Mathematical ad Computatioal Applicatios, Vol. 6, No., pp. 3-38,. Associatio for Scietific Research MULIPLE IME SCALES SOLUION OF AN EQUAION WIH QUADRAIC AND CUBIC NONLINEARIIES HAVING FRAC- IONAL-ORDER DERIVAIVE Fadime Dal Departmet of Mathematics, Ege Uiversity, Izmir,urkey fadimedal@hotmail.com ABSRAC- Noliear vibratios of quadratic ad cubic system are cosidered. he equatio of motio icludes fractioal order term. Multiple time scales (a perturbatio method) solutio of the system is developed. Effect of fractioal order derivative term is discussed. Keywords: Fractioal differetial equatio, Caputo fractioal derivative, Noliear vibratios, Multiple time Scales method. INRODUCION Due to rapid developmet of oliear sciece, may differet methods were used to solve oliear problems. Perturbatio methods are well established ad used for over a cetury to determie approximate aalytical solutios for mathematical models. Algebraic equatios, itegrals, differetial equatios, differece equatios ad itegrodifferetial equatios ca be solved approximately with these techiques. [-4]. Fractioal derivatives appear i differet applicatios such as fluid mechaics, viscoelasticity, biology [5-8]. he asymptotic solutio of va der Pol oscilator with small fractioal dampig was cosidered by Feg Xie ad Xueyua Li [9]. Very recetly, Pakdemirli et al. [] proposed a ew perturbatio method to hadle strogly oliear systems. he method combies Multiple Scales ad Lidstedt Poicare method. he ew method, amely the Multiple Scales Lidstedt Poicare method (MSLP), is applied to free vibratios of a liear damped oscillator, udamped ad damped duffig oscillator. MSLP (a ew perturbatio solutio) was applied to the equatio with quadratic ad cubic oliearities by Pakdemirli ad Karaha []. I this paper, multiple time scales method (a perturbatio method) is used to solve the equatio with quadratic ad cubic oliearities icludig fractioal-order derivative term. Multiple time scales solutio ad umerical solutios of the problem are compared.. MULIPLE IME SCALES (MS) MEHOD he equatio of motio is 3 x ( t) x( t) D x( t) x( t) x( t) () with iitial coditios x ( ), x ( ) () Where the fractioal derivative D x is i the Caputo sese defied as
3 F. Dal t x( s) ds D x( t), < < ( ) ( t s) Fast ad slow time scales are t, t, t (3) he time derivatives, depedet variable ad fractioal derivative are expaded d dt D D D d = D D D D D D... (4) dt d ( ) D D D D D D D dt (5) Where D. he expasio x x (,, ) (,, ) (,, )... o x x (6) is substituted ito equatio () 3 x x x x x x D DD D DD x x x x x x D D D D D D D x x x he equatios at each order are O() D x x (8) O( ) D x x D D x x (9) 3 O( ) D x x DD x x x D x x ( D D D ) x () he solutio at the first order is x i i, e A e A, () where A ad A are complex amplitudes ad their cojugates, respectively. Equatio () is substituted ito (9) ad secular terms are elimiated i i D x x i D A e cc A e AA cc () i D A ( i ( ) a( ) e Where A ) is represeted the complex amplitudes i polar form A Rearraged equatio () i D x x ( A e AA cc) (3) Solutio of the differetial equatio (3) is defied as follows x x x h p (7)
Multiple ime Scales Solutio of a Equatio with Quadratic ad Cubic Noliearities Havig Fractioal-Order Derivative 33 i x h Be he geeral solutio is i i x Be A e 3 AA cc (4) Where i B be Applyig the iitial coditios yields c b() ad ( ) () 3 Let us cosider Eq. () ad use formula i i D e i e (see[]). 4 D x x [ i D A ( i ) A A A AA 3 A A] e cc 3 i 3 3i 3 3i ( A e cc) ( A e cc) (5) 3 ( ) ( ) i i A a e i cos( ) isi( ) e (6) Substitutig relatioships (6) i Eq. (5), whe we separate the real ad the imagiary part of the equatio da a ( si( ) (7) d d a a 3 a a cos( ) (8) 3 3 d 4 8 We obtai the equatios above equatios.he solutio of Eq. (7) is ( ) (.5 si( )) ce a (9) he solutio of Eq. (8) is si( ) c ( 9 ) ( ) ( cos( )) e () 4 si( ) Applyig iitial coditios yield 9 c= 4 si( ) herefore A is defied as follows (.5 si( )) si( ) c ( 9 ) i ( cos( )) ( ) e 4 si( ) A e e ()
34 F. Dal he solutio at the first order is x e (.5 si( )) cos( ( )) () he solutio at order is ( si( )) ( si( )) i i x be e e cos( ( )) e (3) 6 he iitial coditios at this order imply c b(), ( ) 3 x e cos( ( )) e cos( ( )) ( si( )) ( si( )) 3 6 ( si( )) e Fial solutio is obtaied as x x x O( ) 9 9 ( ) ( cos( )) ( ) e ( ) si( ) 4 si( ) 4 si( ) (.5 si( )) ( si( )) x e cos( ( )) e cos( ( )) 3 6 ( si( )) ( si( )) e cos( ( )) e (4) (5) Where t, t, t 3. COMPARISONS WIH HE NUMERICAL SOLUIONS We cosider equatio () with iitial coditios (). I view of the variatioal iteratio method (VIM), we costruct the followig iteratio formulatio: t 3,., x ( t) x ( t) si( s t)[ x ( s) x ( s) D x ( s) ( x ( s)) ( x ( s)) ] ds Where If we begi with x o ( t) x(), we ca obtai a coverget series: x ( t) x ( t)..cost
