A novel solution for fractional chaotic Chen system

Transcription

1 Available online a J. Nonlinear Sci. Appl. 8 (2) Research Aricle A novel soluion for fracional chaoic Chen sysem A. K. Alomari Deparmen of Mahemaics Faculy of Science Yarmouk Universiy 2-63 Irbid Jordan. Communicaed by Wasfi Shaanawi Absrac A novel soluion o he fracion chaoic Chen sysem is presened in his paper by using he sep homoopy analysis mehod. This mehod yields a coninuous soluion in erms of a rapidly convergen infinie power series wih easily compuable erms. Moreover he residual error of he SHAM soluion is defined and compued for each ime inerval. Via he compuing of he residual error we observe ha he accuracy of he presen mehod ends o which is very high. c 2 All righs reserved. Keywords: Chaoic sysem fracional Chen sysem homoopy analysis mehod sep homoopy analysis mehod residual error. 2 MSC: 6P2 26A33 34A8.. Inroducion Naure is inrinsically nonlinear. So i is no surprising ha mos of he sysems we encouner in he real world are nonlinear. And wha is ineresing is ha some of hese nonlinear sysems can be described by fracional-order differenial equaion (FDE) which can display a variey of behaviors including chaos and hyperchaos. The purpose of his paper is o obain a coninuous soluion for fracional chaoic Chen sysem [6 32 3]. D α x = a(y x) (.) y = (c a)x xz + cy (.2) z = xy bz (.3) x() = c y() = c 2 z() = c 3 (.4) D α 2 D α 3 address: abdomari28@yahoo.com (A. K. Alomari) Received

2 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) where D α i i = 2 3 are Capuo fracional derivaives a b c from R and < α i. Finding accurae soluion for FDEs has been an acive research underaking [ ]. Exac soluions of mos of he FDEs canno be found easily hus analyical and numerical mehods mus be used. For example generalized Adams-Bashforh-Moulon mehod (GABMM) is one of he mos used mehod o solve fracional differenial equaions [9 6 34]. Some of he recen analyic mehods for solving nonlinear problems include he Adomian decomposiion mehod (ADM) [2 29] homoopy-perurbaion mehod (HPM) [2 24] and variaional ieraion mehod (VIM) [22 2]. Recenly homoopy analysis mehod (HAM) becomes one of he mos famous echnique o solve such nonlinear problems. he mehod was proposed by Liao [7 8]. Many researchers have applied his mehod for differen class of differenial equaions [ ]. Alomari e al. [4] used he idea of ime sep in he algorihm of HAM o ge mulisage homoopy analysis mehod (MSHAM) and apply o Chen sysem. Recenly Alomari e al. [] inroduce new algorihm o obain he soluion of fracional chaoic sysem using HAM. This paper invesigaes for he firs ime he applicabiliy and effeciveness of HAM when we hybrid he numerical wih analyical in a sequence of inervals (i.e. ime sep) for finding accurae approximae soluions o he fracional Chen sysem. To he bes of our knowledge his is also he firs ime ha he residual error can be calculaed for he analyical soluion a each subinerval of fracional Chen sysem. Numerical resuls are presened graphically and are found o be in excellen agreemen wih he GABMM soluion. 2. Preliminaries and noaions In his secion we give some definiions and properies of he fracional calculus and homoopy-derivaive 2.. Fracional calculus The following properies can found in [26]. Definiion 2.. A real funcion f() > is said o be in he space C µ µ R if here exiss a real number p > µ such ha f() = p f () where f () C( ) and i is said o be in he space C n µ if and only if h (n) C µ n N. Definiion 2.2. The Riemann-Liouville fracional inegral operaor (J α ) of order α of a funcion f C µ µ is defined as J α f() = ( s) α f()ṣ Γ(α) (α > ) (2.) J f() = f() (2.2) where Γ(α) is he well-known gamma funcion. Some of he properies of he operaor J α which we will need here are as follows: For f C µ µ α β and γ :. J α J β f() = J α+β f() 2. J α J β f() = J β J α f() 3. J α γ = Γ(γ+) Γ(α+γ+) α+γ.

