The Chebyshev Wavelet Method for Numerical Solutions of A Fractional Oscillator

Transcription

1 Shiraz University of Tehnology From the SeleteWorks of Habibolla Latifizaeh 01 The Chebyshev Wavelet Metho for Numerial Solutions of A Frational Osillator E. Hesameini, Shiraz University of Tehnology S. Shekarpaz, Shahi Beheshti University Habibolla Latifizaeh, Shiraz University of Tehnology Available at:

2 International Journal of Applie Mathematial Researh, 1 (4) (01) Siene Publishing Corporation The Chebyshev Wavelet Metho for Numerial Solutions of A Frational Osillator E. Hesameini, S. Shekarpaz, Habibolla Latifizaeh Department of Mathematis, Faulty of Basi Sienes, Shiraz University of Tehnology, Moarres Bl. P.O. Box, , Shiraz, Iran s: Hesameini@suteh.a.ir, S.Shekarpaz@suteh.a.ir H.Latifi.6.math@suteh.a.ir Abstrat Wavelet transform or wavelet analysis has been reently evelope as a mathematial tool for many problems. This paper is onerne with the wavelet numerial metho for solving partial ifferential equations (PDE s). The metho is base on isrete wavelet transform, using Chebyshev Wavelet Metho (CWM) whih an be use for solving frational ifferential equations. Interest in solving the problem using the Chebyshev wavelet basis is ue to its simpliity an effiieny in numerial approximations. Four numerial examples were shown an the results emonstrate that the propose way an be quite reasonable while ompare with exat solutions. Keywors: Chebyshev Wavelet Metho, Frational Osillator 000 mathematial Subjet lassifiation: Primary 65T60. Seonary 34A34. 1 Introution Mathematial moel of a system usually onsists of solving orinary an partial ifferential equations [1,]. Aaptive numerial tehniques for approximating the numerial solutions of these equations have attrate the attention of researhers sine many years ago [3]. Several ifferent methos have been applie for solving these problems, whih inlue finite ifferene metho, finite element metho, Laplae transform metho an other numerial methos. During previous years, the issue of wavelets has influene major regions of pure an applie

3 494 E. Hesameini, S. Shekarpaz, H. Latifizaeh mathematis, espeially in the numerial analysis of ifferential equations [4,5]. Wavelets are establishe as a strong novel mathematial implement in signal proessing, turbulene problem, simulation an time-series analysis [6,7,8]. In the reent years, strenuous ation an interest have been shown in the usage of wavelet theory an it s relate multiresolution analysis [4,9]. The substantial sheme bak of the wavelet eomposition is the ompressing representation of wavelet-base funtions. In wavelet methos, the geometri region an funtions are represente in terms of wavelet series efine in a ertain omain. Also, by transforming the ifferential equations to some weak forms, all of them along with the unknown funtions an be represente on the same basis [10,11,1]. Another region of remarkable interest is the stuy of frational ifferential equations [9].In this paper, the ynamis of the so-alle riven frational osillator is probe. This frational osillators are obtaine by replaing the seon-time erivative term in the orresponing harmoni osillator by a frational erivative of the orer onsiering that 1<. The frational erivatives were onsiere in the Caputo sense [13,14]. The general response expression ontains a parameter esribing the orer of the frational erivative that an be varie in orer to obtain various responses. In the ase of, the frational system of osillators reues to the stanar system of simple harmoni osillators. Some aspets of suh a system have been stuie previously by other researhers [14]. The aim of this paper is to solve these problems via a novel metho base on Chebyshev wavelets [15,16,17,18]. So, in orer to obtain solutions for frational osillators, ompatly supporte, orthonormal an ontinuous wavelets are applie, whih are espeially onstrute for the boune interval. Sine hebyshev wavelets ombine orthogonality with loalization an saling properties, naturally, there is an attempt to use these funtions for the numerial approximation solutions of PDE s [19]. The metho presente here onsists of reuing frational ifferential equation to a set of nonlinear equations by expaning unknown funtions in terms of wavelets with unknown oeffiients. The properties of saling funtions an wavelets [0] are then utilize for evaluating the unknown oeffiients. This paper is organize as follows: in Setion, some preliminary efinitions of the frational alulus are presente. Setion 3, is evote to the multiresolution approximations an wavelets to formulate the hebyshev wavelets. In Setion 4, the propose metho is use for approximating the solutions of frational ifferential equations. An finally, the numerial results are expresse in Setion 5. Some Preliminary Definition of the Frational Calulus In this setion, some require efinitions an mathematial notations of frational alulus theory whih are applie in this paper are given.

