Hdaw Publshg Corporato Mathematcal Problems Egeerg Volume 2013, Artcle ID 391901, 7 pages http://dx.do.org/10.11/2013/391901 Research Artcle O Approxmate Solutos for Fractoal Logstc Dfferetal Equato M. M. Khader 1,2 ad Mohammed M. Babat 1 1 Departmet of Mathematcs ad Statstcs, College of Scece, Al-Imam Mohammed Ib Saud Islamc Uversty (IMSIU), P.O.Box6892,Ryadh1166,SaudAraba 2 Departmet of Mathematcs, Faculty of Scece, Beha Uversty, Beha, Egypt Correspodece should be addressed to M. M. Khader; mohamedmbd@yahoo.com Receved 6 March 2013; Revsed 1 Aprl 2013; Accepted 2 Aprl 2013 Academc Edtor: Guo-Cheg Wu Copyrght 2013 M. M. Khader ad M. M. Babat. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. A ew approxmate formula of the fractoal dervatves s derved. The proposed formula s based o the geeralzed Laguerre polyomals. Global approxmatos to fuctos defed o a sem-fte terval are costructed. The fractoal dervatves are preseted terms of Caputo sese. Specal atteto s gve to study the covergece aalyss ad estmate a error upper boud of the preseted formula. The ew spectral Laguerre collocato method s preseted for solvg fractoal Logstc dfferetal equato (FLDE). The propertes of Laguerre polyomals approxmato are used to reduce FLDE to solve a system of algebrac equatos whch s solved usg a sutable umercal method. Numercal results are provded to cofrm the theoretcal results ad the effcecy of the proposed method. 1. Itroducto Ordary ad partal fractoal dfferetal equatos (FDEs) have bee the focus of may studes due to ther frequet appearace varous applcatos flud mechacs, vscoelastcty, bology, physcs, ad egeerg [1]. Fractoal calculus s a geeralzato of ordary dfferetato ad tegrato to a arbtrary oteger order. May physcal processes appear to exhbt fractoal order behavor that mayvarywthtmeorspace.mostfdesdoothaveexact solutos, so approxmate ad umercal techques [2 8] must be used. Several umercal ad approxmate methods tosolvefdeshavebeegvesuchasvaratoalterato method [9 12], homotopy perturbato method [13], Adoma s decomposto method [14, 1], ad collocato method [16, 17]. The fractoal Logstc model ca be obtaed by applyg the fractoal dervatve operator o the Logstc equato. The model s tally publshed by Perre Verhulst 1838 [18, 19]. The cotuous Logstc model s descrbed by frstorder ordary dfferetal equato. The dscrete Logstc model s smple teratve equato that reveals the chaotc property certa regos [20]. There are may varatos of the populato modelg [19, 21]. The Verhulst model s the classc example to llustrate the perodc doublg ad chaotc behavor dyamcal system [20]. The model whch s descrbed the populato growth may be lmted by certa factors lke populato desty [18, 19, 21]. Applcatos of Logstc Equato. A typcal applcato of the Logstc equato s a commo model of populato growth. Aother applcato of Logstc curve s medce, where the Logstc dfferetal equato s used to model the growth of tumors. Ths applcato ca be cosdered a exteso of the above-metoed use the framework of ecology. Deotg by u(t) the sze of the tumor at tme t. The soluto of Logstc equato s explaed the costat populato growth rate whch ot cludes the lmtato o food supply or spread of dseases [19]. The soluto curve of the model s creasg expoetally from the multplcato factor up to saturato lmt whch s maxmum carryg capacty [19], dn/dt = ρn(1 (N/K)) where N s the populato sze wth respect to tme, ρ s the rate of maxmum populato growth, ad K s the carryg capacty.
