Chin. Phys. B Vol., No. (0) 004 Exct solutions for nonliner prtil frctionl differentil equtions Khled A. epreel )b) nd Sleh Omrn b)c) ) Mthemtics Deprtment, Fculty of Science, Zgzig University, Egypt b) Mthemtics Deprtment, Fculty of Science, Tif University, Sudi Arbi c) Mthemtics Deprtment, Fculty of science, South Vlley University, Egypt (Received Februry 0; revised mnuscript received 7 My 0) In this rticle, we use the frctionl complex trnsformtion to convert nonliner prtil frctionl differentil equtions to nonliner ordinry differentil equtions. We use the improved ( /)-expnsion function method to clculte the exct solutions to the time- nd spce-frctionl derivtive fom dringe eqution nd the time- nd spce-frctionl derivtive nonliner KdV eqution. This method is efficient nd powerful for solving wide clsses of nonliner evolution frctionl order equtions. Keywords: frctionl clculus, complex trnsformtion, modified Riemnn Liouville derivtive, improved ( /)-expnsion function method PACS: 0.30.Jr DOI: 0.088/674-056///004. Introduction The seeds of frctionl clculus (tht is the theory of integrls nd derivtives of rbitrry rel or complex orders) were plnted over 300 yers go. Since then, mny reserchers hve contributed to this field. Recently, it turns out tht differentil equtions involving derivtives of non-integer orders cn be dequte models for vrious physicl phenomen. ] For exmple, the nonliner oscilltion of n erthquke cn be modeled with the frctionl derivtives. ] There hs been some ttempt to solve the liner problems with the multiple frctionl derivtives (the soclled multi-term equtions).,3] Little work hs been done for the nonliner problems, nd only few numericl schemes hve been proposed to solve the nonliner frctionl differentil equtions. The pplictions hve included clsses of nonliner equtions with multi-order frctionl derivtives; this motivtes us to develop numericl scheme for their solutions. 4] Numericl nd nlyticl methods include the Adomin decomposition method, 5,6] the vritionl itertion method, 7] nd the homotopy perturbtion method. 80] Deng ] hs discussed the smoothness nd the stbility of the solutions for the nonliner frctionl differentil equtions. The numericl solutions Corresponding uthor. E-mil: kgepreel@yhoo.com 0 Chinese Physicl Society nd IOP Publishing Ltd of the nonliner time frctionl Fokker Plnck differentil equtions hve been given in Ref. ]. In this pper, we will implement the ( /)- expnsion function method 35] to obtin the exct solutions for the following time- nd spce-frctionl derivtive nonliner differentil equtions: (i) the time- nd spce-frctionl derivtive fom dringe eqution 6] α u t α = u β u x β + u β u x β + ( β ) u x β, t > 0, 0 < α, β, () (ii) the nonliner Korteweg de Vries (KdV) eqution with time- nd spce-frctionl derivtives 7] α u t α + u u β x β + 3β u = 0, t > 0, 0 < α, β. () x3β. Preliminries In recent yers, in order to investigte the locl behvior of frctionl models, severl locl versions of frctionl derivtives hve been proposed, i.e., the Kolwnkr ngl locl frctionl derivtive, 8] Chen s frctl derivtive, 9] Cresson s derivtive, 0] nd Jumrie s modified Riemnn http://iopscience.iop.org/cpb http://cpb.iphy.c.cn 004-