3 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) Definiion 2.3. The fracional derivaive (D α ) of f() in he Capuo sense is defined as D α f() = ( s) n α f (n) ()ṣ (2.3) Γ(n α) for n < α < n n N > f C n. The following are wo basic properies of he Capuo fracional derivaive:. Le f C n n N hen Dα f α n is well defined and D α f C. 2. Le n α n n N and f C n µ µ. Then n (J α D α )f() = f() f (k) ( + ) k k!. (2.4) 2.2. homoopy-derivaive k= The following properies can found in [9] Definiion 2.4. Le ϕ be a funcion of he homoopy-parameer q hen D m (ϕ) = m ϕ m! q m. q= (2.) is called he mh-order homoopy-derivaive of ϕ where m is an ineger. For homoopy-series he below ϕ = hold and i= u i q i ϕ 2 = i= v i q i. D m (ϕ ) = u m. 2. D m (qϕ ) = D m (ϕ ) 3. If L be a linear operaor independen of he homoopy-parameer q. For homoopy-series hen D m (Lϕ ) = LD m (ϕ ). 4. If f and g be funcions independen of he homoopy-parameer q hen. D m (ϕ ϕ 2 ) = m i= ϕ iϕ 2m i 3. Soluion approaches To solve (.) (.3) we choose he base funcion as D m (fϕ ± gϕ 2 ) = fd m (ϕ ) ± gd m (ϕ 2 ). { nα +mα 2 +kα 3 n m k } (3.)

4 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) so he soluions are in he form x() = a + y() = b + z() = c + n= m= k= n= m= k= n= m= k= a nmk nα +mα 2 +kα 3 (3.2) b nmk nα +mα 2 +kα 3 (3.3) c nmk nα +mα 2 +kα 3 (3.4) where a nmk b nmk and c nmk are he coefficiens. I is sraighforward o choose x () = c y () = c 2 z () = c 3 (3.) as our iniial approximaions of x() y() and z() and he linear operaor should be L α [ˆx] = D α ˆx L α2 [ŷ] = D α2ŷ L α3 [ẑ] = D α3ẑ (3.6) (3.7) (3.8) since we used Capuo fracional derivaive hen we have he propery L α [A ] = L α2 [A 2 ] = L α3 [A 3 ] = (3.9) where A i i = 2 3 are he inegraion consans ha will be deermined by he iniial condiions. If q [ ] and indicae he embedding and non-zero auxiliary parameers respecively hen he zeroh-order deformaion problems are of he following form: ( q)l α [ˆx(; q) x ()] = q N x [ˆx(; q) ŷ(; q)] (3.) ( q)l α2 [ŷ(; q) y ()] = q N y [ˆx(; q) ŷ(; q) ẑ(; q)] (3.) ( q)l α3 [ẑ(; q) z ()] = q N z [ˆx(; q) ŷ(; q) ẑ(; q)] (3.2) subjec o he iniial condiions ˆx(; q) = c ŷ(; q) = c 2 ẑ(; q) = c 3 (3.3) in which we define he nonlinear operaors N x N y and N z as N x [ˆx(; q) ŷ(; q)] = α ˆx(; q) α N y [ˆx(; q) ŷ(; q) ẑ(; q)] = α2ŷ(; q) α 2 N z [ˆx(; q) ŷ(; q) ẑ(; q)] = α3ẑ(; q) α 3 a(ŷ(; q) ˆx(; q)) (c a)ˆx(; q) + ˆx(; q)ẑ(; q) cŷ(; q) ˆx(; q)ŷ(; q) + bẑ(; q). For q = and q = he above zeroh-order equaions (3.)-(3.2) have he soluions ˆx(; ) = x () ŷ(; ) = y () ẑ(; ) = z () (3.4) and ˆx(; ) = x() ŷ(; ) = y() ẑ(; ) = z(). (3.)

5 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) When q increases from o hen ˆx(; q) ŷ(; q) and ẑ(; q) vary from x () y () and z () o x() y() and z() respecively. Expanding ˆx ŷ and ẑ in Taylor series wih respec o q we have in which ˆx(; q) = x () + ŷ(; q) = y () + ẑ(; q) = z () + x m ()q m y m ()q m z m ()q m (3.6) (3.7) (3.8) x m () = D m (ˆx(; q)) y m () = D m (ŷ(; q)) z m () = D m (ẑ(; q)) (3.9) where is chosen in such a way ha hese series are convergen a q =. Therefore hrough Eqs. (3.4) (3.9) we have x() = x () + y() = y () + z() = z () + x m () y m () z m (). (3.2) (3.2) (3.22) Take he mh-order homoopy-derivaive of zeroh-order equaions (3.)-(3.2) and used he properies () (4) hen we have he mh-order deformaion equaions L α [x m () χ m x m ()] = R x m() (3.23) L α2 [y m () χ m y m ()] = R y m() (3.24) L α3 [z m () χ m z m ()] = Rm() z (3.2) wih he following iniial condiions: x m () = y m () = z m () = (3.26) where Rm() x Rm() y and Rm() z can be found by used he properies ()(4) and () as and Rm() x = D α x m a(y m x m ) (3.27) Rm() y = m D α 2 y m (c a)x m + x i ()z m i () cy m () (3.28) i= m Rm() z = D α 3 z m x i ()y m i () + bz m (). (3.29) i= χ m = { m m >.