2 Mathematcal Problems Egeerg The soluto of cotuous Logstc equato s the form of costat growth rate as formula N(t) = N 0 e ρt where N 0 s the tal populato [22]. I ths paper, we cosder FLDE of the form D ] u (t) =ρu(t)(1 u(t)), t > 0, ρ > 0, (1) here, the parameter ] refers to the fractoal order of tme dervatve wth 0<] 1. We also assume a tal codto u (0) =u 0, u 0 >0. (2) For ] =1,(1) s the stadard Logstc dfferetal equato du (t) dt =ρu(t)(1 u(t)). (3) The exact soluto to ths problem s u(t) = u 0 /((1 u 0 )e ρt + u 0 ). The exstece ad the uqueess of the proposed problem (1) are troduced detals [23, 24]. The ma am of the preseted paper s cocered wth a exteso of the prevous work o FDEs ad derve a approxmate formula of the fractoal dervatve of the Laguerre polyomals ad the we apply ths approach to obta the umercal soluto of FLDE. Also, we preset study of the covergece aalyss ad estmate a error upper boud of the proposed formula. The structure of ths paper s arraged the followg way: Secto 2, we troduce some basc deftos about Caputo fractoal dervatves ad propertes of the Laguerre polyomals. I Secto 3, wegveaapproxmateformula ofthefractoaldervatveoflaguerrepolyomalsad ts covergece aalyss. I Secto4, we mplemet the proposed method for solvg FLDE to show the accuracy of the preseted method. Fally, Secto, thepapereds wth a bref cocluso ad some remarks. 2. Prelmares ad Notatos I ths secto, we preset some ecessary deftos ad mathematcal prelmares of the fractoal calculus theory that wll be requred the preset paper. 2.1. The Caputo Fractoal Dervatve Defto 1. The Caputo fractoal dervatve operator D ] of order ] sdefedthefollowgform: D ] x 1 f (x) = Γ (m ]) f (m) (t) 0 (x t) ] m+1 dt, ] >0, x>0, where m 1<] m, m N. Smlar to teger-order dfferetato, Caputo fractoal dervatve operator s lear (4) D ] (λf (x) +μg(x)) =λd ] f (x) +μd ] g (x), () where λ ad μ are costats. For the Caputo s dervatve we have D ] C=0, Cs a costat, (6) D ] x = { 0, for N 0,< ] ; { Γ (+1) { Γ (+1 ]) x ], for N 0, ]. We use the celg fucto ] to deote the smallest teger greaterthaorequalto], adn 0 = {0, 1, 2,...}. Recallthat for ] N, the Caputo dfferetal operator cocdes wth the usual dfferetal operator of teger order. For more detals o fractoal dervatves deftos ad ther propertes see [1, 2 28]. 2.2. The Defto ad Propertes of the Geeralzed Laguerre Polyomals. Spectral collocato methods are effcet ad hghly accurate techques for umercal soluto of olear dfferetal equatos. The basc dea of the spectral collocato method s to assume that the ukow soluto u(t) ca be approxmated by a lear combato of some bass fuctos, called the tral fuctos, such as orthogoal polyomals. The orthogoal polyomals ca be chose accordg to ther specal propertes, whch make them partcularly sutable for a problem uder cosderato. I [16], Khader troduced a effcet umercal method for solvg the fractoal dffuso equato usg the shfted Chebyshev polyomals. I [29] thegeeralzedlaguerrepolyomals were used to compute a spectral soluto of a olear boudary value problems. The geeralzed Laguerre polyomals costtute a complete orthogoal sets of fuctos o the sem-fte terval [0, ). Covoluto structures of Laguerre polyomals were preseted [30]. Also, other spectral methods based o other orthogoal polyomals are used to obta spectral solutos o ubouded tervals [31]. The geeralzed Laguerre polyomals [L (α) (x)] =0, α> 1 are defed o the ubouded terval (0, ) ad ca be determed wth the ad of the followg recurrece formula: (+1) L (α) +1 (x) + (x 2 α 1) L(α) (x) (7) + (+α) L (α) 1 (x) =0, =1,2,..., (8) where L (α) 0 (x) = 1 ad L(α) 1 (x)=α+1 x. The aalytc form of these polyomals of degree s gve by L (α) (x) = ( 1) k ( +α k! k )xk k=0 = ( +α ( ) k x k (α+1) k k!, ) k=0 (9)