Multiple ime Scales Solutio of a Equatio with Quadratic ad Cubic Noliearities Havig Fractioal-Order Derivative 35 x ( t)..794796 cos( t).75485t si( t).349775cos( t).967393si( t) 5 3 5 7.46597t cos( t).339568t cos( t).5738637697t cos( t).33cos( t) 3 4.998473t cos( t).998473t si( t).46597t si( t).4453cos( t) 7 5 5.349775cos(t) -.5738637697si(t) t -.967393si(t) -.339568si(t) t.75485si(t) cos(t)-.4453 si(t) cos(t) -.88798885si(t)FreselC(.797884568t ).88798885cos(t)FreselS(.797884568t )-.539897cos(t)FreselC(.797884568t )-.539897si(t)FreselS(.797884568t ) Figure, Figure ad Figure3 show multiple time scales solutio of the system for.,. 5 ad respectively. Figure4 shows compariso of approximate aalytical ad (VIM) umerical solutios for,.,,. As see from figure4, MS method is more suitable tha VIM method. As time icreases, VIM method fails i our problem. Figure 5 shows compariso of approximate aalytical ad fiite differece method umerical solutios for,.,,. Figure. Approximate aalytical solutios (MS) for,.,,
36 F. Dal Figure. Approximate aalytical solutios (MS) for,. 5,, Figure3. Approximate aalytical solutios (MS) for,,,
Multiple ime Scales Solutio of a Equatio with Quadratic ad Cubic Noliearities Havig Fractioal-Order Derivative 37 Figure4. Compariso of approximate aalytical (MS) ad VIM umerical solutios for,.,,.5 X -.5 - Fiite differece method Multiple time scale -.5 4 6 8 ime icremetal Figure5. Compariso of approximate aalytical (MS) ad fiite differece method umerical solutios for,.,, 4. CONCLUDING REMARKS I this paper, multiple time scales method is successfully applied to fid the solutio of the equatio with quadratic ad cubic oliearities havig frac-
38 F. Dal tioal order derivative which correspods to uharmoic oliear oscillator. he fractioal derivative is cosidered i the Caputo sese which is more physical tha other derivatives [3]. he solutio of equatio is made by usig variatioal iteratio method (VIM) ad Multiple scale method (MS). he solutios of VIM, fiite differece method ad MS methods are compared. MS method produced solutios with good agreemet with the umerical solutios. It is cocluded that fractioal derivative term effects as dampig due to fractio. hat is, the amplitudes decrease by icreasig time. 5.REFERENCES. A. H. Nayfeh ad D.. Mook, Noliear Oscillatios, Wiley classics Library Editio, New York, USA, 995.. J.H. He, Modified Listedt-Poicare methods for some strogly oliear oscillatio, Part III: double series expasio, It. J. Noliear Sci. Numer. Simul.,, 37-3,. 3. J.H. He, Modified Listedt-Poicare methods for some strogly oliear oscillatio,part I : expasio of a costat, It J. Noliear Mech., 37, 39-34,. 4. J.H. He, No-perturbative methods for some strogly oliear problems, Dissertatio, Der Verlag im Iteret GmbH, Berli, 6. 5.Barbosa R S, ereiro Machado J A, Viagre B M ad Caldero A J, Aalysis of the Va der Pol oscillator cotaiig derivatives of fractioal order, J. Vib. Cotrol, 3, 9-3, 7. 6.Ge Z M ad Zhag A R, chaos i a modified Va der Pol system ad i its fractioal order systems, Chaos Solitos Fractals, 3, 79-8, 7. 7.Che J H ad Che W C, chaotic Dyamics of the fractioally damped Va der Pol equatio, Chaos solitos Fractals 35, 88-98, 8. 8.Gafiychuk V, Datsko B ad Meleshkov, Aalysis of fractioal order Bohoeffer Va der Pol oscillator, Physica A 387, 48-44, 8. 9. Xie F ad Li x,, Asymptotic solutio of the Va der Pol oscillator with small fractioal dampig, Phys. Scr., 9433.. Pakdemirli M, Karaha M M F, Boyacı, H, A, ew perturbatio algorithm with better covergece properties : Multiple scales Lidstedt poicare method, Math. Ad Comp. Appl., Vol 4, No:, 3-44, 9.. Pakdemirli M, Karaha M M F, A ew perturbatio solutio for systems with strog quadratic ad cubic oliearities, Mathematical Methods i the Applied Scieces, 33, 747,,.Samko, S G, Kilbas, A A ad Marichov, OI, Fractioal Itegrals ad derivatives heory ad Applicatios (i Russia), Nauka ; ekhika, Misk (Eglish traslatio by Gordo ad Breach sciece publ.,amsterdam, 999), 988. 3. Rossikhi, YA, Shitikova, M, Applicatio of Fractioal calculus for Dyamic problems of solid Mechaics : Novel reds ad Recet results,applied Mechaics Reviews, vol 63-8-,.