Gepreel Advnces in Difference Equtions 20, 20:286 R E S E A R C H Open Access Explicit Jcobi elliptic exct solutions for nonliner prtil frctionl differentil equtions Khled A Gepreel * * Correspondence: kgepreel@yhoo.com Mthemtics Deprtment, Fculty of Science, Zgzig University, Zgzig, Egypt Mthemtics Deprtment, Fculty of Science, Tif University, Tif, Sudi Arbi Abstrct In this rticle, we use the frctionl complex trnsformtion to convert nonliner prtil frctionl differentil equtions to nonliner ordinry differentil equtions. An lgebric method is improved to construct uniformly series of exct solutions for some nonliner time-spce frctionl prtil differentil equtions. We construct successfully series of some exct solutions including the elliptic doubly periodic solutions with the id of computerized symbolic computtion softwre pckge such s Mple or Mthemtic. This method is efficient nd powerful in solving wide clsses of nonliner prtil frctionl differentil equtions. The Jcobi elliptic doubly periodic solutions re generted by the trigonometric exct solutions nd the hyperbolic exct solutions when the modulus m 0ndm, respectively. Keywords: frctionl clculus; frctionl complex trnsformtion; modified Riemnn-Liouville derivtive; Jcobi elliptic functions; nonliner frctionl differentil equtions Introduction Nonliner prtil frctionl equtions re very effective for the description of mny physicl phenomen such s rheology, the dmping lw, diffusion processes, nd the nonliner oscilltion of n erthquke cn be modeled with frctionl derivtives [, 2]. Also mny pplictions of nonliner prtil frctionl differentil equtions cn be found in turbulence nd fluid dynmics nd nonliner biologicl systems [ 0]. There re mny methods for finding the pproximte solutions for nonliner prtil frctionl differentil equtions such s the Adomin decomposition method [5, 6], the vritionl itertion method [7], the homotopy perturbtion method [8, 9], nd the homotopy nlysis method [3, 0, ] nd so on. No nlyticl method hd been vilble before 998 for nonliner frctionl differentil equtions. Li nd He [2] hve proposed the frctionl complex trnsformtion to convert the nonliner prtil frctionl differentil equtions into ordinry differentil equtions so tht ll nlyticl methods devoted to dvnced clculus cn be pplied to frctionl clculus. Recently Zhng nd Zhng [3]hveintroduced direct method clled the sub-eqution method to look for the exct solutions for nonliner prtil frctionl differentil equtions. He [] hs extended the exp-function method to frctionl prtil differentil equtions in the sense of modified Riemnn- Liouville derivtive bsed on the frctionl complex trnsform. There re mny methods 20 Gepreel; licensee Springer. This is n Open Access rticle distributed under the terms of the Cretive Commons Attribution License (http://cretivecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 2 of for solving the nonliner prtil frctionl differentil equtions such s those in [5, 6]. Fn [7], Zyed et l. [8] nd Hong nd Lu [9, 20]hveproposednlgebricmethodfor nonliner prtil differentil equtions to obtin series of exct wve solutions including the soliton, nd rtionl, tringulr periodic, Jcobin, nd Weierstrss doubly periodic solutions. In this pper, we will improve the extended