4 495 Definitions.1: A real funtion f(x), x 0 is sai to be in the spae C μ, μ R if there exists a real number p(>μ) suh that f (x)=x f (x) where f 1 (x) C[0, ) an is sai to be in the spae m μ C, m N U0, if p μ 1 (m) f C,m N. Definitions.: The Riemann-Liouville frational integral operator of orer 0, of a funtional f C, μ -1 is efine as μ Properties of the operator 1 x -1 J f(x)= (x-ξ) f(ξ) ξ Γ() 0 > 0, x > 0. 0 J f(x)=f(x). for f C μ, μ -1,, β > 0, μ -1 an 1, 1) ) 3) 4) J [18,19] are mentione in the following way: J f(x) exists for almost every x [a,b] β +β J J f(x)=j f(x). β β J J f(x)=j J f(x), γ Γ(γ+1) +γ J a (x-a) = (x-a). Γ(+γ+1) Definitions.3: The frational erivative of f(x) in Caputo sense is efine as 1 - D f(x)=j D f(x)= (x-ξ) f (ξ)ξ, x m- m m -1 (m) (1) Γ(m-) 0 for m -1< m, m N, x 0, m f C -1 [17]. Also, two basi properties of the Capoto s frational erivative are neee whih are esribe in the following forms; Lemma.1: If m-1< an m, m N an m f C μ,μ -1 then D J f(x)=f(x), m-1 (k) + x k J D f(x)=f(x)- f (0 ), x>0 k=0 k! The Caputo frational erivative is peruse here whereas it allows traitional initial an bounary onitions to be inlue in the formulation of the problem.

5 496 E. Hesameini, S. Shekarpaz, H. Latifizaeh Definitions.4. For m to be the smallest integer that exees, the Caputo time-frational erivative operator of the orer 0. is efine as follows; m 1 m--1 u(ξ) (t-ξ) ξ, m-1 < < m, 0 m u(t) t Γ(m- ) ξ Dt u (t)= = t m u(t) m, = m N. t () For more information on the mathematial properties of frational erivatives an integrals, see [9,13,14,16,17,18,19]. For real >0 (later, only for 1< ), onsier the frational ifferential equation u +ω u (t)=f(t), m-1< m, (3) t subjet to the initial onitions k u (0)= k, k=0,1,,m-1, (4) where ω is an arbitrary onstant an f (t) a given ontinuous funtion. Here, m is an integer uniquely efine by m-1< m, whih provies the number of the presribe initial values k u (0)= k, k=0,1,...,m-1. Equation (3) is alle the frational osillation equation for 1<, the frational relaxation equation for 0< 1an the frational growing osillation equation orresponing to < 3. The frational erivative in Equation (3) is onsiere in Caputo sense. 3 Chebyshev Wavelets an their Properties In this setion, the general efinitions of wavelets are introue. Also, the Chebyshev wavelets an their main properties are illustrate Wavelets an Chebyshev Wavelets Wavelets are a family of funtions whih are erive from the family of saling funtions { jk, : k Z} where (t)= a (t-k). (5) k k

6 497 For the ontinuous wavelet, the following form an be presente: 1 - t-b ψ a,b (t)= a ψ( ) a,b R,a 0, a where a an b are ilation an translation parameters, respetively, suh that ψ(x) is a single wavelet funtion. If a an b are isrete values in the initial form of the ontinuous wavelets, (6) -j a=a, a >1, b >0, b=n b a -j 0 0, j,k Z, (7) then, the family of isrete wavelets are shown as follows: j,k j j ψ (t)= a ψ(a t-nb ), (8) where, {ψ j,k} j Z forms a wavelet basis for L (R). Also, an orthonormal basis is onstrute for a0 an b 0=1. Four parameters ontribute to the general form of Chebyshev Wavelets, m is the egree of Chebyshev polynomials of the first kin, n=1,,,, k is any positive integer an t enotes the time. Consequently, Chebyshev Wavelets are in the following form: k k n-1 n T m ( t-n+1) k x< k, ψ (t)= 0 otherwise, where m=0,1,,..., M-1, k=1,,..., T m (t)= j-1 an, 1, m = 0, π T m (t), m > 0, π where T m (t ) are the Chebyshev polynomials. Then, by (8), the wavelets {ψ } form an orthonormal basis for L [0,1]. In the above formulation of Chebyshev wavelets, the Chebyshev polynomials are shown as follows:

