FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR

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1 Romnin Repors in Physics, Vol. 64, Supplemen, P. 7 77, Dediced o Professor Ion-Ioviz Popescu s 8 h Anniversry FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR D. BALEANU,,3, J. H. ASAD 4, I. PETRAS 5, S. ELAGAN 6, A. BILGEN King Abdulziz Universiy, Fculy of Engineering, Deprmen of Chemicl nd Merils Engineering, P.O. Box: 84, Jeddh, 589, Sudi Arbi Cnky Universiy, Deprmen of Mhemics nd Compuer Science, 653, Ankr, Turkey dumiru@cnky.edu.r 3 Insiue of Spce Sciences, Mgurele-Buchres, Romni 4 Tbuk Universiy, Deprmen of Physics, P.O. Box 74, Tbuk 749, Sudi Arbi 5 Technicl Universiy of Kosice, BERG Fculy, B. Nemcovej 3, 4 Kosice, Slovki 6 Tif Universiy, Fculy of Science, Mhemics & Sisics Deprmen, P.O. Box 888, Sudi Arbi Received Sepember, Absrc. A sudy of he frcionl Lgrngin of he so-clled Cldirol-Kni oscillor is presened. The frcionl Euler-Lgrngin equions of he sysem hve been obined, nd he obined Euler-Lgrngin equions hve been sudied numericlly. The numericl sudy is bsed on he so-clled Grünwld-Lenikov pproch, which is power series expnsion of he genering funcion (bckwrd nd forwrd difference) nd i cn be esy derived from he Grünwld-Lenikov definiion of he frcionl derivive. This pproch is bsed on he fc, h Riemmn-Liouville frcionl derivive is equivlen o he Grünwld-Lenikov derivive for wide clss of he funcions. Key words: Riemnn-Liouville derivives, Cldirol-Kni oscillor, Grünwld-Lenikov pproch.. INTRODUCTION Hmilonin formulion plys n imporn role in clssicl nd qunum mechnics [, ]. As n exmple he dissipive s well s he non-conservive sysems cn be consruced nd reed using he Lgrngin nd he Hmilonin formulions [3]. Usully, dissipive sysems re scribed s hving microscopic nure [4 8]. Dmped hrmonic oscillor ws invesiged by Cldirol nd Kni [4 5]. The dmped qunum hrmonic oscillion wih one or wo degree of freedom in he frmework of Cldirol-Kni oscillor cn be considered s one of he bsic models of dissipion [8 9].

2 7 D. Blenu e l. When sudying he dmped qunum hrmonic oscillor in he cnonicl pproch, wo differen Hmilonin represenions hve been inroduced. In he firs represenion one-dimensionl sysem wih n exponenilly incresing mss ws considered (i.e., he so-clled he Cldirol-Kni oscillor) [ 9]. On he oher hnd, he second represenion he so-clled Bemn Feshbch- Tikochinsky oscillor, which consiss of dmped nd n mplified oscillor [, 6 9] is considered. The Cldirol-Kni oscillor is n open sysem whose prmeers such s mss nd frequency re ll ime dependen, while he Bemn Feshbch-Tikochinsky oscillor is closed sysem whose ol energy is conserved nd he dissiped energy from he dmped oscillor is rnsferred o mplified one. Recenly, frcionl clculus hs found mny pplicions in mny brnches of science nd engineering [ ]. Frcionl differenil equions hve been reed numericlly nd nlyiclly in mny ppers [ 9]. Decomposiion mehod is one of he mehods used o solve such equions numericlly nd nlyiclly [3 34]. Recenly new mehod clled mrix pproch hs been inroduced nd used [35, 36]. On he oher hnd, nlyic soluions o frcionlorder deferenil equions re ofen expressed in erms of he Mig-Leffler funcion [37 39]. In his pper, we py enion o sudy numericlly he frcionl Euler- Lgrnge equion of he so-clled Cldirol-Kni oscillor. This work is orgnized s follows. In secion we discussed briefly he bsic definiions of he frcionl derivives. In secion 3 we presened our model. In secion 4 numericl resuls of he obined Euler-Lgrnge equion of he model re depiced. Finlly, he concluding remrks re illusred.. BASIC DEFINITIONS In his secion some fundmenl formuls of frcionl clculus re presened. The firs one is he lef Riemnn-Liouville frcionl inegrl defined s follows [, ] () ( ) ( )d ( ) I x = τ x τ τ Γ. () The form of he righ Riemnn-Liouville frcionl inegrl is b () ( ) ( )d ( ) Ib x = τ x τ τ Γ. () The expression of he lef Riemnn-Liouville frcionl derivive reds