proposed lgebric method to solve the nonliner prtil frctionl differentil equtions. Also we use the improved extended proposed lgebric method to construct the Jcobi elliptic exct solutions for the following nonliner time-spce prtil frctionl differentil equtions: (i) First, we hve the spce-time frctionl derivtive nonliner Korteweg-de Vries (KdV) eqution [2] α u t u α u x 3 u =0, t >0,0<α,, () x3 where is constnt. When α,, the KdV eqution hs been used to describe wide rnge of physics phenomen of the evolution nd interction of nonliner wves. It ws derived from the propgtion of dispersive shllow wter wves nd is widely used in fluid dynmics, erodynmics, nd continuum mechnics, nd s model for shock wve formtion, solitons, turbulence, boundry lyer behvior, mss trnsport, nd the solution representing the wter s free surfce over flt bed [22]. (ii) Then we hve the spce-time frctionl derivtive nonliner frctionl Zkhrov- Kunzetsov-Benjmin-Bon-Mhony (ZKBBM) eqution [23], α u t α u x 2u u x b α t α ( 2 u x 2 ) =0, t >0,0<α,, (2) where, b re rbitrry constnts. When α,, this system hs been investigted by Benjmin et l. [23] for the first time, s n lterntive model to the KdV eqution for long wves nd it plys n importnt role in the modeling of nonliner dispersive systems. The Benjmin-Bon-Mhony eqution is pplicble to the study of drift wves in plsm or Rossby wves in rotting fluids. 2 Preliminries There re mny types of the frctionl derivtives such s the Kolwnkr-Gngl locl frctionl derivtive [2], Chen s frctl derivtive [25], Cresson s derivtive [26], nd Jumrie s modified Riemnn-Liouville derivtive [27, 28]. In this section, we give some bsic definitions of frctionl clculus theory which will be used in this work. Jumrie s modified Riemnn-Liouville derivtive of order α is defined s [28, 29] D α x f (x)= d Ɣ( α) dx x 0 (x ξ) α( f (ξ) f (0) ) dξ, 0<α <, (3) where f : R R, x f (x) denotes continuous (but not necessrily first order differentible) function. Also the inverse of Jumrie s modified Riemnn-Liouville derivtive to f (x)oforderα in the intervl [, b] is defined by I α x f (x)= Ɣ(α) x 0 (x ξ) α f (ξ) dξ = Ɣ(α ) x 0 f (ξ)(dx) α, 0<α. ()
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 3 of Some properties for the proposed Jumrie s modified Riemnn-Liouville derivtive re listed from [27 30] s follows: f (α)[ x(t) ] = df dx x(α) (t), (5) D α d x x = Ɣ( α) dx x 0 ξ (x ξ) α dξ = B(, α) Ɣ( α) d dx ( x α ) Ɣ( ) d ( = x α ) Ɣ( ) = Ɣ( α 2) dx Ɣ( α ) x α, > α, (6) D α x ( f (x) g(x) ) = g(x)d α x ( f (x) ) f (x)d α x ( g(x) ), (7) where B is the bet function. The function f (x) should be differentible with respect to x(t) nd x(t) is frctionl differentible in (6).The bove resultsre employed in the following sections. The Leibniz rule is given (7) for modified Riemnn-Liouville derivtive which is modified by Wu in [30]. The modified Riemnn-Liouville derivtive hs been successfully pplied in probbility clculus [3], frctionl Lplce problems [32], the frctionl vritionl pproch with severl vribles [33], the frctionl vritionl itertion method [3], the frctionl vritionl pproch with nturl boundry conditions [35], nd the frctionl Lie group method [36]. 