iteratio formula (.7), we obtai the followig ucceive approximatio

5 Reliable aalyi for delay differetial equatio ariig i mathematical biology 6 Figure (a) Lie: exact olutio (.6) of Eq. (.4), poit: ecod-order approximatio olutio u (t), (b) error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t). u ðtþ ¼ 1 þ 1 6 t; u 1 ðtþ ¼u ðtþþ ð tþ u ðþ ðþ e u ð1þ 1 1Þu ðþþu 88 þ t 144 t þ 1 ¼ 46 þ e t 1 6 þ 8 þ 9 1 i t 1 1 u ðþ fðþ d 6 t þ 7 4 t4 7 co t 1 þ 1 i t 1 1 co t 1 þ 1 ad o o, th approximatio ca be calculated uig (.7). Gettig =, we obtai the ecod-order approximatio, u (t). The ecod-order approximatio olutio, exact olutio (.6) ad error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t) are plotted i Fig.. I the ame maer, we obtaied the higher-order approximatio olutio of Eq. (.4) with high accuracy by uig the iteratio formula (.7) ad Maple. Example.. We ow coider Eq. (1.1) with m =, J = 1 ad the followig ozero coefficiet l jk, a jk ad b jk (Shakeri ad Dehgha, 8): l 1 ðtþ ¼e t 1 ; l 11 ðtþ ¼ ; l ðtþ ¼ t ; a 1 ¼ 1 ; a 11 ¼ 1 ; a ¼ 1; ad fðtþ ¼ e t t7 þ 7 t þ 81 t t þ 1 t 4 t þ þ t t 48 t4 þ 74t þ 4 t 9 t þ 9; ad the ozero coefficiet c ik ad k i i the iitial coditio are give a c ¼ 1; c 1 ¼ 1; c ¼ ; c 11 ¼ 1; c 1 ¼ 1; c 1 ¼ ; c ¼ ; k ¼ ; k 1 ¼ 7; k ¼ 6: Thee coefficiet reult i the followig problem: u ðþ ðtþ ¼ t uðþ ðtþþe t 1 u ð1þ t u ð1þ t þ fðtþ; with iitial coditio 8 >< uðþ u ð1þ ðþ u ðþ ðþ ¼; u ð1þ ðþ u ðþ ðþ ¼7; >: u ð1þ ðþþu ðþ ðþ ¼ 6: ð:8þ ð:9þ The exact olutio of thi problem i uðtþ ¼t t þ t 4t 4 þ t þ t 6 t 8 : ð:1þ To olve thi equatio uig the variatioal iteratio method, accordig to (.6), we have 1 u þ1 ðtþ ¼u ðtþ ð tþ u ðþ ðþ uðþ ðþ e 1 u ð1þ þ u ð1þ fðþ d: ð:11þ We tart with iitial approximatio u (t)=a + bt + ct.by the ame maipulatio a i the previou example, we fid the coefficiet a, b ad c uch that u (t) atifie the iitial coditio (.9). I thi maer, we will get a ¼ ; b ¼ ; c ¼ ; ad therefore u (t) =t t. Begiig with above iitial approximatio, by the iteratio formula (.11), we obtai the followig ucceive approximatio