Chin. Phys. B Vol., No. (0) 004 Liouville derivtive.,] Jumrie s derivtive is defined s D α xf(x) = x 0 d Γ( α) dx (x ξ) α (f(ξ) f(0))dξ, 0 < α <, (3) f : R R, x f(x) denotes continuous (but not necessrily first-order-differentible) function. We cn obtin the following properties. Property Let f(x) stisfy the definition of the modified Riemnn Liouville derivtive nd f(x) be kα order differentible function. The generlized Tylor series is given by,3] f(x + h) = Property k=0 h αk (αk)! f (αk) (x), 0 < α <. (4) Assume tht f(x) denotes continuous R R function. We use the following equlity for the integrl with respect to (dx) α :,3] I α x f(x) = Γ(α) = x 0 Γ(α + ) (x ξ) α f(ξ)dξ x 0 f(ξ)(dx) α, 0 < α. (5) Property 3 Some useful formuls include f (α) x(t)] = df dx x(α) (t), (6) D α xx β Γ( + β) = Γ( + β α) xβα, (7) (dx) β = x β. (8) Function f(x) should be differentible with respect to x(t), nd x(t) is frctionl differentible in Eq. (6). the following sections. The bove results re employed in The modified Riemnn Liouville derivtive hs been successfully pplied in the probbility clculus, 4] the frctionl Lplce problems, 5] the frctionl vritionl pproch with severl vribles, 6] the frctionl vritionl itertion method, 7] the frctionl vritionl pproch with nturl boundry conditions, 8] nd the frctionl Lie group method. 9] 3. The improved ( /)-expnsion function method Consider the following nonliner prtil frctionl differentil eqution: F (u, D α t u, D β xu, D γ yu, D δ zu, D α t D α t u, D α t D β xu, D β xd β xu, D β xd γ y, D γ yd γ yu,...) = 0, 0 < α, β, δ, γ <, (9) u is n unknown function, nd F is polynomil of u nd its prtil frctionl derivtives, in which the highest order derivtives nd the nonliner terms re involved. In the following, we give the min steps of the improved ( /)-expnsion method. Step Li et l. 30] proposed frctionl complex trnsform to convert frctionl differentil equtions into ordinry differentil equtions (ODE), so ll nlyticl methods devoted to the dvnced clculus cn be esily pplied to the frctionl clculus. The trveling wve vrible u(x, y, z, t) = u(ξ), K xβ Γ(β + ) + N yγ Γ(γ + ) + M zδ Γ(δ + ) + Ltα Γ(α + ), (0) K L, M, nd N re non zero rbitrry constnts, permits us to reduce Eq. (9) to n ODE of u = u(ξ) in the form P (u, u, u, u,...) = 0. () If possible, we should integrte Eq. () term by term one or more times. Step Suppose the solution of Eq. () cn be expressed s polynomil of ( /) in the form 35] u(ξ) = m i=m i α i, α m 0, () α i (i = 0, ±,..., ±m) re constnts, while (ξ) stisfies the following second-order liner ODE (ξ) + λ (ξ) + µ(ξ) = 0, (3) with λ nd µ being constnts. Step 3 The positive integer m cn be determined by considering the homogeneous blnce between the 004-