6 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) In his way i is easy o solve he linear non-homogeneous Eqs. (3.23) (3.2) a iniial condiions (3.26) for all m and now we successfully obain x () = α a ( c 2 + c ) Γ (α + ) y () = α 2 ( c c + c a cc 2 + c 3 c ) Γ (α 2 + ) z () = α 3 c 2 (b c ) Γ (α 3 + ) ec. Then he 3-erm of he approximae soluions of Eqs. (.) (.4) are x() = x () + y() = y () x m () y m () z() = z () + z m (). To deermine he value of we plo he -curves for Eqs. (3.3) (3.32) in Fig. (). (3.3) (3.3) (3.32) x( 7 ) y( 7 ) z( 7 ) h h h Figure : -curves of 3h-order approximaion for (.9.9.9) From his figure i is noed ha he valid regions of correspond o he line segmens nearly parallel o he horizonal axis. If = we ge he homoopy perurbaion mehod (HPM) soluion when α = α 2 = α 3 = which is no effecive for large values of (for more deail see [8]). 3.. SHAM The HAM soluion for Eqs. (3.3) (3.32) is no effecive for larger. In case if we need he soluion for [ 7] hen he simple idea is o divide he inerval [ 7] o subinervals wih ime sep and we ge he soluion a each subinerval. So in his case we have o saisfy he iniial condiion a each of he subinerval. Accordingly he iniial values x y z will be changed for each subinerval i.e. x( ) = c = x y( ) = c 2 = y and z( ) = c 3 = z and we should saisfy he iniial condiions x m ( ) = y m ( ) = and z m ( ) = for all m so x () = ( ) α a ( c 2 + c ) Γ (α + ) y () = ( ) α 2 ( c c + c a cc 2 + c 3 c ) Γ (α 2 + )

7 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) z () = ( ) α 3 c 2 (b c ) Γ (α 3 + ). So he soluion will be as follows: x() = c + 2 x m ( ) (3.33) y() = c y m ( ) (3.34) z() = c z m ( ) (3.3) where saring from = unil n = T = 7. To carry ou he soluion on every subinerval of equal lengh we need o know he values of he following iniial condiions: c = x( ) c 2 = y( ) c 3 = z( ). In general we do no have hese informaion a our clearance excep a he iniial poin = = bu we can obain hese values by assuming ha he new iniial condiion is he soluion in he previous inerval. (i.e. If we need he soluion in inerval [ i i+ ] hen he iniial condiions of his inerval will be as c = x( i ) = 2 m= 2 x m ( i i ) (3.36) c 2 = y( i ) = y m ( i i ) (3.37) c 3 = z( i ) = m= 2 m= z m ( i i ) (3.38) where c c 2 and c 3 are he iniial condiions in he inerval [ i i+ ]). By his way we ge modified homoopy perurbaion mehod (MHPM) soluion as a special case when = and α = α 2 = α 3 = [8] Error analysis for SHAM The differen beween he exac soluion and he given soluion which we will so-call residual error can be define as E x = D α X a(y X) (3.39) E y = D α 2 Y (c a)x + XZ cy (3.4) E z = D α 3 Z XY + bz. (3.4) where X Y and Z are he HAM soluion for he equaions (.) (.3) respecively. Since he SHAM soluion is analyic a each ime sep hen i is easy o obain he residual error a each ime sep. According Eqs. (3.39) (3.4) we can find he residual error on each ime sep by applying he ha equaions and using c c 2 and c 3 which is defined as in SHAM soluion. We noed ha he orders of magniude of he errors in SHAM soluion depend on he order of approximaion and he lengh of he subinervals.

