Research Article Improved (G /G)-Expansion Method for the Space and Time Fractional Foam Drainage and KdV Equations

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1 Abstrct nd Applied Anlysis Volume 213, Article ID , 7 pges Reserch Article Improved (G /G)-Expnsion Method for the Spce nd Time Frctionl Fom Dringe nd KdV Equtions Ali Akgül, 1,2 Adem KJlJçmn, 3 nd Mustf Inc 4 1 Deprtment of Mthemtics, Eduction Fculty, Dicle University, 2128 Diyrbkır, Turkey 2 Deprtment of Mthemtics nd Sttistics, Missouri University of Science nd Technology, Roll, MO , USA 3 Deprtment of Mthemtics nd Institute for Mthemticl Reserch, University Putr Mlysi, 434 Serdng, Selngor, Mlysi 4 Deprtment of Mthemtics, Science Fculty, Fırt University, Elzığ, Turkey Correspondence should be ddressed to Mustf Inc; minc@firt.edu.tr Received 1 June 213; Accepted 17 July 213 Acdemic Editor: Sntnu Sh Ry Copyright 213 Ali Akgül et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. The frctionl complex trnsformtion is used to trnsform nonliner prtil differentil equtions to nonliner ordinry differentil equtions. The improved (G /G)-expnsion method is suggested to solve the spce nd time frctionl fom dringe nd KdV equtions. The obtined results show tht the presented method is effective nd pproprite for solving nonliner frctionl differentil equtions. 1. Introduction The soliton solutions of nonliner evolution equtions hve mde mjor impct in the flesh. These solitons pper in vrious res of physicl nd biologicl sciences. They show up in nonliner optics, plsm physics, fluid dynmics, biochemistry, nd mthemticl chemistry. Frctionl prtil differentil equtions (FPDEs) hve received considerble interest in recent yers nd hve been extensively investigted. These equtions were pplied for mny rel problems which re modeled in vrious res, for exmple, in mthemticl physics [1], fluid nd continuum mechnics [2], viscoplsticndviscoelsticflow[3], biology, chemistry, coustics, nd psychology [4, 5]. Some FPDEs do not hve exct solutions, so pproximtion nd numericl techniques must be used. There re severl pproximtion nd numericl methods. The most commonly used ones re the homotopy perturbtion method [6, 7], the Adomin decomposition method [8, 9], the vritionl itertion method [1 12], the homotopy nlysis method [13, 14], the generlized differentil trnsform method [15], the finite difference method [16], nd the finite element method [17]. In recent yers, some uthors hve got exct solutions of FPDEs by using nlyticl methods. S. Zhng nd H.-Q. Zhng [18]proposed to solve the nonliner time frctionl biologicl popultion model nd (4 + 1)- dimensionl spce-time frctionl Foks eqution by using the frctionl subeqution method. Guo et l. [19]presented the improved subeqution method to solve the spce-time frctionl Whithm-Broer-Kup nd the generlized Hirot- Stsum coupled KdV equtions. Tng et l. [2] usedthe generlized frctionl subeqution method to obtin exct solutions of the spce-time frctionl Grdner eqution with vrible coefficients. Lu [21] investigted the exct solutions of the nonliner frctionl Klein-Gordon eqution, the generlized time frctionl Hirot-Stsum coupled KdV system, nd the nonliner frctionl Shrm-Tsso-Olver eqution. Bin [22] solved the time-spce frctionl generlized Hirot- Stsum coupled KdV equtions nd the time frctionl fifthorder Swd-Koter eqution by using the (G /G)-expnsion method. Omrn nd Gepreel [23] used the improved (G /G)-expnsion method to clculte the exct solutions to the time-spce frctionl fom dringe nd KdV equtions. In this pper, we will pply the improved (G /G)-expnsion method to obtin the exct solutions for the time-spce frctionl fom dringe nd KdV equtions with the modified Riemnn-Liouville derivtive defined by Jumrie [24 27]: α u = 1 2 u 2β u β 2 u u x 2β +2u2 x β +( β x ), β t >, α >, β 1, (1)

