Research Article An Especial Fractional Oscillator

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1 Inernaional Saisical Mechanics Volume 3, Aricle ID 7573, 5 pages hp://dx.doi.org/.55/3/7573 Research Aricle An Especial Fracional Oscillaor A. Tofighi Deparmen of Physics, Faculy of Basic Science, Universiy of Mazandaran, P.O. Box , Babolsar, Iran Correspondence should be addressed o A. Tofighi; a.ofighi@umz.ac.ir Received 5 March 3; Revised June 3; Acceped 3 June 3 Academic Edior: Anonina Pirroa Copyrigh 3 A. Tofighi. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We propose a peculiar fracional oscillaor. By assuming ha he moion akes place in a complex media he level of fracionaliy is low, we find ha he ime rae of change of he energy of his sysem has an oscillaory behavior.. Inroducion In complex media such as glasses, liquid crysals, polymers, and biopolymers, he dynamical variable of ineres ofen obeys fracional differenial equaions [ 6]. For insance, he mean squared displacemen of a Brownian paricle is given by x () ; his linear dependence on ime is referred o as normal diffusion. In complex media his kind of behavior is ofen violaed, leading o anomalous diffusion. For a subdiffusive process x () ],wih< ] <. For his process, fracional dynamic equaions emerge naurally in he physical concep of coninuous ime random walks [6, 7]. Fracional differenial equaions have many applicaions in applied science and engineering [8 ]. Fracional differenial equaions have been invesigaed in pure sciences, such as pure mahemaics []. As a fracional generalizaion of he oscillaion phenomena, one can consider [3] d α x () d α +x() =, <α<3, >. () The case wih < α < corresponds o aenuaed oscillaion phenomenon [4 9]. Wih he iniial condiions x() = and x () =, hesoluionisx() = E α, ( α ),he so-called Miag-Leffler funcion. The case wih < α < 3 corresponds o amplified oscillaion phenomenon. Wih he iniial condiions x() =, x () =, andx () =, hesoluionisx() = E α, ( α ). In [4] i has been discarded on he ground ha i signifies he insabiliy of he sysem. In a recen sudy [] ihas been proven ha he Miag-Leffler funcion of his order is a Nussbaum funcion. Hence i may have applicaions in he conrolheoryofelecricalengineering.inanohersudy[] general equaions of he ype d α x () d α +f(x ()) =, <α<3, > () have been considered; however in his work he emphasis is on he exisence and uniqueness of soluions. There are various represenaions of he Miag-Leffler funcion: (i) series represenaion [3], (ii) inegral represenaion [4], (iii) approximae represenaion for a medium wih low level of fracionaliy [7, 9, ]. In his work we use he approximae represenaion o probe fracional differenial equaions of order <α<3. Theplanofhispaperisasfollows. In Secion we describe his approximae represenaion. In Secion 3 we obain he soluion for ()andwecomparei wih he exac resuls of [, 4]. In Secion 4 we propose a new fracional oscillaor of he form d α x () d α +b dβ x () d β +x() =, <α<3, >, (3) β = α/. We obain he soluion for his damped oscillaor, and we discuss he ime rae of change of his oscillaor. Finally in Secion 5 we presen our conclusions.