3 The improved extended proposed lgebric method for nonliner prtil frctionl differentil equtions Consider the following nonliner prtil frctionl differentil eqution: F ( u, D α t u, D x u, Dγ y u, Dδ z u, Dα t Dα t u, Dα t D x u, D x D x u, D x Dγ y, Dγ y Dγ y u,...) =0, 0<α,, δ, γ, (8) where u is n unknown function, F is polynomil in u nd its prtil frctionl derivtives in which the highest order frctionl derivtives nd the nonliner terms re involved. We give the min steps of the modified extended proposed lgebric method for nonliner prtil frctionl differentil equtions. Step. The frctionl complex trnsformtion u(x, y, z, t)=u(ξ), ξ = Kx Ɣ( ) Nyγ Ɣ(γ ) Mzδ Ɣ(δ ) Lt α Ɣ(α ), (9) where K, L, M,ndN re nonzero rbitrry constnts, permits us to reduce (8) to nonliner ODE for u = u(ξ) in the following form: P ( u, u, u, u,... ) =0. (0) If possible, we should integrte (0) term by term one or more times. Step 2. Suppose the solutions of (0) cn be expressed by polynomil in φ in the form [7, 8]: u(ξ)= m α i φ i (ξ), α m 0, () i= m
GepreelAdvnces in Difference Equtions 20, 20:286 Pge of Tble The exct solutions of the Jcobi elliptic differentil eqution (2)whene 0, e nd e 2 tke specil vlues e 0 e e 2 φ(ξ) (m 2 ) m 2 sn(ξ)orcd(ξ) m 2 2m 2 m 2 cn(ξ) m 2 2 m 2 dn(ξ) m 2 (m 2 ) ns(ξ)ordc(ξ) m 2 2m 2 m 2 nc(ξ) 2 m 2 m 2 nd(ξ) m 2 2 m 2 cs(ξ) 2 m 2 m 2 sc(ξ) 2m 2 m 2 (m 2 ) sd(ξ) m 2 (m 2 ) 2m 2 ds(ξ) 2 ( 2m2 ) ns(ξ) ± cs(ξ) ( m2 ) 2 ( m2 ) ( m2 ) nc(ξ) ± sc(ξ) m 2 2 (m2 m 2) sn(ξ) ± icn(ξ) m 2 2 (m2 2) ns(ξ) ± ds(ξ) m 2 2 (m2 m 2) m 2 sd(ξ) ± cd(ξ) 2 ( m2 ) mcd(ξ) ± i m 2 nd(ξ) 2 ( 2m2 ) msn(ξ) ± idn(ξ) 2 ( m2 ) m 2 sc(ξ) ± dc(ξ) (m2 ) 2 ( m2 ) (m2 ) msd(ξ) ± nd(ξ) 2 (m2 2) m 2 sn(ξ)/( ± dn(ξ)) where α i (i =0,±,...,±m) re rbitrry constnts to be determined lter, while φ(ξ)stisfies the following nonliner first order Jcobi elliptic differentil eqution: [ φ (ξ) ] 2 = e0 e φ 2 (ξ)e 2 φ (ξ), (2) where e 0, e,nde 2 re rbitrry constnts. Step 3. The positive integer m cn be determined by considering the homogeneous blnce between the highest order derivtives nd the nonliner terms ppering in Eq. (0). Step. We must substitute () into(0)ndusing(2), collect ll terms with the sme order of φ(ξ) together, then equting ech coefficient of the resulting polynomil to be zero.thisyieldssetoflgebricequtionsforα i (i =0,±,...,±m), e 0, e, e 2, K, L, M, nd N. We then solve this system of lgebric eqution with the help of Mple softwre pckge to determine α i (i =0,±,...,±m), e 0, e, e 2, K, L, M,ndN. Step 5. The generl solutions of (2) hve been discussed in [37, 38]. We put some of the generl solutions of (2)from[37]inTble. There re other cses which re omitted here for convenience; see [37]. Step 6. Since the generl solutions of (2) re discussed in Tble, then substituting α i (i =0,±,...,±m), e 0, e, e 2, K, L, M,ndN nd the generl solutions of (2)into(), we obtin more new Jcobi elliptic exct solutions for the nonliner prtil frctionl derivtive eqution (8). Applictions In this section, we construct some new Jcobi elliptic exct solutions of some nonliner prtil frctionl differentil equtions vi the time-spce frctionl nonliner KdV eqution nd the time-spce frctionl nonliner Zkhrov-Kunzetsov-Benjmin-Bon- Mhomy eqution using the modified extended proposed lgebric method which hs been pid ttention to by mny uthors.