7 498 E. Hesameini, S. Shekarpaz, H. Latifizaeh T (t) =1, 0 T (t) =t, 1 T ( t) = t T (t) - T (t), m= 1,,... m+1 m m-1 whih are orthogonal with respet to the weight funtion 1 ω(t)= on the 1- t interval [-1,1]. Beause of the orthogonality, in this form of Chebyshev wavelets the weight k funtion ω(t)=ω ( t-1) shoul be ilate an translat to ω (t)=ω ( t- n+1). 3.. Funtion Approximation n A given funtion u(t) with the omain [0,1] may be approximate as: u(t )= ψ (t) (9) If the infinite series in Eq. (9) is trunate, then this equation an be written as: j-1 u(t ) ψ (t)=c Ψ(t), T (10) where C an Ψ(t) are matries of size 1,0 ( M 1) as follows: C=[,,...,,,,...,,...,,,..., ], 1,1 1,M-1,0,1,M-1,0,1,M-1 T 1,0 1,1,.. 1, M-1,0,1, M-1,0,...,ψ,1,M-1 Ψ(t)=[ψ,ψ.,ψ,ψ,ψ,...,ψ,...,ψ,ψ ]. T Let M t be a set of olloation points as follows: i i=1 (i-1) t i =, i=1,,..., M k M (11) Thus, the Chebyshev wavelet matrix Ψ an be efine as: m m 1 3 m-1 Ψ m m =[Ψ( ) Ψ( ) Ψ( )]. m m m (1) For instane, when k= an M=3, the following matrix an be presente:

8 Appliation of the CWM for Solving Frational Differential Equations In this setion, the CWM is applie for solving a system of riven frational osillators with the ommon form as follows; u +ω u (t)=f (t), 1<, (13) t with the initial onitions; u(0)=a, u (0)=b, (14) where is the natural frequeny an f(t) is a given ontinuous funtion. By applying the CWM for approximating the solutions of this problem, (13) an be rewritten as follows; n=1 m= 0 ( ψ (t)) + ω ( ψ (t)) = f ( t), 1<, t (15) with:

9 500 E. Hesameini, S. Shekarpaz, H. Latifizaeh ψ (0) =a, ( ψ (t))(0) = b, t (16) where f(t)= n',m' ψ n',m' (t), an u(t) is the frational erivative of a given n'=1 m'=0 t funtion u(t), efine in Setion(). Here, let M t be a set of olloation points, then Eq. (15) an be alulate at i i=1 t i olloation points, suh that i i n=1 m= 0 ( ψ (t )) ω ( ψ (t ))=f( t 1< t i ) an the initial onitions are in the following form: +,, ψ (t ) =a, 0 ( ψ (t))( t=t ) =b. t 0 Thus, by evaluating the oeffiients nm,, u(t) an be obtaine at every hosen point. 5 Numerial Results In orer to emonstrate the valiity of the propose metho, three examples are examine an the approximations of solutions are ompare with the exat solutions or solutions obtaine using other methos. Example 1. Consier the following system: u +ω u(t)=0, 1< t (17) subjet to the initial onitions;

10 501 u(0)=1, u (0)=0, (18) A simple harmoni frational osillator is esribe using this equation an the foring funtion in this ase is f(t)=0. Thus by inserting Eq. (10) in Eq. (17), the following an be presente, t n=1 m= 0 + ( ( ψ (t)) ω ψ (t))=0, 1< (19) with the initial onitions; ψ (0) =1, ( ψ (t))(t=0) =0, t (0) By olloating the above equation in olloation points t M i i=1, one an gets: with t i i n= 1 m=0 + =, ( ψ (t )) ω ( ψ (t )) 0 1< ψ (0) =1, ( ψ (t))(t=0) = 0, t Next, by omputing the oeffiients, the solutions u(t) of this equation are nm, obtaine. The numerial results of this problem with =1.7, =1.9 an = are presente in Table 1. By setting = in Eq. (17), the solution of a simple harmoni osillator is obtaine an expresse by u(t)=os(ωt).

11 50 E. Hesameini, S. Shekarpaz, H. Latifizaeh t Table 1: Numerial results ompare with the results in [1] by ω = 0.5. (Example 1) = 1.7 = 1.9 = Our metho Ref[1] Our metho Ref[1] Our metho Ref[1] Figure 1 emonstrates the progress results for =, =1.9 an =1.7. The omparison of these results shows how the reloation of the frational osillator varies as a funtion of time an how this time hange epens on the parameter. Also, the behavior of the riven frational osillator is similar to that of the ampe harmoni osillator, where the loomotion is yet osillatory, whereas the assemble energy ereases an the phase plane iagram is no longer a lose urve, but a logarithmi spiral. The results of these omputations, for ifferent values of, are onvergent to the solutions obtaine by setting =. 1.a 1.b Figure 1. a. The solution obtaine by the wavelet metho (--) an analytial metho (-) with ω = 0.5. Figure 1. b. The exat solution of a simple harmoni osillator =( ),=1.9( ),=1.7(--) by ω = 0.5.