Chin. Phys. B Vol., No. (0) 004 highest order derivtives nd the nonliner terms ppering in Eq. (). More precisely, we define the degree of u(ξ) s Du(ξ)] = m, which gives rise to the degrees of the other expressions s follows: d q ] u D dξ q = m + q, ( d D u p q ) s ] u dξ q = mp + s(q + m). (4) Therefore, we cn obtin the vlue of m in Eq. (). Step 4 Substituting Eq. () into Eq. (), using Eq. (3), collecting ll terms with the sme order of ( /) together, nd then equting ech coefficient of the resulting polynomil to zero, we obtin set of lgebric equtions for α i (i = 0, ±,..., ±m), λ, µ, K, L, M, nd N. Step 5 Since the generl solutions to Eq. (3) re well known, then substituting α i (i = 0, ±,..., ±m), λ, µ, K, L, M, N nd the generl solutions of Eq. (3) into Eq. (), we get more trveling wve solutions of the nonliner prtil frctionl derivtives, Eq. (9). 4. Applictions In this section, we use the improved ( /)- expnsion method to construct the exct solutions for some nonliner prtil frctionl differentil equtions, i.e., the nonliner fom dringe eqution with time- nd spce-frctionl derivtives nd the nonliner KdV eqution with time- nd spce-frctionl derivtives. Such equtions hve received wide ttention. 4.. The time- nd spce-frctionl derivtive nonliner fom dringe eqution In this section, to demonstrte the effectiveness of our pproch, we will pply the modified ( /)- expnsion function method to construct the exct solutions for the time- nd spce-frctionl derivtive nonliner fom dringe eqution. 6] We cn see tht the frctionl complex trnsform u(x, t) = U(ξ), Γ( + β) + Ltα Γ( + α), (5) K nd L re constnts, permits us to reduce Eq. () into the following ODE: LU + K UU + KU U + K U = 0. (6) Considering the homogeneous blnce between the highest order derivtive nd the nonliner term in Eq. (6), we deduce tht n =. Thus, we get u(ξ) = α + α 0 + b, (7) α, α 0, b, L, nd K re rbitrry constnts to be determined lter. Substituting Eq. (7) into Eq. (6), collecting ll the terms of powers of ( /), nd setting ech coefficient 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 α 0 = Kλ, L = K3 µ+ K3 λ, = K, b = 0, (8) 4 K, λ, nd µ re rbitrry constnts. Substituting Eq. (8) into Eq. (7) yields u(ξ) = K + Kλ. (9) From the solutions of Eqs. (3) nd (9), we deduce the following types of trveling wve solutions for Eq. (). Fmily If λ 4µ > 0, then we hve the hyperbolic solution u(ξ) = K A cosh( λ 4µ (λ 4µ) ξ) + B sinh( (λ 4µ) ξ), (0) A sinh( (λ 4µ) ξ) + B cosh( (λ 4µ) ξ) Fmily If λ 4µ < 0, then we hve the trigonometric solution u(ξ) = K 4µ λ A sin( 4µ λ ξ) + B cos( 4µ λ ξ), () A cos( 4µ λ ξ) + B sin( 4µ λ ξ) 004-3
Chin. Phys. B Vol., No. (0) 004 Γ( + β) + K3 (λ 4µ) t α. () 4 Γ( + α) In prticulr, if we set B = 0, A 0, λ > 0, nd µ = 0 in Eq. (0), then we get the solitry wve solutions to Eq. () u(ξ) = Kλ If B 0, A < B, λ > 0, nd µ = 0, then we get u(ξ) = Kλ ( tnh λ + K3 λ t α 4 Γ( + α) Kx β ( Kx β coth λ Γ( + β) + K3 λ t α ])]. (3) 4 Γ( + α) Γ( + β) ] )] + ξ 0, (4) Cse α 0 = Kλ, L = (λ 4µ)K 3, = 0, b = Kµ, 4 (5) K, λ, nd µ re rbitrry constnts. Substituting Eq. (5) into Eq. (7) yields ξ 0 = tnh (A/B). Similrly we cn clculte the solitry wve solutions to Eq. (). Remrk We neglect the cse with λ 4µ = 0, becuse if λ 4µ = 0, we get L = 0. This yields contrdiction with L being nonzero constnt. u(ξ) = Kµ Kλ. (6) From the solutions of Eqs. (3) nd (6), we deduce the following types of trveling wve solutions of Eq. (). Fmily If λ 4µ > 0, then we hve the hyperbolic solution u(ξ) = Kλ Kµ A cosh( λ 4µ (λ 4µ) ξ) + B sinh( (λ 4µ) ξ) A sinh( (λ 4µ) ξ) + B cosh( (λ 