8 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) erm 2-erm 3 2 -erm 2-erm erm 2-erm x() - y() - x() Figure 2: Time series of he SHAM soluion Using and 2 order of approximaion for (.9.9.9) 4. Resuls and discussion In his par we se a = 3 c = 28 b = 3 and we ake he iniial condiions x() = y() = and z() = 37 as in [4] afer he sandard case (α α 2 α 3 ) as ( ). To observe he convergen of he soluion we plo he -erm and 2-erm of SHAM soluion wih =. in Fig. 2. I is clear ha he soluion of -erm like he soluion of 2-erm hen we can consider 2-erm as good approximae soluion. The phase porrais of he SHAM soluion and GABMM soluion are given in Fig. 3 and Fig. 4 a differen fracional derivaive. The figure gives ha SHAM soluion have good agreemen wih GABMM soluion. z (a) y z x (b) y x Figure 3: Comparison he phase porrais of x y z using 2-erm SHAM in (b) wih GABMM in (a) when (α α 2 α 3 ) as (.9.9.9) z z (a) y x (b) y x Figure 4: Comparison he phase porrais of x y z using 2-erm SHAM in (b) wih GABMM in (a) when (α α 2 α 3 ) as (.9.9.9) and The residual error of he SHAM soluion is presened in Fig. and 6 for (.9.9.9) ( ) respecively. Table () and (2) give he residual error for he given soluion a several poins. We observe ha a higher accuracy of he given soluion is cied which is no exended han. On he oher hand he GABMM usually used accuracy wih 6.

9 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) E x E y E z (a) (b) (c) Figure : Residual error for SHAM soluion using 8-erms wih =. when (α α 2 α 3 ) as (.9.9.9) E z E z E z (a) (b) (c) Figure 6: Residual error for SHAM soluion using 8-erms wih =. when (α α 2 α 3 ) as ( ) Table : The Residual error for α i =.9 wih 8-erms and =. using 2 Digi E x E y E z.68434e e e E E E E E E E E E E E E E E E- Table 2: The Residual error for α i =.99 wih 8-erms and =. using 2 Digi E x E y E z E E E E E E E-4.429E-3.429E E-3-4.9E E E E E-4.429E E E E-3. Conclusions In his presen work coninuous soluion for fracional Chen sysem is obained by SHAM. The modified mehod has he advanage of giving an analyical form of he soluion wihin each ime inerval which is no possible in purely numerical echniques like fourh order Runge Kua mehod RK4 or ABMM. The residual error for subinervals soluion is defined and calculaed. We also noe ha he SHAM soluions