2 2 Abstrct nd Applied Anlysis α u u +αu β x + 3β u =, β x3β t >, α >, β 1, where α is rbitrry constnt. This pper is orgnized s follows. In Section 2, we introduce some bsic definitions of Jumrie s modified Riemnn-Liouville derivtive. In Section 3, the min steps of the improved (G /G)-expnsion method re given. In Section 4, we construct the exct solutions of (1) nd(2) bytheproposedmethod.some conclusions re given in Section Preliminries There re severl definitions for frctionl differentil equtions. These definitions include Riemnn-Liouville, Weyl, Grünwld-Letnikov, Riesz, nd Jumrie frctionl derivtives. The Riemnn-Liouville frctionl derivtive of constnt is not zero. So the frctionl derivtive is only defined for differentible function. In order to del with nondifferentible functions,jumrie [24 27] presented modifiction of the Riemnn-Liouville definition which ppers to provide frmework for frctionl clculus. This modifiction ws successfully pplied in the probbility clculus, frctionl Lplce problem, exct solutions of the nonliner frctionl differentil equtions, nd mny other types of liner nd nonliner frctionl differentil equtions [28 3]. Definition 1. The Riemnn-Liouville frctionl integrl is defined [31]s I α x f (x) =Iα f (x) x 1 = Γ (α) f (ξ)(x ξ) α 1 dξ, α >, I f (x) =f(x). Definition 2. Jumrie [24 27] defined the frctionl derivtive in the limit form by f (α) Δ α [f (x) f()] (x) = lim h h α, (4) where f(x) should be continuous (but not necessrily differentible) function nd h>denotes constnt discretiztion spn. So, the modified form of the Riemnn-Liouville derivtive is defined s D α x f (x) = 1 d n x Γ (n α) dx n (x ξ) (n α) [f (ξ) f()], (5) where x [,1], n 1 α<nnd n 1. Lemm 3. The integrl with respect to (dx) α is defined by Jumrie [24, 25]sfollows: x d α x f (ξ)(dξ) α =α (x ξ) f (ξ) dξ, < α 1, u(x) dx α f (ξ)(dξ) α =Γ(α+1) f[u(ξ)][u (ξ)] α, <α 1. (2) (3) (6) Theorem 4. Assume tht the continuous function f(x) hs frctionl derivtive of order α;then d α dx α Iα f (x) =f(x), I α d α (7) f (x) =f(x) f(), dxα <α 1, hold. 3. Description of the Improved (G /G)- Expnsion Method In this section, we give the description of the improved (G /G)-expnsion method for solving the nonliner FPDEs s F(u,D α t u, Dβ x u, Dγ y u, Dδ z u, Dα t Dα t u, D α t Dβ x u, Dβ x Dβ x u,...)=, <α,β,γ,δ 1, where u is n unknown function nd F is polynomil of u nd its prtil frctionl derivtives, in which the highest order derivtives nd nonliner terms re involved. We offer n improved (G /G)-expnsion method [32]. The essentil steps of this method re described s follows. Step 1. Li nd He [33] ndhendli[34] presentedfrctionl complex trnsform to trnsform frctionl differentil equtions into ordinry differentil equtions. So, ll nlyticl methods devoted to dvnced clculus cn be esily dedicted to frctionl clculus. The trveling wve vrible is given s u(x,y,z,t)=u(ξ), Γ(β+1) + Ny γ Γ(γ+1) + (8) L Γ (α+1), (9) where K, N, ndl re nonzero rbitrry constnts. So, (9) is reduced to (1): p (u, u,u,u,...) =, (1) where u=u(ξ). Step 2. Suppose tht (1)hs the solution(11): u (ξ) = n i= i F i (ξ), (11) where i (i=,1,...,n)re rel constnts to be determined, the blncing number n is positive integer which cn be determined by blncing the highest derivtive terms with the highest power nonlinerterms in (1). More precisely, we define the degree of u(ξ) s D[u(ξ)] = m,whichgivesriseto the degrees of other expressions, s follows: D[ dq u ]=m+q, dξq D[u p ( dq s (12) u dξ q ) ]=mp+s(q+m). Therefore, we cn obtin the vlue of m in (11).