12 503 Example. Now, the following frational ifferential equation is onsiere: u +ω u (t)=0, 1< t (1) Whih is subjet to the following initial onitions u(0)=, u (0)=0, () where the foring funtion is the step funtion that is efine by f (t)= A, t>0 0, t<0. Similar to the proeure use in Example.1, (1) an be written as with; t +ω t =, ( ψ (t)) ( ψ ( )) A 1<, t >0, ψ (0) ψ t ( (t))(t=0) = 0. =, Then, by substituting the olloation points presente: M t, this formula an be i i=1 with t i i n= 1 m=0 t +ω t, ( ψ ( )) ( ψ ( ))=0 1< ψ (0) =, ( ψ (t))(t=0) = 0. t After omputing the oeffiients, solutions u(t) for this equation are obtaine. The numerial results of this problem with =1.7, =1.9 an = are shown in Table.

13 504 E. Hesameini, S. Shekarpaz, H. Latifizaeh t Table. Numerial results ompare with the results in [1] by ω = 0.5. = 1.7 = 1.9 = Our metho Ref[1] Our metho Ref[1] Our metho Ref[1] a.b Figure. a. The solutions obtaine by the wavelet metho (- -) an analytial metho ( ).with ω = 0.5. Figure. b. The exat solution of a simple harmoni osillator =( ),=1.9(- -),=1.7( ) by ω = 0.5. Figure shows the numerial results for =,=1.9 an =1.7. To try the impression of the funtion f(t)= A, the numerial solutions are evaluate by hoosing u( 0) = = 0 an A=1. The behavior of the riven frational osillator for the step funtion is similar to that of the ampe osillator. Example 3. In this example, assuming that the foring funtion is f(t)=sin(ωt), the frational ifferential equation an be rewritten as:

14 505 u +ω u (t)=sin (ω t), 1<, t (3) by the initial onitions u(0)=, u (0)=0, (4) By expressing f(t) in terms of Taylor series at x=0, one obtains; (ω t) (ω t) (ω t) f (t)=ω t L 3! 5! 7! Thus, pursuant to CWM, the following is obtaine; (ω t) (ω t) (ω t) ( ψ (t) )+ ω ( ))=ω t- + -! ψ (t +, 1<, t 3! 5 7! with; ψ (0) =, ( ψ (t))(t= 0) 0 t =, M The olloation points t are replae at the above equations, suh that; i i= ( ψ (t i )) + ω ψ ( t) i = t 3! 5! 7! (ω t) (ω t) (ω t) ( ) ω t , 1< with the following initial onitions ψ (0) =, ( ψ (t))(t=0) = 0. t

15 506 E. Hesameini, S. Shekarpaz, H. Latifizaeh After omputing the oeffiients nm,, the solutions u(t) are obtaine for this equation. The numerial results of this problem with =1.7, =1.9 an = are shown in Table 3. t Table 3. Numerial results ompare with the results in [1] by ω = = 1.7 = 1.9 = Our metho Ref[1] Our metho Ref[1] Our metho Ref[1] a 3.b Figure 3. a. The solutions of obtaine by the wavelet metho (- -) an analytial metho ( ) with ω = 0.5. Figure 3. b. The exat solution of a simple harmoni osillator =( ),=1.9(- -),=1.7( ) by ω = 0.01.