4µ) ξ) λ ], (7) Fmily If λ 4µ < 0, then we hve the trigonometric solution ] u(ξ) = Kλ Kµ 4µ λ A sin( 4µ λ ξ) + B cos( 4µ λ ξ) A cos( 4µ λ ξ) + B ( λ, (8) 4µ λ ξ) Γ( + β) + K3 (λ 4µ) t α. (9) 4 Γ( + α) Cse 3 α 0 = 0, L = 4µK 3, λ = 0, = K, b = Kµ, (30) K nd µ re rbitrry constnts. Substituting Eq. (30) into Eq. (7) yields u(ξ) = K Kµ. (3) From the solutions to Eqs. (3) nd (3), we deduce the following types of trveling wve solutions to Eq. (). Fmily If µ < 0, then we hve the hyperbolic solution u(ξ) = K A cosh( µ ξ) + B sinh( µ ξ) µ A sinh( µ ξ) + B cosh( µ ξ) ] + K µ ] A sinh( µ ξ) + B cosh( µ ξ) A cosh( µ ξ) + B sinh(, (3) µ ξ) Fmily If µ > 0, then we hve the trigonometric solution u(ξ) = K ] A sin( µξ) + B cos( µξ) µ A cos( µξ) + B sin( K ] A cos( µξ) + B sin( µξ) µ µξ) A sin( µξ) + B cos(, (33) µξ) Γ( + β) 4µK3 t α Γ( + α). (34) 004-4
Chin. Phys. B Vol., No. (0) 004 4.. The nonliner KdV eqution with time- nd spce-frctionl derivtives In this section, to demonstrte the effectiveness of our pproch, we use the complex trnsformtion (5) to convert the nonliner KdV eqution with time- nd spce-frctionl derivtives () to n ordinry differentil eqution. After tht, we integrte twice to obtin LU + 6 KU 3 + K3 U ] + C U + C = 0, (35) C nd C re the integrtion constnts. Considering the homogeneous blnce between the highest order derivtives nd the nonliner terms in Eq. (35), we hve U(ξ) = α 0 + α + α + α 3 + α 4, (36) α 0, α α, b, b, L, nd K re rbitrry constnts to be determined lter. Substituting Eqs. (3) nd (36) into Eq. (35), collecting ll the terms of powers of ( /), nd setting ech coefficient 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 L = 8K 3 µ K 3 λ Kα 0, α = K λ, α = K, C = K (4K4 λ µ + α 0 + K λ α 0 + 48K 4 µ + 6α 0 K µ), C = K 6 (3 α 3 0 + 43K 6 λ µ + 44α 0 K 4 µ + 3α 0 K λ + 4α 0 K µ + 7α 0 K 4 λ µ), α 3 = α 4 = 0, (37) α 0, K,, λ, nd µ re rbitrry constnts. Substituting Eq. (37) into Eq. (36) yields u(ξ) = α 0 K λ K. (38) From the solutions to Eqs. (3) nd (38), we deduce the following types of trveling wve solutions to Eq. (). Fmily If λ 4µ > 0, then we hve the hyperbolic solution u(ξ) = 3K (λ 4µ) A cosh( (λ 4µ) ξ) + B sinh( (λ 4µ) ξ) + α A sinh( (λ 4µ) ξ) + B cosh( 0 + 3K λ, (39) (λ 4µ) ξ) Fmily If λ 4µ < 0, then we hve the trigonometric solution u(ξ) = 3K (4µ λ ) A sin( 4µ λ ξ) + B cos( 4µ λ ξ) + α A cos( 4µ λ ξ) + B sin( 0 + 3K λ, (40) 4µ λ ξ) Γ( + β) (8K3 µ + K 3 λ + K α 0 ) t α. Γ( + α) (4) Fmily 3 If λ 4µ = 0, then we hve the rtionl solution u(η) = K B (Bη + A) + α 0 + 3K λ, (4) η = Γ( + β) (3K3 λ + K α 0 ) t α. Γ( + α) (43) In prticulr, if we set B = 0, A 0, λ > 0, nd µ = 0, in Eq. (39), then we get the solitry wve solutions to Eq. () s follows: u(ξ) = 3K λ ( Kx coth β λ Γ( + β) K 3 λ t α ]) + α 0 + 3K λ. (44) 4 Γ( + α) While if B 0, A < B, λ > 0, nd µ = 0, then we 004-5