10 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) were compued via a simple algorihm wihou any need for perurbaion echniques special ransformaions linearizaion or discreizaion. The SHAM soluions are in excellen agreemen wih he GABMM soluion. Moreover The HPM and MHPM soluion is a special case when = and α = α 2 = α 3 =. Acknowledgemens The auhor would like o hank he edior and reviewers for heir valuable commens and suggesions o improve he paper. Also we hank he research council a Yarmouk universiy for heir financial suppor. References [] S. Abbasbandy The applicaion of homoopy analysis mehod o solve a generalized Hiroa-Sasuma coupled KdV equaion Phys. Le. A 36 (27) [2] A. K. Alomari M. S. M. Noorani R. Nazar On he homoopy analysis mehod for he exac soluions of Helmholz equaion Chaos Solions Fracal 4 (29) [3] A. K. Alomari F. Awawdeh N. Taha F. Bani Ahmad Shaanawi W Muliple soluions for fracional differenial equaions: Analyic approach Appl. Mah. Compu. 29 (23) [4] A. K. Alomari M. S. N. Noorani R. Nazar Adapaion of homoopy analysis mehod for he numeric-analyic soluion of Chen sysem Comm. Nonlinear Sci. Numer. Simul. 4 (29) [] A. K. Alomari M. S. N. Noorani R. Nazar C. P. Li Homoopy analysis mehod for solving fracional Lorenz sysem Comm. Nonlinear Sci. Numer. Simul. (2) [6] M. A. F. Araghi A. Fallahzadeh Discree Homoopy Analysis Mehod for Solving Linear Fuzzy Differenial Equaions Adv. Environ. Biol. 9 (2) 9 2. [7] A. S. Baaineh M. S. N. Noorani I. Hashim Solving sysems of ODEs by homoopy analysis mehod Comm. Nonlinear Sci. Numer. Simul. 3 (28) [8] M. S. H. Chowdhury I. Hashim Applicaion of mulisage homoopy-perurbaion mehod for he soluions of he Chen sysem Nonlinear Anal. Real. World Appl. (29) [9] K. Diehelm N. J. Ford Analysis of fracional differenial equaions J. Mah. Anal. Appl. 26 (22) [] K. Diehelm N. J. Ford A. D. Freed A predicor-correcor approach for he numerical soluion of fracional differenial equaions Nonlinear Dyn. 29 (22) [] A. Fallahzadeh K. Shakibi A mehod o solve Convecion-Diffusion equaion based on homoopy analysis mehod J. Inerpola. Approx. Sci. Compu. 2 (2) 8 pages. [2] A. Golbabai K. Sayevand An efficien applicaions of hes variaional ieraion mehod based on a reliable modificaion of Adomian algorihm for nonlinear boundary value problems J. Nonlinear Sci. Appl. 3 (2) 2 6. [3] T. Haya M. Sajid On analyic soluion for hin film flow of a fourh grade fluid down a verical cylinder Phys. Le. A 36 (27) [4] T. Haya M. Khan Homoopy soluions for a generalized second-grade fluid pas a porous plae Nonlinear Dyn. 42 (2) [] H. Jafari V. Dafardar-Gejji Solving a sysem of nonlinear fracional differenial equaions using Adomian decomposiion J. Compu. Appl. Mah. 96 (26) [6] C. Li C G. Peng Chaos in Chen s sysem wih a fracional order Chaos Solions Fracals 22 (24) [7] S. Liao The proposed homoopy analysis echniques for he soluion of nonlinear problems Ph.D. Disseraion. Shanghai Jiao Tong Universiy Shanghai (in English) (992). [8] S. Liao Beyond perurbaion: Inroducion o he homoopy analysis mehod. CRC Press Chapman and Hall Boca Raon (23). [9] S. Liao Noes on he homoopy analysis mehod: Some definiions and heorems Commun. Nonlinear Sci. Numer. Simul. 4 (29) [2] Y. Liu Y H. Shi Exisence of unbounded posiive soluions for BVPs of singular fracional differenial equaions J. Nonlinear Sci. Appl. (22) [2] S. Momani Z. Odiba Homoopy perurbaion mehod for nonlinear parial differenial equaions of fracional order Phys. Le. A. 36 (27) [22] S. Momani Z. Odiba Numerical comparison of mehods for solving linear differenial equaions of fracional order Chaos Solions Fracals 3 (27) [23] M. Inc On exac soluion of Laplace equaion wih Dirichle and Neumann boundary condiions by he homoopy analysis mehod Phys. Le. A. 36 (27) [24] Z. Odiba S. Momani Modified homoopy perurbaion mehod: applicaion o quadraic Riccai differenial equaion of fracional order Chaos Solions Fracals 36 (28)

11 A. K. Alomari J. Nonlinear Sci. Appl. 8 (2) [2] Z. Odiba S. Momani Applicaion of variaional ieraion mehod o nonlinear differenial equaion of fracional order In. J. Nonlinear Sci. Numer. Simul. 7 (26) [26] I. Podlubny Fracional differenial equaions Academic Press New York (999). 2. [27] T. Qiu Z. Bai Posiive soluions for boundary value problem of nonlinear fracional differenial equaion J. Nonlinear Sci. Appl. (28) [28] M. Sajid T. Javed T. Haya MHD roaing flow of a viscous fluid over a shrinking surface Nonlinear Dyn. (28) [29] N. T. Shawagfeh Analyical approximae soluions for nonlinear fracional differenial equaions Appl. Mah. Compu. 3 (22) [3] T. Wang F. Xie Exisence and uniqueness of fracional differenial equaions wih inegral boundary condiions J. Nonlinear Sci. Appl. (28) [3] Y. Wang Y. Yang Posiive soluions for Capuo fracional differenial equaions involving inegral boundary condiions J. Nonlinear Sci. Appl. 8 (2) [32] J. Wanga X. Xionga Y. Zhang Exending synchronizaion scheme o chaoic fracional-order Chen sysems Physica A 37 (26) [33] W. Yang Posiive soluions for singular coupled inegral boundary value problems of nonlinear Hadamard fracional differenial equaions J. Nonlinear Sci. Appl. 8 (2) 29. [34] T. S. Zhou C. Li Synchronizaion in fracional-order differenial sysems Phys. D. 22 (2) 2. [3] H. Zhu S. Zhou Z. He Chaos synchronizaion of he fracional-order Chen s sysem Chaos Solions Fracals 4 (29)