16 507 The results of this metho for ifferent values of (1< ) an the sinusoial funtion are onvergent to the analyti solutions with =. Also, Figure 3 shows the numerial results for =, =1.9 an = Disussion The prinipal target of this paper was to ompute numerial solutions for a system of riven osillators. Several numerial methos were use for solving frational ifferential equations, an the wavelet tehniques were esribe as a numerial tool for the fast an aurate solution of these ifferential equations [9,19]. In the present work, the Chebyshev wavelet metho was exeute for solving frational ifferential equations. Note that, in the Chebyshev wavelet metho, the values of integrals were evaluate very aurately beause; the bases of Chebyshev wavelets were polynomials of ifferent orers. Thus, using this metho, the problem was transforme to a system of nonlinear equations an satisfatory approximations were obtaine for the solutions. 7 Conlusions Numerial results prove higher ability of the propose tehnique ompare with that of the other methos. All the examples showe that the numerial results of the CWM were onvergent to the exat solutions with =. On the other han, the obtaine approximations emonstrate that the behavior of the riven frational osillator was similar to that of the ampe harmoni osillator. It an be onlue that the isplaement funtions are able to esribe the intermeiate proesses between exponential eay ( =1) an pure sinusoial osillation ( = ). Referenes [1] P. Flether, S. Haswell, an V. Paunov. Theoritial onsierations of hemial reations in miro-reators operating uner eletro-osmoti an eletrophoreti ontrol, Analyst 14 (1991) [] S. Muller. Aaptive Multisale Shemes for Conversation Laws, Leture Notes in Computational Siene an Engineering, vol. 7, Springer, Berlin, 003. [3] G. Beylkin, an J. M. Keiser. On the Aaptive Numerial Solution of Nonlinear Partial Differential Equations in Wavelet Bases Can, Applie Math. So, JCP, 13 (1997)(CP96556), [4] J. M. Alam, N. K. R-. Kevlahan, an O. V. Vasilyev. Simultaneous Spae time Aaptive Wavelet Solution of Nonlinear Paraboli Differential Equations, Journal of Computational Physis, 14 (006)

17 508 E. Hesameini, S. Shekarpaz, H. Latifizaeh [5] E. Hesameini, an S. Shekarpaz. Wavelet solutions of the seon painleve equation.,iranian. J. of Siene an Tehnology. prepare to publish, Vol In press. [6] C. K. Chui. Wavelets: A Mathematial Tool for Signal Analysis, SIAM, Philaelphia, PA, [7] A. e Vries. Wavelets. FH Süwestfalen University of Applie Sienes, Halener Straße 18, D Hagen, Germany Version: September 4, 006. [8] Oleg V. Vasilyev an W. Kenal Bushe. On the Use of a Dynamially Aaptive Wavelet Colloation Algorithm in Diret Numeria Simulations of Non-Premixe Turbulent Combustion, Center for Turbulene Researh, Annual Researh Briefs, [9] E. Hesameini, S. Shekarpaz, H. Latifizaeh an F. Fotros. An profitable algorithm to frational ifferential equation: espeially wave equations.,int. J. of Differential equations. In press. [10] L. Gagnon, an J. M. Lina. Wavelets an numerial split-step metho: A global aaptive sheme, Opt. So. Am. B, to appear. [11] J. Lianrat, V. Perrier, an Ph. Thamithian. Numerial Resolution of Nonlinear Partial Differential Equations using the Wavelet Approah. Wavelets an Their Appliations, eite by M. B. Ruskai, G. Raphael (Jones & Bartlett, Boston, 199). print, [1] R. L. Shult, an H. W. Wyl. Using wavelets to solve the Burgers equation: A omparative stuy, Phys. Rev.A46, 1 (199). [13] Y. Luhko, an R. Gorne o. The initial value for some frational ifferential equations with the aputo erivate, Preprint series A08-98, Fahbreih Mathemati an Informati, Frei Universittal Berlin, [14] K. S. Miller an B. Ross. An Introution to the Frational Calulus an Frational Differential Equations, Wiley, New York, [15] L. Blank. Numerial treatment of ifferential equations of frational orer, MCCM Numerial Analysis Rep. 87, The University of Manhester, [16] R. Goreno, an F. Mainari. Frational aluls. Integral an ifferential equations of frational orer in Fratals an Frational Calulus in Continuum Mehanis, A Carpinter an F. Mainari, es., Springer-Verlag. New Nork [17] F. Mainrai. Frational relaxation-osillation an frational iffusion-wave phenomena, Chaos Solitons Fratals 7(9) (1996) pp [18] F. Mainrai. Frational alulus: Some basi problems in ontinuum an statistial mehanis, in Fratals an Frational Calulus in Continuum Mehanis, A Carpinteri an F. Mainrai, es., Springer-Verlag, New York, 1997, pp [19] Yuanlu Li. Solving a nonlinear frational ifferential equation using Chebyshev wavelets., J. Commun Nonlinear Si Numer Simulat 15 (010) 84-9.

18 509 [0] E. Babolian an F. Fattahzaeh. Numerial omputation metho in solving integral equations by using Chebyshev wavelet operational matrix of integration., Applie Math an Comput. 188 (007) [1] A. Yilirim an S. Momani. Series solutions of a frational osillator by means of the homotopy perturbation metho, Int. J. of Computer Mathematis. Vol 87, No 5, April 010,