6 64 A. Yıldırım et al. Figure (a) Lie: exact olutio (.1) of Eq.(.8), poit: ecod-order approximatio olutio u (t), (b) error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t). u ðtþ ¼t t ; 1 u 1 ðtþ ¼u ðtþ ð tþ u ðþ ðþ uðþ ðþ e 1 u ð1þ þ u ð1þ fðþ d: ¼ t t þ t 4t 4 þ t þ t t t8 þ t1 þ e et 1 þ ad o o, th approximatio ca be calculated uig (.11). Gettig =, we obtai the ecod-order approximatio, u (t). The ecod-order approximatio olutio, exact olutio (.1) ad error betwee the ecod-order approximatio olutio ad exact olutio, u exact (t) u (t) are plotted i Fig.. I the ame maer, we obtaied the higher-order approximatio olutio of Eq. (.8) with high accuracy by uig the iteratio formula (.11) ad Maple. The reult how that the variatioal iteratio method produce highly accurate approximatio with oly a few iteratio. 4. Cocluio Our tudy deal with the umerical olutio of delay differetial equatio uig variatioal iteratio method. Thi techique wa teted o ome example ad wa ee to produce atifactory reult. The reliability of the method ad the reductio i the ize of computatioal domai give thi method a wider applicability. Furthermore thi techique, i cotrat to the traditioal perturbatio method, doe ot require a mall parameter i the ytem ad the approximatio obtaied by the propoed method are uiformly valid ot oly for mall parameter, but alo for very large parameter. Referece Abdou, M.A., 7. O the variatioal iteratio method. Phy. Lett. A 66, Abdou, M.A., Solima, A.A., a. Variatioal iteratio method for olvig Burger ad coupled Burger equatio. J. Comput. Appl. Math. 181 (), 4 1. Abdou, M.A., Solima, A.A., b. New applicatio of variatioal iteratio method. Phyica D 11, 1 8. Dugard, L., Verriet, E.I., Stability ad cotrol of time-delay ytem. I: Lecture Note i Cotrol ad Iformatio Sciece, vol. 8. Spriger. Gla, L., Mackey, C., A imple model for phae lockig of biological ocillator. J. Math. Bio. 7, 9. He, J.H., 1997a. A ew approach to oliear partial differetial equatio. Commu. Noliear Sci. Numer. Simulat. (4),. He, J.H., 1997b. Variatioal iteratio method for delay differetial equatio. Commu. Noliear Sci. Numer. Simulat. (4), 6. He, J.H., 1998a. Approximate olutio of oliear differetial equatio with covolutio product o-liearitie. Comput. Method Appl. Mech. Eg. 167, He, J.H., 1998b. Approximate aalytical olutio for eepage flow with fractioal derivative i porou media. Comput. Method Appl. Mech. Eg. 167, He, J.H., Variatioal iteratio method a kid of o-liear aalytical techique: ome example. It. J. Noliear Mech. 4, He, J.H.,. Variatioal iteratio method for autoomou ordiary differetial ytem. Appl. Math. Comput. 114, He, J.H., 7. Variatioal iteratio method ome recet reult ad ew iterpretatio. J. Comput. Appl. Math. 7, 17. Kuag, Y., 199. Delay Differetial Equatio with Applicatio i Populatio Dyamic. Academic pre, Ic, New York. Momai, S., Abuaad, S., 6. Applicatio of He variatioal iteratio method to Helmholtz equatio. Chao, Solito & Fractal 7 (), Momai, S., Abuaad, S., Odibat, Z., 6. Variatioal iteratio method for olvig oliear boudary value problem. Appl. Math. Comput. 18,

7 Reliable aalyi for delay differetial equatio ariig i mathematical biology 6 Öziß, T., Yıldırım, A., 7. A tudy of oliear ocillator with u 1/ force by He variatioal iteratio method. J. Soud Vib. 6, Ro, R., The Prevetio of Malaria. Murray, Lodo. Shakeri, F., Dehgha, M., 8. Solutio of delay differetial equatio via a homotopy perturbatio method. Math. Comput. Modell. 48, Tatari, M., Dehgha, M., 7a. O the covergece of He variatioal iteratio method. J. Comput. Appl. Math. 7, Tatari, M., Dehgha, M., 7b. Solutio of problem i calculu of variatio via He variatioal iteratio method. Phy. Lett. A 6, Wag, S.Q., He, J.H., 7. Variatioal iteratio method for olvig itegro-differetial equatio. Phy. Lett. A 67, Wazwaz, A.M., 7a. A compario betwee the variatioal iteratio method ad Adomia decompoitio method. J. Comput. Appl. Math. 7, Wazwaz, A.M., 7b. The variatioal iteratio method for exact olutio of Laplace equatio. Phy. Lett. A 6, 6 6. Xu, L., He, Ji-Hua, Wazwaz, Abdul-Majid, 7. Variatioal iteratio method reality, potetial, ad challege. J. Comput. Appl. Math. 7, 1. Yıldırım, A., 8a. Applyig He variatioal iteratio method for olvig differetial-differece equatio. Math. Prob. Eg., 8:Article ID Yıldırım, A., 8b. Variatioal iteratio method for modified Camaa Holm ad Degaperi Procei equatio. Commu. Numer. Meth. Eg.. doi:1.1/cm.114. Yıldırım, A., Öziß, T., 8. Solutio of igular IVP of Lae Emde type by the variatioal iteratio method. Noliear Aal. Ser. A: Theory Method. Appl.. doi:1.116/j.a.8..1.