2 Inernaional Saisical Mechanics. The Perurbaion Scheme There exiss a muliude of definiions [ 5] like Riemann- Lioville, Weyl, Reisz, and Capuo for he fracional derivaive. However, he fracional operaor is uniquely defined. For insance, Riemann-Liouville and Capuo formulaions emerge by choice (how o include he iniial values) and are fully equivalen. In his paper we only use he Capuo fracional derivaive as i is easier o apply he iniial condiions in his ype. The lef(forward) Capuo fracional derivaive for α>is defined by D α f () = C Dα f () = Γ (n α) f (n) (τ) dτ, α+ n ( τ) n <α<n, >, n is an ineger number and f (n) (τ) denoes he nh derivaive of he funcion f(τ). To consruc a soluion for a process described by an equaion wih Capuo fracional derivaive, one needs he iniial condiions ha can be wrien as (4) f (k) () =c k, k=,,...,(n ). (5) In his work we seek causal soluion. Hence we require he condiion f() = for < [3]. A he ime = he soluion is deermined by iniial condiion equaion (5). And for >iis obained from he fracional differenial equaion describing he process under consideraion. Now if he order of he fracional derivaive α is close o a posiive ineger, namely, α = n ε, wihsmallposiiveε[7], hen we will have D n ε f () = Γ (ε) f (n) (τ) dτ, <ε<, >. (6) ε ( τ) As noed in [7] his represenaion is no convenien in he limi ε,since Γ (ε) =ε+o(ε ). (7) However, in his case from inegraion by pars we find D n ε f () = f(n) () ε Γ (+ε) + Γ (+ε) Now from he expansion ( τ) ε =+ε[γ+ln ( τ)] Γ (+ε) f (n+) (τ)( τ) ε dτ, <ε<, >. (8) +ε [ ln ( τ) +γln ( τ) + γ π ] +, (9) τ < and γ = is he Euler consan. Henceweobain D n ε f () =f(n)() +εd n f (), () D (n) f() is D n f () =f(n) () ln +γf (n) () + f (n+) (τ) ln ( τ) dτ. () In order o ge a good approximaion from (6)wefindhe following condiions: (i) ε, (ii) ε ln( τ). Nex we generalize his expansion for he case, α= n+ε,wihsmallposiiveε.wenoehainhiscasen<α< n+; herefore So D α f () = Γ (n+ α) f (n+) (τ) ( τ) n<α<n+, >. D n+ε f () = Γ ( ε) (α n) dτ, f (n+) (τ) ( τ) ε dτ, <ε<, >. By expansion in erms of he parameer ε we ge () (3) D n+ε f () =f (n) () f (n) () εd n f (). (4) Bu he process described by fracional derivaive D n+ε has one exra iniial condiion more han he process described by D n ε, namely, he condiion f n () = c n. However, if we assume c n =, hen our expansion and hence he soluions will have a coninuous dependence on he parameer α in ransiion from α=n εo α=n+ε.inoher words if we se c n =,henwege D α f () =f (n) () + (n α) D n f (), (5) wih α=n ε,<ε<. Now we consider () wihα = n + ε and he iniial condiions which are specified by x (k) () =η k, k=,,...(n ). (6) The perurbaion expansion of he dynamical variable is given by x () =x () +εx (), (7) x () is he soluion of he ineger case, and i is described by d n x () d n +x () =, >, (8)

3 Inernaional Saisical Mechanics 3 and i is subjec o x (k) =η k, k=,,...(n ). (9) The erm εx () is a correcion erm, and i designaes he deviaion from he ineger case. By subsiuing (5), (7) in ()weobain d n x () d n D n x () +x () =, >, () subjec o x () =, k=,,...(n ). () By assuming hese se of iniial condiions we obain he correcresulinhelimiofε. 3. Amplified Oscillaions The phenomena of amplified oscillaions correspond o he case <α<3,in(). I has been discussed in [,, 4]. In conrol heory, a branch of elecrical engineering, one uilizes a closed feedback loop o guaranee he sabiliy of he sysem. In [] iisprovedhahesoluionof() isa Nussbaum funcion. Hence when i is used as a par of a general sysem, i can guaranee he asympoic sabiliy of his larger sysem. Bu his discussion is resriced o a specific field of applied science, namely, conrol heory. In [] he general properies of () such as uniqueness and exisence have been invesigaed. Bu i does no consider he behavior of he soluion. In [4]() is briefly discussed; however i was discarded on he grounds ha i signals he insabiliy of he sysem under consideraions. Oursraegyoovercomehisdifficulyis () o consider he media wih low-level fracionaliy, () o add a damping erm o remove his insabiliy. In his secion we invesigae he limi, α=+ε, so d +ε x () d +ε +x() =, >. () For he normalized iniial condiions x () =, x () =,andx () = one obains [9] x () = cos () +εx (), ε, (3) x () = [ cos () +Si () cos () Ci () sin ()]. (4) Here Si() and Ci() are sine and cosine inegral