Chin. Phys. B Vol., No. (0) 004 get u(ξ) = 3K λ + K3 λ t α 4 Γ( + α) ( Kx tnh β λ Γ( + β) ] ) + ξ 0 + α 0 + 3K λ, (45) ξ 0 = tnh (A/B). Similrly we cn clculte the solitry wve solutions to Eq. (40). Cse L = 8K 3 µ K 3 λ Kα 0, α 3 = K λµ, α 4 = K µ, C = K (4K4 λ µ + α 0 + K λ α 0 + 48K 4 µ + 6α 0 K µ), C = K 6 (3 α 3 0 + 43K 6 λ µ + 44α 0 K 4 µ + 3α 0 K λ + 4α 0 K µ + 7α 0 K 4 λ µ), α = α = 0, (46) α 0, K,, λ, nd µ re rbitrry constnts. Substituting Eq. (46) into Eq. (36) yields u(ξ) = α 0 K λµ K µ. (47) From the solutions to Eqs. (3) nd (47), we deduce the following types of trveling wve solutions to Eq. (). Fmily If λ 4µ > 0, then we hve the hyperbolic solution ] u(ξ) = α 0 K λµ A cosh( λ 4µ (λ 4µ) ξ) + B sinh( (λ 4µ) ξ) A sinh( (λ 4µ) ξ) + B cosh( λ (λ 4µ) ξ) ] K µ A cosh( λ 4µ (λ 4µ) ξ) + B sinh( (λ 4µ) ξ) A sinh( (λ 4µ) ξ) + B cosh( λ. (48) (λ 4µ) ξ) Fmily If λ 4µ < 0, then we hve the trigonometric solution ] u(ξ) = α 0 K λµ 4µ λ A sin( 4µ λ ξ) + B cos( 4µ λ ξ) A cos( 4µ λ ξ) + B sin( λ 4µ λ ξ) ] K µ 4µ λ A sin( 4µ λ ξ) + B cos( 4µ λ ξ) A cos( 4µ λ ξ) + B sin( λ, (49) 4µ λ ξ) Γ( + β) (8K3 µ + K 3 λ + K α 0 ) t α. (50) Γ( + α) Fmily 3 If λ 4µ = 0, then we hve the rtionl solution ] u(η) = α 0 K λµ (Bη + A) (B Aλ) λbη K µ ] (Bη + A), (5) (B Aλ) λbη η = Γ( + β) (3K3 λ + Kα 0 ) t α. (5) Γ( + α) 5. Conclusion In this pper, we use the improved ( /) - expnsion function method to clculte the exct solutions for the time- nd spce-frctionl derivtive fom dringe eqution nd the time- nd spcefrctionl derivtive nonliner KdV eqution. This method is relible, simple nd gives mny new exct solutions for the nonliner frctionl differentil equtions. When the prmeters tke certin specil vlues, the solitry wves re derived from the trveling wves. This method is very efficient nd powerful in finding the exct solutions for the nonliner frctionl differentil equtions. 004-6
Chin. Phys. B Vol., No. (0) 004 References ] Podlubny I 999 Frctionl Differentil Equtions (Sn Diego: Acdemic Press) ] He J H 004 Bull. Sci. Technol. 5 86 3] Diethelm K nd LuchkoY 008 J. Comput. Anl. Appl. 6 43 4] Erturk V S, Momni S nd Odibt Z 008 Commun. Nonliner Sci. Numer. Simult. 3 64 5] Dftrdr-ejji V nd Bhlekr S 008 Appl. Mth. Comput. 0 3 6] Dftrdr-ejji V nd Jfri H 007 Appl. Mth. Comput. 89 54 7] Sweilm N H, Khder M M nd Al-Br R F 007 Phys. Lett. A 37 6 8] olbbi A nd Syevnd K 0 Comput. Mth. Appliction 6 7 9] olbbi A nd Syevnd K 00 Nonliner Science Lett. A 47 0] epreel K A 0 Applied Mth. Lett. 4 48 ] Deng W H 00 Nonliner Anlysis: TMA 7 768 ] Deng W H 007 Journl of Computtionl Physics 7 50 3] Wng M L, Li X Z nd Zhng J L 008 Phys. Lett. A 37 47 4] Zyed E M E nd epreel K A 009 J. Mth. Phys. 50 0350 5] Zhng H 009 Commun. Nonliner Sci. Numer. Simult. 4 30 6] Dhmni Z nd Anber A 00 Inter. J. Nonliner Sci. 0 39 7] Momni S, Odibt Z nd Alwneh A 008 J. Numer. Method. Prtil Diff. Equ. 4 6 8] Kolwnkr K M nd ngl A D 998 Phys. Rev. Lett. 80 4 9] Chen W nd Sun H 009 Mod. Phys. Lett. B 3 449 0] Cresson J 005 J. Mth. Anl. Appl. 307 48 ] Jumrie 006 Comput. Mth. Appl. 5 367 ] Jumrie 006 Appl. Mth. Lett. 9 873 3] Wu C 0 Appl. Mth. Lett. 4 046 4] Jumrie 006 Mth. Comput. Modelling 44 3 5] Jumrie 009 Appl. Mth. Lett. 659 6] Almeid R, Mlinowsk A B nd Torres D FM 00 J. Mth. Phys. 5 033503 7] Wu C nd Lee E W M 00 Phys. Lett. A 374 506 8] Mlinowsk A B, Sidi Ammi M R nd Torres D F M 00 Commun. Frc. Clc. 3 9] Wu C 00 Commun. Frc. Clc. 3 30] Li Z B nd He J H 00 Mth. Comput. Applictions 5 970 004-7