Mathematcal Problems Egeerg 3 L (α) +α (0) = ( ). Thesepolyomalsareorthogoalothe terval [0, ) wth respect to the weght fucto w(x) = (1/Γ(1 + α))x α e x.theorthogoaltyrelatos 1 Γ (1+α) 0 x α e x L (α) m (x) L(α) (x) dx=(+α Also, they satsfy the dfferetato formula D k L (α) )δ m. (10) (x) = ( 1)k L (α+k) k (x), k=0,1,...,. (11) Ay fucto u(x) belogs to the space L 2 w [0, ) of all square tegrable fuctos o [0, ) wth weght fuctow(x) ca be expaded the followg Laguerre seres: u (x) = =0 c L (α) (x), (12) where the coeffcets c are gve by Γ (+1) c = Γ (+α+1) x α e x L (α) (x) u (x) dx, =0,1,... 0 (13) Cosder oly the frst (m + 1) terms of geeralzed Laguerre polyomals, so we ca wrte u m (x) = m =0 c L (α) (x). (14) For more detals o Laguerre polyomals, ts deftos, ad propertes, see [29, 31, 32]. 3. A Approxmate Fractoal Dervatve of (x) ad Its Covergece Aalyss L (α) The ma goal of ths secto s to troduce the followg theorems to derve a approxmate formula of the fractoal dervatves of the geeralzed Laguerre polyomals ad study the trucatg error ad ts covergece aalyss. Lemma 2. Let L (α) (x) be a geeralzed Laguerre polyomal the (x) =0, =0,1,..., ] 1, ] >0. (1) Proof. Ths lemma ca be proved drectly by applyg (6)-(7) o (9). The ma approxmate formula of the fractoal dervatve of u(x) s gve the followg theorem. Theorem 3. Let u(x) be approxmated by the geeralzed Laguerre polyomals as (14) ad also suppose ] > 0;the ts approxmated fractoal dervatve ca be wrtte the followg form: D ] (u m (x)) m = ] k= ] c w (]), k xk ], (16) where w (]), k s gve by w (]), k = ( 1) k Γ (k+1 ]) (+α ). (17) k Proof. Sce the Caputo s fractoal dfferetato s a lear operato, we obta D ] (u m (x)) = m =0 c D ] (L (α) (x)). (18) Also, from (9)wecaget (x) =0, =0,1,..., ] 1, ] >0. (19) Therefore, for = ], ] +1,...,m,adbyusg(6)-(7) (9), we get ( 1) k (x) = ( +α k! k )D] x k k=0 = k= ] ( 1) k Γ (k+1 ]) (+α k )xk ]. (20) A combato of (18) (20) leads to the desred result (16) ad eds the proof of the theorem. Test Example.Cosderthefuctou(x) = x 3 wth m=3, ] = 1.,adα= 0., the geeralzed Laguerre seres of x 3 s x 3 = 1.87L (α) 0 (x) 11.2L (α) 1 (x) + 1L (α) 2 (x) 6L (α) 3 (x). (21) Now, by usg formula (16), we obta D 1. x 3 = 3 =2 c w (1.),k xk 1., (22) k=2 where w (1.) 2,2 = 1.12838, w (1.) 3,2 = 2.8209, w (1.) 3,3 = 0.7223, therefore, D 1. x 3 =c 2 w (1.) 2,2 x0. +c 3 w (1.) 3,2 x0. +c 3 w (1.) 3,3 x1. = Γ (4) Γ (2.) x1., whch agrees wth the exact dervatve (7). (23) Theorem 4. The Caputo fractoal dervatve of order ] for the geeralzed Laguerre polyomals ca be expressed terms of the geeralzed Laguerre polyomals themselves the followg form: where Ω k = r=0 (x) = k ] k= ] =0 Ω k L (α) (x), = ], ] +1,...,m, ( 1) r+k (α+)! ()! (k+α ] +r)! (24) (k ])! ( k)! (α+k)!r! ( r)! (α+r)!. (2)
4 Mathematcal Problems Egeerg Proof. From the propertes of the geeralzed Laguerre polyomals [33] ad expadg x k ] (20) thefollowg form: x k ] = k ] =0 c k L (α) (x), (26) where c k ca be obtaed usg (13), where u(x) = x k ],the Γ(+1) c k = Γ(+1+α) = r=0 0 x k+α ] e x L (α) (x) dx ( 1) r ()! (k ] +α+r)!, r! ( r)! (α+r)! =0,1,..., (27) ths by substtutg from (9) adusgthedeftoof Gamma fucto. Now, we ca wrte (26) thefollowg form: x k ] = k ] =0 r=0 ( 1) r ()! (k ] +α+r)! r! ( r)! (α+r)! L (α) (x). (28) Therefore, the Caputo fractoal dervatve (x) (20) caberewrttethefollowgform: (x) = k= ] k ] =0 r=0 (( 1) r+k (α+)! ()! (k ] +α+r)! ((k ])! ( k)! (α+k)!r! ( r)! (α+r)!) 1 )L (α) (x), (29) for = ], ] +1,...,m.Equato(29) leads to the desred result (24) ad ths completes the proof of the theorem. Theorem. For the Laguerre polyomals L (α) (x), oe has the followg global uform bouds estmates: (α+1)! L(α) (x) { { { e x/2, (2 (α+1) )e x/2,! for α 0,x 0, = 0, 1,... ; for 1<α 0, x 0,=0,1,... (30) Proof. These estmates were preseted [33 3]. Theorem 6. The error approxmatg D ] u(x) by D ] u m (x) s bouded by E T (m) c Π ] (, ) (α+1) e x/2, =m+1! α 0, x 0, =0,1,..., E T (m) =m+1 c Π ] (, ) (2 (α+1) )e x/2,! 