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 5 of. Exmple : Jcobi elliptic solutions for nonliner frctionl KdV eqution In this section, to demonstrte the effectiveness of this method, we use the complex trnsformtion (9) to converting the nonliner KdV eqution with time-spce frctionl derivtives () to n ordinry differentil eqution; nd we integrte twice, to find 2 LU2 6 KU3 K 3 [ U ] 2 C U C 2 = 0, (3) 2 where C nd C 2 re the integrtion constnts. Considering the homogeneous blnce between the highest order derivtives nd the nonliner terms in (3), we get U(ξ)=α 0 α φ(ξ)α 2 φ 2 (ξ) α 3 φ(ξ) α φ 2 (ξ), () where α 0, α, α 2, 3,, L, ndk re rbitrry constnts to be determined lter. Substituting () nd(2) into(3), collectingll termsof φ(ξ), nd then setting ech coefficient φ(ξ) to be zero, we get system of lgebric equtions. With the id of Mple or Mthemtic we cn solve this system of lgebric equtions to obtin the following cses of solutions: Cse. 2C (L K 3 e ) 2C K 3 e 2 0 = 92K 6 e 0 e 2 L 2 6K 6 e 2, 2 = 92K 6 e 0 e 2 L 2 6K 6 e 2, = 92K 6 e 0 e 2 L 2 6K 6 e 2 KC, = 2C K 3 e 0 92K 6 e 0 e 2 L 2 6K 6 e 2, C 2 = 2C2 ( 576Le 2K e 0,608K 9 e 0 e e 2 L 3 8K 6 e 2 L 28K 9 e 3 ) 3(92K 6 e 0 e 2 L 2 6K 6 e 2 )2, (5) = 3 =0, where C, L, K, e 0, e,nde 2 re rbitrry constnts. Cse 2. C = 2( 92K 6 e 0 e 2 L 2 6K 6 e 2 ) 6C 2 576Le 2 K 6 e 0,608K 9 e 2 e e 0 L 3 8K 6 e 2 L 28K9 e 3, 0 = ( L K 3 ) 6C 2 e 576Le 2 K 6 e 0,608K 9 e 2 e e 0 L 3 8K 6 e 2 L 28K9 e 3, 2 =2K 3 6C 2 e 2 576Le 2 K 6 e 0,608K 9 e 2 e e 0 L 3 8K 6 e 2 L 28K9 e 3, (6) = 2K 6 [ 576Le 2 K 6 e 0,608K 9 e 2 e e 0 L 3 8K 6 e 2 C L 28K9 e] 3, 2 =2K 3 6C 2 e 0 576Le 2 K 6 e 0,608K 9 e 2 e e 0 L 3 8K 6 e 2 L 28K9 e 3, = 3 =0, where e 0, e, e 2, K, L,ndC 2 re rbitrry constnts.
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 6 of Note tht there re other cses which re omitted here for convenience. Since the solutions obtined here re so mny, we just list some of the Jcobi exct solutions corresponding to Cse to illustrte the effectiveness of the proposed method. Substituting (5) into ()wehve 2C (L K 3 e ) u = 92K 6 e 0 e 2 L 2 6K 6 e 2 2C K 3 e 2 92K 6 e 0 e 2 L 2 6K 6 e 2 φ 2 (ξ) 2C K 3 e 0 [92K 6 e 0 e 2 L 2 6K 6 e 2 ]φ2 (ξ), (7) where ξ = Kx Ɣ( ) Lt α Ɣ(α ). According to the generl solutions of (2) which re discussed in Tble, we hve the following fmilies of exct solutions: Fmily. If e 0 =,e = (m 2 ), e 2 = m 2 the exct trveling wve solution tkes the form u = 2C [L K 3 ( m 2 )] 92K 6 m 2 L 2 6K 6 ( m 2 ) 2 2C K 3 m 2 92K 6 m 2 L 2 6K 6 ( m 2 ) 2 sn2 2C K 3 [92K 6 m 2 L 2 6K 6 ( m 2 ) 2 ] ns2 [ Kx Ɣ( ) [ Kx Ɣ( ) Lt α ] Ɣ(α ) Lt α Ɣ(α ) To illustrte the behvior of the Jcobi elliptic solution u (8), see Figure. Furthermore, u 2 = 2C [L K 3 ( m 2 )] 92K 6 m 2 L 2 6K 6 ( m 2 ) 2 2C K 3 m 2 92K 6 m 2 L 2 6K 6 ( m 2 ) 2 cd2 2C K 3 [92K 6 m 2 L 2 6K 6 ( m 2 ) 2 ] dc2 [ Kx Ɣ( ) [ Kx Ɣ( ) Lt α ] Ɣ(α ) Lt α Ɣ(α ) ]. (8) ], (9) Figure The Jcobi elliptic doubly periodic solution u (8) nd its position t t = 0, when the prmeters C =,L =2,K =3,m =0.2,α =0.5,nd =0.6.