funcions, respecively [], For his process we define he momena by p () = dβ x (), d β β=+ ε. (6) From (6)wefind[9] p () = sin () +εp (), (7) p () = [sin () Si () sin () Ci () cos ()]. (8) Hence he oal energy of his oscillaor is E F = (x +p )= ε [ cos () Si ()]. (9) I is possible o calculae he ime rae of change of he energy. The resul is de F d = ε Si (), ε >, >. (3) Therefore, he ime rae of change of he energy is always posiive. Hence he oal energy of he sysem is a monoonous increasing funcion of ime, which means ha he sysem absorbs energy from he environmen. For insance, if we ake ε=,henwehaveabouonepercen increase in he value of E F in a year. For ε= 5 we have abouonepercenincreaseinhevalueofe F in a day. These values are chosen in a way ha he condiion ε is saisfied [7]. Thus, our sudy suggess an absorpion inerpreaion for fracional derivaive of order α=+εfor he soluion of (), o be conrased wih he common dissipaion inerpreaion for fracional derivaive of order α= ε. 3.. Comparison wih he Exac Resul. Wih our choice of he iniial condiion, he exac resul is x () = E α ( α ) =f α () +g α (), (3) he one parameer Miag-Leffler funcion is E α = Γ (αk+), (3) k= z k x () =E α ( α )=f α () +g α (), (33) f α () = π r α sin (απ) exp ( r) r α +r α cos (απ) +. (34) The funcion g α () is given by Si () = sin (u) du, u Ci () = cos (u) du. (5) u g α () = α exp ( cos (π α )) cos [ sin (π α )], <α<3. (35)

4 4 Inernaional Saisical Mechanics Using α=+εand he condiion ε,weobain f α () = ε [ sin () Ci () +cos () Si () π cos ()] + g α () = cos () ε[ π 4 ] cos (). (36) By subsiuing (36) ino(33) we recover he resul expressedin(3). 4. An Especial Fracional Oscillaor In his secion we invesigae he damped fracional oscillaor described by (3). For simpliciy we consider a case he damping coefficien is b=cos(π/α).in hemedia wih lowlevel fracionaliyα =+ε. By insering (7)in(3)we ge d x () d +x () =, >. (37) Wih he iniial condiions x () = and x () =, we will have x () = cos () +εx (), >. (38) The perurbaion erm x () saisfies d x () d D x () + π dx () +x d () =, >, (39) wih he iniial condiions x () = and x () =. By assuming hese iniial condiions we obain he correc resul in he limi of ε.from(39) and afer some calculaions we obain x () = [ cos () +Si () cos () Ci () sin ()] + π (4) 4 [sin () cos ()]. For his process we define he momena by p () = dβ x (), d β β=+ ε. (4) From (4)wefind p () = sin () +εp (), (4) p () = [sin () Si () sin () Ci () cos ()] + π (43) 4 sin (). Hence he oal energy of his oscillaor is E F = (x +p )= ε [ cos () Si ()] + επ (44) 4 [sin () cos () ]. I is possible o calculae he ime rae of change of he energy. The resul is de F = ε d Si () επ sin (), ε >, >. (45) The firs erm and he second erm in he righ hand side of (45) are due o he firs and he second erm of (3), respecively. The firs erm is posiive and denoes absorpion of energy from he environmen, and he second erm is always negaive and shows he dissipaion of energy from he sysem o he environmen. For large values of we have Si() π/.hencefrom(45) de F επ cos (), ε>. (46) d 4 Weseehaheimeaverageofimeraeofchangeof energy of his sysem is zero; namely, 5. Conclusions de F =, ε>. (47) d Previous sudies of fracional oscillaion were limied o he case of <α<; see[3 9]. The soluions porray damped moion. A he limi of we have he algebraic decay of he soluions. These soluions are of relevance in vibraion of mechanical sysems or oscillaions of elecrical nework he energy is dissipaed in he form of hea. Hence in his domain of he values of he parameer α, fracional derivaive is a convenien ool for modeling of damping. The resuls of Secion 3 indicae ha we may use fracional derivaive o model amplificaion process as well. In Secion 4 we inroduced a damped oscillaor for a case he order of fracional derivaive is +ε.the model presened in Secion 4 has no been considered for general values of <α<3.iwillbeofineresogobeyondhesmall epsilon expansion. We conjecure ha analyical soluion for his model exiss. We plan o repor on hese and oher relaed issues in he fuure. Acknowledgmen The auhor is graeful o he reviewers for heir useful suggesions and commens. References [] K. B. Oldham and J. Spanier, The Fracional Calculus, Academic Press, New York, NY, USA, 974. [] I. Podlubny, Fracional Differenial Equaions, Acasemic Press, San Diego Calif, USA, 999.