1<α 0, x 0, =0,1,..., (31) where E T (m) = D ] u(x) D ] u m (x) ad Π ] (, ) = k= ] k ] =0 Ω k. Proof. A combato of (12), (14), ad (24)leadsto E T (m) = D] u (x) D ] u m (x) (32) c Π ] (, ) L(α) (x), =m+1 usg (30) ad subtractg the trucated seres from the fte seres, boudg each term the dfferece, ad summg the bouds completes the proof of the theorem. 4. Implemetato of Laguerre Spectral Method for Solvg FLDE I ths secto, we troduce a umercal algorthm usg Laguerre spectral method for solvg the fractoal Logstc dfferetal equato of the form (1). The procedure of the mplemetato s gve by the followg steps. (1) Approxmate the fucto u(t) usg the formula (14) ad ts Caputo fractoal dervatve D ] u(t) usg the preseted formula (16) wthm=, the FLDE (1) s trasformed to the followg approxmated form: =1 c w (]) k=1,k tk ] ρ( =0 (1 =0 c L (α) (t)) c L (α) (t)) =0, (33) where w (]), k s defed (17). We ow collocate (33)at(m+1 ] ) pots t p, p= 0,1,...,m ] as =1 k=1 c w (]),k tk ] p (1 ρ( =0 =0 c L (α) (t p )) c L (α) (t p )) = 0. (34) (2) From the tal codto (2) we obta the followg equato: =0 c L (α) (0) =u 0. (3)
Mathematcal Problems Egeerg u(t) 0.36 0.34 0.32 0.3 0.28 0.26 u(t) 0.38 0.36 0.34 0.32 0.3 0.28 0.26 0.24 0 0.1 0.2 0.3 0.4 0. 0.6 0.7 0.8 0.9 1 t Exact soluto Approxmate soluto (a) 0.24 0 0.1 0.2 0.3 0.4 0. 0.6 0.7 0.8 0.9 1 t (b) Fgure 1: A comparso betwee the approxmate soluto ad the exact soluto at ] =1(a). The behavor of the approxmate soluto usg the proposed method at ] = 0.8 (b). u(t) 0.38 0.36 0.34 0.32 0.3 0.28 0.26 u(t) 0.38 0.36 0.34 0.32 0.3 0.28 0.26 0.24 0 0.1 0.2 0.3 0.4 0. 0.6 0.7 0.8 0.9 1 0.24 0 0.1 0.2 0.3 0.4 0. 0.6 0.7 0.8 0.9 1 (a) t (b) t Fgure 2: The behavor of the approxmate soluto usg the proposed method at ] = 0.6 (a) ad at ] = 0.4 (b). Equatos (34)-(3) represetasystemofolear algebrac equatos whch cotas sx equatos for the ukows c, =0,1,...,. (3) Solve the resultg system usg the Newto terato method to obta the ukows c, = 0,1,...,. Therefore, the approxmate soluto wll take the form u (t) =c 0 L (α) 0 (t) +c 1 L (α) 1 (t) +c 2 L (α) 2 (t) +c 3 L (α) 3 (t) +c 4 L (α) 4 (t) +c L (α) (t). (36) The umercal results of the proposed problem (1) aregve Fgures 1 ad 2 wth dfferet values of ] the terval [0, 1] wth ρ = 0. ad u 0 = 0.2. WhereFgure 1, we preseted a comparso betwee the behavor of the exact soluto ad the approxmate soluto usg the troduced techque at ] = 1 (Fgure 1(a)), ad the behavor of the approxmate soluto usg the proposed method at ] =0.8 (Fgure 1(b)). But, Fgure 2, we preseted the behavor of the approxmate soluto wth dfferet values of ] (] =0.6 (Fgure 2(a))ad] = 0.4 (Fgure 2(b))).. Cocluso ad Remarks I ths paper, we troduced a ew spectral collocato methodbasedolaguerrepolyomalsforsolvgflde.
6 Mathematcal Problems Egeerg We have troduced a approxmate formula for the Caputo fractoal dervatve of the geeralzed Laguerre polyomals terms of geeralzed Laguerre polyomals themselves. I the proposed method we used the propertes of the Laguerre polyomals to reduce FLDE to solve a system of algebrac equatos. The error upper boud of the proposed approxmate formula s stated ad proved. The obtaed umercal resultsshowthattheproposedalgorthmcovergesasthe umber of m termsscreased.thesolutosexpressedas a trucated Laguerre seres ad so t ca be easly evaluated for arbtrary values of tme usg ay computer program wthout ay computatoal effort. From llustratve examples, we ca coclude that ths approach ca obta very accurate adsatsfactoryresults.comparsosaremadebetweethe approxmate soluto ad the exact soluto to llustrate the valdty ad the great potetal of the techque. All computatosaredoeusgmatlab. Refereces [1] I. Podluby, Fractoal Dfferetal Equatos, vol. 198, Academc Press, New York, NY, USA, 1999. 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