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 7 of Figure 2 The Jcobi elliptic doubly periodic solution u 3 (20) nd its position t t = 0, when the prmeters C =,L =2,K =3,m =0.5,α =0.5,nd =0.6. where C 2 = 2C2 [ 576Lm2 K,608K 9 m 2 (m 2 )L 3 8K 6 (m 2 ) 2 L28K 9 (m 2 ) 3 ] 3[92K 6 m 2 L 2 6K 6 (m 2 ) 2 ] 2. Fmily 2. If e 0 = m 2, e =2m 2,e 2 = m 2, the exct trveling wve solution tkes the form u 3 = 2C (L K 3 (2m 2 )) { 92K 6 ( m 2 )m 2 L 2 6K 6 (2m 2 ) 2 } 2C K 3 m 2 cn 2 [ Kx { 92K 6 ( m 2 )m 2 L 2 6K 6 (2m 2 ) 2 } Ɣ() Ɣ(α) ] 2C K 3 ( m 2 )nc 2 [ Kx { 92K 6 ( m 2 )m 2 L 2 6K 6 (2m 2 ) 2 }, (20) Ɣ() Ɣ(α) ] where C 2 = {2C 2 3[ 92K 6 m 2 ( m 2 ) L 2 6K 6 (2m 2 ) 2 ] 2 (576Lm2 K (2m 2 ),608K 9 m 2 (2m 2 )( m 2 )L 3 8K 6 (2m 2 ) 2 L 28K 9 (2m 2 ) 3 )}. To illustrte the behvior of the Jcobi elliptic solution u 3 (20), see Figure 2. Fmily 3. If e 0 = m 2,e =2 m 2, e 2 =, the exct trveling wve solution tkes the form u = 2C [L K 3 (2 m 2 )] [92K 6 ( m 2 ) L 2 6K 6 (2 m 2 ) 2 ] 2C K 3 dn 2 [ Kx [92K 6 ( m 2 ) L 2 6K 6 (2 m 2 ) 2 ] Ɣ() Ɣ(α) ] Ɣ() Ɣ(α) ] 2C K 3 (m 2 )nd 2 [ Kx [92K 6 ( m 2 ) L 2 6K 6 (2 m 2 ) 2 ], (2) where C 2 = 2C2 (576LK (m 2 ),608K 9 (m 2 )(2 m 2 )L 3 8K 6 (2 m 2 ) 2 L 28K 9 (2 m 2 ) 3 ) 3[92K 6 ( m 2 ) L 2 6K 6 (2 m 2 ) 2 ] 2. Fmily. If e 0 = ( m2 ), e = 2 (m2 ), e 2 = ( m2 ), the exct trveling wve solution tkes the form u 5 = 2C (L 2K 3 ( m 2 )) [2K 6 ( m 2 ) 2 L 2 K 6 ( m 2 ) 2 ] 6C K 3 ( m 2 )[nc( Kx Ɣ() Kx ) ± sc( Ɣ(α) Ɣ() [2K 6 ( m 2 ) 2 L 2 K 6 ( m 2 ) 2 ] Ɣ(α) )]2
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 8 of Figure 3 The Jcobi elliptic doubly periodic solution u 5 (22) nd its position t t = 0, when the prmeters C =,L =2,K =3,m =0.5,α =0.5,nd =0.6. 6C K 3 ( m 2 ) [2K 6 ( m 2 ) 2 L 2 K 6 ( m 2 ) 2 ] [nc( Kx Ɣ() Kx ) ± sc( Ɣ(α) Ɣ(), (22) )]2 Ɣ(α) where C 2 = 2C2 ( 36LK ( m 2 ) 2 K 9 ( m 2 ) 2 (m 2 )L 3 2K 6 (m 2 ) 2 L 6K 9 (m 2 ) 3 ) 3[2K 6 ( m 2 ) 2 L 2 K 6 (m 2 ) 2 ] 2. To illustrte the behvior of the Jcobi elliptic solution u 5 (22), see Figure 3. Fmily 5. If e 0 = m2, e = 2 (m2 2),e 2 = m2, the exct trveling wve solution tkes the form u 6 = 2C (L 2K 3 (m 2 2)) 2K 6 m L 2 K 6 (m 2 2) 2 6C K 3 m 2 [sn( Kx Ɣ() Kx ) ± icn( Ɣ(α) [2K 6 m L 2 K 6 (m 2 2) 2 ] 2C K 3 m 2 [2K 6 m L 2 K 6 (m 2 2) 2 ] [sn( Kx Ɣ() Kx ) ± icn( Ɣ(α) Ɣ() Ɣ() Ɣ(α) )]2, (23) )]2 Ɣ(α) where C 2 = 2C2 ( 36LK m K 9 m (m 2 2)L 3 2K 6 (m 2 2) 2 L 6K 9 (m 2 2) 3 ) 3[2K 6 m L 2 K 6 (m 2 2) 2 ] 2. To illustrte the behvior of the Jcobi elliptic solution u 6 (23), see Figures nd 5. Similrly, we cn write down the other fmilies