3 Abstrct nd Applied Anlysis 3 Step 3. F(ξ) is F (ξ) = G (ξ) G (ξ), (13) where G(ξ) expresses the solution of the following uxiliry ordinry differentil eqution G (ξ) G (ξ) =AG 2 (ξ) +BG(ξ) G (ξ) +C[G(ξ)] 2, (14) where the prime denotes derivtive with respect to ξ. A, B, nd C re rel prmeters. Step 4. Substituting (13) into(1), using (14), collecting ll terms with the sme order of (G (ξ)/g(ξ)) together, nd then equting ech coefficient of the resulting polynomil to zero, we obtin set of lgebric equtions for i (i =,1,...,n), A, B, C, K, N,ndL. Step 5. Using the generl solutions of (14), with the id of Mthemtic, we hve the following four solutions of (13). Cse 1. If B =nd Δ=B 2 +4A 4AC,then F (ξ) = B 2 (1 C) + B Δ 2 (1 C) c 1 exp (( Δ/2) ξ) + c 2 exp (( Δ/2) ξ) c 1 exp (( Δ/2) ξ) c 2 exp (( Δ/2) ξ). Cse 2. If B =nd Δ=B 2 +4A 4AC<,then F (ξ) = B 2 (1 C) + B Δ 2 (1 C) ic 1 cos (( Δ/2) ξ) c 2 sin (( Δ/2) ξ) ic 1 sin (( Δ/2) ξ) + c 2 cos (( Δ/2) ξ). (15) (16) Cse 3. If B=nd Δ=A(C 1),then F (ξ) = Δ c 1 cos ( Δξ) + c 2 sin ( Δξ) (1 C) c 1 sin ( Δξ) c 2 cos ( Δξ). (17) Cse 4. If B=nd Δ=A(C 1)<,then F (ξ) = Δ ic 1 cosh ( Δξ) c 2 sinh ( Δξ) (1 C) ic 1 sinh ( Δξ) c 2 cosh ( Δξ), (18) where Γ(β+1) + Ny γ Γ(γ+1) + nd A, B, C, c 1,ndc 2 re rel prmeters. 4. Applictions L Γ (α+1) (19) We use the improved (G /G)-expnsion method on the timespce frctionl nonliner fom dringe eqution nd the time-spce frctionl nonliner KdV eqution in this section. 4.1.TheTimendSpce-FrctionlNonlinerFomDringe Eqution. We pply the improved (G /G)-expnsion method to construct the exct solutions for the time-spce frctionl nonliner fom dringe eqution in this subsection. Foms re of gret importnce in mny technologicl processes nd pplictions. Their properties re subject of intensive studies from prcticl nd scientific points of view [27, 35 37]. Liquid fom is n exmple of soft mtter with very well-defined structure, described by Joseph Plteu in the 19th century. Fomsrecommoninfoodsndpersonlcreproducts such s lotions nd crems. They hve importnt pplictions in food nd chemicl industries, minerl processing, fire fighting, nd structurl mteril sciences [27, 35 37]. This eqution is numericlly nd nlyticlly tken into ccount by different uthors [38 4]. The spce-time frctionl nonliner fom dringe eqution is solved nlyticlly only by Omrn nd Gepreel [23]. We cn see the frctionl complex trnsform s u (x, t) =u(ξ), Γ(β+1) + L (2) Γ (α+1), where K nd L re constnts. So, (2)reducesto(21): Lu K2 uu +2Ku 2 u +K 2 (u ) 2 =. (21) Blncing the highest order nonliner term nd the highest order liner term, we get n=1.thus,weobtin u (ξ) = + 1 F (ξ), F(ξ) = G (ξ) G (ξ), (22) where nd 1 will be determined constnts. Substituting (22)into(21), using (14), collectingll theterms ofpowers of (G /G), nd setting ech coefficient to zero, we hve the following system of lgebric equtions: G ) :2A 2 1K+A K A 1 BK 2 A 1 L=, G ) 1 :2 2 1BK A 1 K A2 1 BK B 2 K 2 +A 1 CK 2 1 BL =, G ) 2 : 2 2 1K+2A 3 1 K BK CK 3A 2 1 K BK B2 K 2 +3A 2 1 CK BCK B2 K 2 +3A 2 1 CK BCK L 1 CL =,

4 4 Abstrct nd Applied Anlysis G ) 3 : K+23 1 BK CK + 1 K BK CK BCK2 + 1 C 2 K 2 =, G ) 4 : K+23 1 CK K CK C2 K 2 =. (23) Solving the set of the bove lgebric equtions, we get the following result: = 8ABK (C 1) B 2 +6A 6AC, 1 =K(C 1), L= 1 2 (42 K+ BK 2 +2AK 3 2ACK 3 ), KB (C 1) =. (24) Substituting this vlue in (22)ndbyCses1 4,weobtin the following exponentil, hyperbolic nd tringulr function solutions of (1). (1) If we choose B =nd Δ=B 2 +4A 4AC,then the exponentil function solutions cn be found s u (x, t) = where 16ABK (C 1) KB[Δ+2A(C 1)] 2Δ + 2A (C 1) KB Δ 2 c 1 exp (( Δ/2) ξ) + c 2 exp (( Δ/2) ξ) c 1 exp (( Δ/2) ξ) c 2 exp (( Δ/2) ξ), (25) Γ(β+1) (42 K+ BK 2 +2AK 3 2ACK 3 ) Γ (α+1). (26) (2) If we choose B =nd Δ=B 2 +4A 4AC<,then the tringulr function solution will be u (x, t) = 16ABK (C 1) KB[Δ+2A(C 1)] 2Δ + 2A (C 1) KB Δ 2 ic 1 cos (( Δ/2) ξ) c 2 sin (( Δ/2) ξ) ic 1 sin (( Δ/2) ξ) + c 2 cos (( Δ/2) ξ), (27) where Γ(β+1) (42 K+ BK 2 +2AK 3 2ACK 3 ) Γ (α+1). (28) (3) If we choose B=nd Δ 1 = A(C 1), thenwe get nother tringulr function solution u (x, t) = K Δ 1 c 1 cos ( Δ 1 ξ) + c 2 sin ( Δ 1 ξ) c 1 sin ( Δ 1 ξ) c 2 cos ( Δ 1 ξ), (29) where /Γ(β + 1) Δ 1 (K 3 /Γ(α + 1)). (4) If we choose B=nd Δ 1 = A(C 1) <, thenwe obtin the hyperbolic function solution u (x, t) = K Δ 1 ic 1 cosh ( Δ 1 ξ) c 2 sinh ( Δ 1 ξ) ic 1 sinh ( Δ 1 ξ) c 2 cosh ( Δ 1 ξ), where /Γ(β + 1) Δ 1 (K 3 /Γ(α + 1)). (3) If we tke c 1 = c 2 nd c 1 =c 2 in (25), respectively, then we get u (x, t) = u (x, t) = 16ABK (C 1) KB[Δ+2A(C 1)] 2Δ + 2A (C 1) KB Δ 2 tnh ( Δ 2 ξ), 16ABK (C 1) KB[Δ+2A(C 1)] 2Δ + 2A (C 1) KB Δ 2 coth ( Δ 2 ξ). (31) 4.2. The Nonliner Spce-Time Frctionl KdV Eqution. The KdV eqution is the most populr soliton