5 Inernaional Saisical Mechanics 5 [3] R. Hilfer, Applicaion of Fracional Calculus in Physics, World Scienific, Singapore,. [4] B. J. Wes, M. Bologna, and P. Grigolini, Physics of Fracal Operaors, Springer, New York, NY, USA, 3. [5] G. M. Zaslavski, Hamilonian Chaos and Fracional Dynamics, Oxford Universiy Press, Oxford, UK, 5. [6] R. Mezler and J. Klafer, The random walk s guide o anomalous diffusion: a fracional dynamics approach, Physics Repor, vol. 339, no., pp. 77,. [7] R. Mezler, E. Barkai, and J. Klafer, Anomalous diffusion and relaxaion close o hermal equilibrium: a fracional Fokker- Planck equaion approach, Physical Review Leers,vol.8,no. 8, pp , 999. [8] G. Failla and A. Pirroa, On he sochasic response of a fracionally damped Duffing oscillaor, Communicaions in Nonlinear Science and Numerical Simulaion, vol.7,no.,pp ,. [9] M. DiPaola, A. Pirroa, and A. Valenza, Visco-elasic behavior hrough fracional calculus: an easier mehod for bes fiing experimenal resuls, Mechanics of Maerials, vol. 43, no.,pp ,. [] C. Celauro, C. Fecaroi, A. Pirroa, and A. C. Collop, Experimenal validaion of a fracional model for creep/recovery esing of asphal mixures, Consrucion and Building Maerials, vol. 36, pp ,. [] Y. Li and Y. Chen, When is a Miag-Leffler funcion a Nussbaum funcion? Auomaica, vol. 45, no. 8, pp , 9. [] J. R. Wang, L. Lv, and W. Wei, Differenial equaions of fracional order α (, 3) wih boundary value condiions in absrac Banach spaces, Mahemaical Communicaions,vol. 7, no., pp ,. [3] F. Mainardi, Fracional relaxaion-oscillaion and fracional diffusion-wave phenomena, Chaos, Solions and Fracals, vol. 7, no. 9, pp , 996. [4] R. Gorenflo and F. Mainardi, Fracional calculus: inegral and differenial equaions of fracional order, in Fracals and Fracional Calculus in Coninuum Mechanics,A.Carpineriand F. Mainardi, Eds., pp. 3 76, Springer, New York, NY, USA, 997. [5] B. N. N. Achar, J. W. Hanneken, T. Enck, and T. Clarke, Dynamics of he fracional oscillaor, Physica A, vol. 97, no. 3-4, pp ,. [6] B. N. N. Achar, J. W. Hanneken, and T. Clarke, Damping characerisics of a fracional oscillaor, Physica A,vol.339,no. 3-4, pp. 3 39, 4. [7] V. E. Tarasov and G. M. Zaslavsky, Dynamics wih low-level fracionaliy, Physica A,vol.368,no.,pp ,6. [8] A. Tofighi, The inrinsic damping of he fracional oscillaor, Physica A, vol. 39, no. -, pp. 9 34, 3. [9] A. Tofighi and H. N. Pour, ε-expansion and he fracional oscillaor, Physica A,vol.374,no.,pp.4 45,7. [] A. Tofighi and A. Golesani, A perurbaive sudy of fracional relaxaion phenomena, Physica A, vol. 387, no. 8-9, pp , 8. [] M. A. Abramowiz and I. A. Segun, Eds., Handbook of Mahemaical Funcions wih Formulas, Graphs, and Mahemaical Tables,Dover,NewYork,NY,USA,9hediion,97.

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