of exct solutions of ()whichreomitted for convenience..2 Exmple 2: Jcobi elliptic solutions for nonliner frctionl ZKBBM eqution In this section we use the proposed method to find the Jcobi elliptic solutions for the nonliner frctionl ZKBBM eqution with time nd spce frctionl derivtives (2). The complex frctionl trnsformtions (9) convert the nonliner frctionl ZKBBM eqution (2) to the following nonliner ordinry differentil eqution: (L K)U KU 2 bk 2 U C =0, (2)
GepreelAdvnces in Difference Equtions 20, 20:286 Pge 9 of Figure The rel prt of Jcobi elliptic doubly periodic solution u 6 (23) nd its position t t =0,when the prmeters C =,L =2,K =3,m =0.6,α =0.5,nd =0.6. Figure 5 The imginry prt of Jcobi elliptic doubly periodic solution u 6 (23) nd its position t t =0, when the prmeters C =,L =2,K =3,m =0.6,α =0.5,nd =0.6. where C is the integrtion constnt. Considering the homogeneous blnce between the highest order derivtive nd the nonliner term in (2), we hve U(ξ)=α 0 α φ(ξ)α 2 φ 2 (ξ) α 3 φ(ξ) α φ 2 (ξ), (25) where α 0, α, α 2, 3,, L, ndk re rbitrry constnts to be determined lter. Substituting (25) nd(2) into(2), collecting ll the terms of powers of φ(ξ) ndsettingech coefficient φ(ξ) to zero, we get system of lgebric equtions. With the id of Mple or Mthemtic we cn solve this system of lgebric equtions to obtin the following sets of solutions: Cse. 0 = bk 2 Le K L, 2 = 6KbLe 2, = 6KbLe 0, C = { K 2 2LK 92b 2 K L 2 e 0 e 2 L 2 6b 2 K L 2 e 2 }, (26) K = 3 =0, where b, L, K, e 0, e,nde 2 re rbitrry nonzero constnts. There re mny other cses which re omitted for convenience.
Gepreel Advnces in Difference Equtions 20, 20:286 Pge 0 of Substituting (26)into(25)wehve where u = bk 2 Le K L 6KbLe 2 φ 2 (ξ) 6KbLe 0 φ 2 (ξ), (27) ξ = Kx Ɣ( ) Lt α Ɣ(α ) (28) nd C = { K 2 2LK 92b 2 K L 2 e 0 e 2 L 2 6b 2 K L 2 e 2 }. K According to the generl solutions of (2) which re discussed in Tble, we hve the following fmilies of Jcobi elliptic exct solutions to the nonliner ZKBBM eqution: Fmily. If e 0 =,e = (m 2 ), e 2 = m 2, the exct trveling wve solution tkes the form u = bk 2 L( m 2 [ )K L 6KbLm2 Kx sn 2 Ɣ( ) 6KbL [ Kx ns2 Ɣ( ) Lt α Ɣ(α ) Lt α ] Ɣ(α ) ]. (29) To illustrte the behvior of the Jcobi elliptic solution u (29), see Figure 6. Furthermore u 2 = bk 2 L( m 2 [ )K L 6KbLm2 Kx cd 2 Ɣ( ) 6KbL [ Kx dc2 Ɣ( ) Lt α Ɣ(α ) Lt α ] Ɣ(α ) ], (30) where C = K { K 2 2LK 92b 2 K L 2 m 2 L 2 6b 2 K L 2 ( m 2 ) 2 }. Figure 6 The Jcobi elliptic doubly periodic solution u (29) nd its position t t = 0, when the prmeters =,b =2,L =3,K =,m =0.2,α =0.5,nd =0.6.