3 3 Frcionl Euler-Lgrnge equion of Cldirol-Kni oscillor 73 n x d f () τ Dx f x n+ Γ( n ) d x ( x τ) ( ) = dτ. (3) The righ Riemnn-Liouville frcionl derivive is given by n b d f () τ xdb f x n+ Γ( n ) d x ( τ x) x ( ) = dτ. (4) Here is he order of he derivive such h n n nd is differen o zero. If is n ineger, hese derivives become he clssicl ones. The generlized Mig-Leffler funcion is defined s E β, n z ( z) =, (5) Γ( n +β) n= such h, nd β. Thus, he exponenil funcion is specil cse for Mig-Leffler funcion, nmely n n z z E, ( z) = = = exp( z) Γ ( n+ ) n!. (6) n= n= 3. THE MODEL We sr our formlism by considering hrmonic oscillor whose mss depends on ime such h m () = mexp(in s βγ ), nd described by he following Hmilonin: p m H q m = exp( sin βγ ) + exp(sin βγ ). (7) The mss depends explicily on ime, β, γ re vrible prmeer nd dmping fcors, while p, nd q re cnonicl conjuge. If exp( sin βγ ) is Tylor expnded o firs order in incresing power of βγ wih β. Then, Eq. (7) is reduces o Cldirol-Kni Oscillor. The Lgrngin corresponding o he Hmilonin given by Eq. (7) is given s: L= exp( sin βγ)[ mq mw ( ) q ]. (8) The clssicl equion of moion is h of dmped oscillor q () +βγcos βγ q () + w() q () =. (9)

4 74 D. Blenu e l. 4 The frcionl counerpr of Eq. (8) cn be wrien s: F L = E, (in s βγ)[ m( D q) mw () q ]. () Using he generl form for he frcionl Euler-Lgrnge equion, nmely L L β L + Db + D =, () β q D q Dq i i b i he frcionl Euler-Lgrnge equions hve he following form Db ( E, ( sin ) D q( )) E, ( sin ) w ( ) q( ) βγ βγ =. () Now, our gol is o solve Eq. () numericlly for differen vlues of. 4. NUMERICAL ANALYSIS For numericl soluion of he liner frcionl-order equions () we cn use he decomposiion o is cnonicl form wih subsiuions q () x () nd Dq () x(). As resul, we obin he se of equion in he following form: D x () = x (), D( E (sin βγ x ) ()) = E (sin βγw ) x(). b,, (3) Insed lef nd righ side Riemnn-Liouville frcionl derivives (3) nd (4) in he se of equions (3) cn be used he lef nd righ Grünwld-Lenikov derivives, which re equivlen o he Riemnn-Liouville frcionl derivives for wide clss of he funcions [4]. The Grünwld-Lenikov derivives cn be defined by using upper nd lower ringulr srip mrices (Podlubny s mrix pproch) or we cn direcly pply he formul derived from he Grünwld- Lenikov definiions, bckwrd nd forwrd, respecively, for discree ime sep kh, k =,,3, Le us consider he second pproch, which works very well for liner s well s for nonliner frcionl differenil equions [4]. Time inervl [, b] is discreized by (N + ) equl grid poins, where N = ( b )/ h. Thus, we obin he following formul for discree equivlens of lef nd righ frcionl derivives: k k i k i i= D x = h c x, k =,..., N, (4) N k b k i k+ i i= D x = h c x, k = N,...,, (4b)

5 5 Frcionl Euler-Lgrnge equion of Cldirol-Kni oscillor 75 respecively, where xk x ( k) nd k = kh. The binomil coefficiens c i, i =,,3,, cn be clculed ccording o relion + ci = ci (5) i for c =. Then, generl numericl soluion of he frcionl liner differenil equion wih lef side derivive in he form cn be expressed for discree ime k D x() = f ( x (),) (6) = kh in he following form: k x( ) = f( x( ), ) h c x( ), (7) k k k i k i i= m where m = if we do no use shor memory principle, oherwise i cn be reled o memory lengh. Similrly we cn derive soluion for n equion wih righ side frcionl derivive. Le us consider he differen vlue of order for simulion ime second nd ime sep h =.5. The prmeers se up re he following: β =, γ =, nd iniil condiion q() =. nd w() =. Fig. Numericl soluion of equion (8) for ime second. Le us consider he differen funcion w() in he form w() = 4 + nd prmeers: β =, γ =, nd iniil condiion q() =..

6 76 D. Blenu e l. 6 Fig. Numericl soluion of equion (8) for ime second. In Fig. nd Fig. re depiced he simulion resuls of equions (7) for prmeers β =, γ =, nd vrious order, where derivive inervl is = nd b =, iniil condiion q() =., for ol simulion ime second nd compuionl ime sep h = CONCLUSIONS In his pper we invesiged he numericl soluions of he frcionl Cldirol-Kni Euler-Lgrnge equions. We sred wih he clssicl Cldirol- Kni Lgrngin nd hen we frcionlized i nd we obined he frcionl Euler-Lgrnge equions. Finlly, we invesiged numericlly he soluion of he frcionl Euler-Lgrnge equions obined. Our numericl mehod used is bsed on he formul derived from he Grünwld-Lenikov definiions, bckwrd nd forwrd, respecively. The numericl resuls re shown in Fig., nd Fig.. For exmple, Fig. shows he numericl soluion of Eq. (7) for prmeers β =, γ =, nd vrious order, nd iniil condiion q() =. nd w() =. While, Fig. shows he numericl soluion of Eq. (7) for prmeers β =, γ =, nd vrious order, iniil condiion q() =., nd w() = 4 +. I is cler from he figures h he behviors of he frcionl Euler-Lgrnge equion srongly depend on he order of he frcionl derivive, in ddiion o he form of he funcion w(). For ech grph we provided he clssicl soluion of he equions ( = ) nd hree differen cses for.

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