Gepreel Advnces in Difference Equtions 20, 20:286 Pge of Figure 7 The Jcobi elliptic doubly periodic solution u 3 (3) nd its position t t = 0, when the prmeters =,b =2,L =3,K =,m =0.2,α =0.5,nd =0.6. Fmily 2. If e 0 = m 2, e =2m 2,e 2 = m 2, the exct trveling wve solution tkes the form u 3 = bk 2 L(2m 2 ) K L 6KbL( ( m2 ) nc 2 6KbLm2 Kx Ɣ( ) Lt α Ɣ(α ) ( Kx cn 2 Ɣ( ) Lt α ) Ɣ(α ) ), (3) where C = K { K 2 2LK 92b 2 K L 2 ( m 2 )m 2 L 2 6b 2 K L 2 (2m 2 ) 2 }. To illustrte the behvior of the Jcobi elliptic solution u 3 (3), see Figure 7. Fmily 3. If e 0 = m 2,e =2 m 2, e 2 =, the exct trveling wve solution tkes the form u = bk 2 L(2 m 2 ) K L [ 6KbL(m2 ) nd 2 6KbL [ Kx dn2 Ɣ( ) Kx Ɣ( ) Lt α Ɣ(α ) Lt α ] Ɣ(α ) ], (32) where C = K { K 2 2LK 92b 2 K L 2 (m 2 ) L 2 6b 2 K L 2 (2 m 2 ) 2 }. Fmily. If e 0 = ( m2 ), e = 2 (m2 ), e 2 = ( m2 ), the exct trveling wve solution tkes the form u 5 = 2bK 2 L( m 2 ) K L 3KbL( m2 ) 2 2[nc( Kx Ɣ() [ ( Kx nc Ɣ( ) 3KbL( m 2 ) Kx ) ± sc( Ɣ(α) Ɣ() Lt α ) ( Kx ± sc Ɣ(α ) Ɣ( ) Lt α )] 2 Ɣ(α ), (33) )]2 Ɣ(α) where C = K { K 2 2LK 2b 2 K L 2 ( m 2 ) 2 L 2 b 2 K L 2 ( m 2 ) 2 }. To illustrte the behvior of the Jcobi elliptic solution u 5 (33), see Figure 8.
Gepreel Advnces in Difference Equtions 20, 20:286 Pge 2 of Figure 8 The Jcobi elliptic doubly periodic solution u (32) nd its position t t = 0, when the prmeters =,b =2,L =3,K =,m =0.2,α =0.5,nd =0.6. Figure 9 The rel prt of Jcobi elliptic doubly periodic solution u 6 (3) nd its position t t =0,when the prmeters =,b =2,L =3,K =,m =0.2,α =0.5,nd =0.6. Fmily 5. If e 0 = m2, e = 2 (m2 2),e 2 = m2, the exct trveling wve solution tkes the form u 6 = 2bK 2 L(m 2 2) K L [ ( 3KbLm2 Kx sn 2 Ɣ( ) 2[sn( Kx Ɣ() 3KbLm 2 Lt α ) ( Kx ± icn Ɣ(α ) Ɣ( ) Kx ) ± icn( Ɣ(α) Ɣ() Lt α )] 2 Ɣ(α ), (3) )]2 Ɣ(α) where C = K { K 2 2LK 2b 2 K L 2 m L 2 b 2 K L 2 (m 2 2) 2 }. To illustrte the behvior of the Jcobi elliptic solution u 6 (3), see Figures 9 nd 0. Similrly, we cn write down the other fmilies of exct solutions of (2)whichreomitted for convenience. 5 Conclusion In this rticle we constructed the Jcobi elliptic exct solutions for the nonliner prtil frctionl differentil equtions with the help of the complex frctionl trnsformtion nd the improved extended proposed lgebric method. This method is effective nd powerful for finding the Jcobi elliptic solutions for nonliner frctionl differentil equtions.
Gepreel Advnces in Difference Equtions 20, 20:286 Pge 3 of Figure 0 The imginry prt of Jcobi elliptic doubly periodic solution u 6 (3) nd its position t t = 0, when the prmeters =,b =2,L =3,K =,m =0.2,α =0.5,nd =0.6. Jcobi elliptic solutions re generlized to the hyperbolic exct solutions nd trigonometric exct solutions when the modulus m ndm 0, respectively. Competing interests Theuthordeclrestohvenocompetinginterests. Author s contributions The uthor red nd pproved the finl mnuscript. Acknowledgements The uthor would like to thnk the editors nd the nonymous referees for their creful reding of the pper. Received: 20 August 20 Accepted: 2 October 20 Published: 0 Nov 20 References. Podlubny, I: Frctionl Differentil Equtions. Acdemic Press, Sn Diego (999) 2. He, JH: Some pplictions of nonliner frctionl differentil equtions nd their pplictions. Bull. Sci. Technol. 5, 86-90 (999) 3. Mohmed, SM, Al-Mlki, F, Tlib, R: Approximte nlyticl nd numericl solutions to frctionl Newell-Whitehed eqution by frctionl complex trnsform. Int. J. Appl. Mth. 26, 657-669 (203). Erturk, VS, Odibt, SM: Appliction of generlized differentil trnsform method to multi-order frctionl differentil equtions. Commun. Nonliner Sci. Numer. Simul. 3, 62-65 (2008) 5. Dftrdr-Gejji, V, Bhlekr, S: Solving multi-term liner nd non-liner diffusion wve equtions of frctionl order by Adomin decomposition method. Appl. Mth. Comput. 202, 3-20 (2008) 6. Herzllh, MA, Gepreel, KA: Approximte solution to the time-spce frctionl cubic nonliner Schrodinger eqution. Appl. Mth. Model. 36, 5678-5685 (202) 7. Sweilm, NH, Khder, MM, Al-Br, RF: Numericl studies for multi-order frctionl differentil eqution. Phys. Lett. A 37, 26-33 (2007) 8. Golbbi, A, Syevnd, K: Frctionl clculus - new pproch to the nlysis of generlized fourth-order diffusion-wve equtions. Comput. Mth. Appl. 6, 2227-223 (20) 9. Gepreel, KA: The homotopy perturbtion method to the nonliner frctionl Kolmogorov-Petrovskii-Piskunov equtions. Appl. Mth. Lett. 2,28-3 (20) 0. Gepreel, KA, Mohmed, SM: Anlyticl pproximte solution for nonliner spce-time frctionl Klein-Gordon eqution. Chin. Phys. B 22, 0020 (203). Mohmed, SM, Al-Mlki, F, Tlib, R: Jcobi elliptic numericl solutions for the time frctionl dispersive long wve eqution. Int. J. Pure Appl. Mth. 80, 635-66 (202) 2. Li, ZB, He, JH: Frctionl complex trnsformtion for frctionl differentil equtions. Mth. Comput. Appl. 5, 970-973 (200) 3. Zhng, S, Zhng, HQ: Frctionl sub-eqution method nd its pplictions to nonliner frctionl PDEs. Phys. Lett. A 375, 069-073 (20). He, JH: Exp-function method for frctionl differentil equtions. Int. J. Nonliner Sci. Numer. Simul. 3, 363-366 (203) 5. Gepreel, KA, Al-Thobiti, AA: Exct solution of nonliner prtil frctionl differentil equtions using the frctionl sub-eqution method. Indin J. Phys. 88, 293-300 (20) 6. Bekir, A, Güner, O: Exct solutions of nonliner frctionl differentil equtions by (G /G)-expnsion method. Chin. Phys. B 22, 0202 (203) 7. Fn, E: Multiple trvelling wve solutions of nonliner evolution equtions using unified lgebric method. J. Phys. A, Mth. Gen. 35, 6853-6872 (2002) 8. Zyed, EME, Gepreel, KA, El Horbty, MM: Extended proposed lgorithm with symbolic computtion to construct exct solutions for some nonliner differentil equtions. Chos Solitons Frctls 0,36-52 (2009) 9. Hong, B, Lu, D: New Jcobi elliptic function-like solutions for the generl KdV eqution with vrible coefficients. Mth. Comput. Model. 55, 59-600 (202)
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