eqution, nd it hs been lrgely investigted. In ddition, the spce nd time frctionl KdV equtions with initil conditions were widely worked by [27, 38, 39]. Integrting (2) withrespecttou nd ignoring the integrl constnts leds to 1 2 Lu Ku K3 (u ) 2 =. (32) Considering the homogeneous blnce between the highest order derivtives nd the nonliner term in (32), we get n=2.so,wecnsupposetht(32)hsthefollowingnstz: u (ξ) = + 1 F (ξ) + 2 F 2 (ξ), (33) where, 1, 2, L, ndk re rbitrry constnts to be determined lter. Substituting (33) nd(14), long with (13),

5 Abstrct nd Applied Anlysis 5 into (32) nd using Mthemtic yields system of Equtions of (G /G): G ) : K+A2 2 1 K3 + 2 L=, G ) 1 : K+2A K 3 +A 2 1 BK3 + 1 L=, G ) 2 : K K A 2 1 K3 +2A K3 +4A 1 2 BK 3 + +A 2 1 CK L+ 2 L=, G ) 3 : K+ 1 2 K 4A 1 2 K B2 K BK3 +4A 2 2 BK B 2 K 3 +4A 1 2 CK BCK L=, 4 G ) : K K K3 4A 2 2 K BK B2 K CK BCK C2 K L=, G ) 5 : K K BK CK BCK C 2 K 3 =, 6 G ) : K+22 2 K CK C2 K 3 =. (34) Solving the set of the bove lgebric equtions by use of Mthemtic, we get the following results: =, = 12AK2 (C 1), 1 = 12BK2 (C 1), 2 = 12K2 (C 1), L= K 3 (B 2 4AC+4A). (35) Substituting (35) into(33) nd ccording to (15) (18), we obtin the following exponentil function solutions, hyperbolic function solutions, nd tringulr function solutions of (2), respectively. (1) If we choose B =nd Δ=B 2 +4A 4AC,then the exponentil function solution cn be found s u (x, t) = 3K2 (Δ 2C) 6CK2 Δ c 1 exp ( Δξ/2) + c 2 exp (( Δ/2) ξ) c 1 exp (( Δ/2) ξ) c 2 exp (( Δ/2) ξ) 6CK2 Δ [ c 2 1 exp (( Δ/2) ξ) + c 2 exp ( ( Δ/2) ξ) c 1 exp (( Δ/2) ξ) c 2 exp ( ( Δ/2) ξ) ], where /Γ(β + 1) K 3 Δ( /Γ(α + 1)). (36) (2) If we choose B =nd Δ=B 2 +4A 4AC<,then the tringulr function solution will be u (x, t) = 3K2 (Δ 2C) + 6CK2 Δ ic 1 cos ( Δξ/2) c 2 sin (( Δ/2) ξ) ic 1 sin (( Δ/2) ξ) + c 2 cos (( Δ/2) ξ) + 6CK2 Δ [ ic 2 1 cos (( Δ/2) ξ) c 2 sin (( Δ/2) ξ) ic 1 sin (( Δ/2) ξ) + c 2 cos (( Δ/2) ξ) ], where /Γ(β + 1) K 3 Δ( /Γ(α + 1)). (37) (3) If we choose B=nd Δ 1 = A(C 1), then the tringulr function solution is given s u (x, t) = 12K2 Δ 1 12K2 Δ 1 [ c 2 1 cos ( Δ 1 ξ) + c 2 sin ( Δ 1 ξ) c 1 sin ( Δ 1 ξ) c 2 cos ( Δ 1 ξ) ], where /Γ(β + 1) + 4K 3 Δ 1 ( /Γ(α + 1)). (38)

6 6 Abstrct nd Applied Anlysis (4) If we choose B=nd Δ 1 = A(C 1) <, then the hyperbolic function solution is given s u (x, t) = 12K2 Δ K2 Δ 1 [ ic 2 1 cosh ( Δ 1 ξ) c 2 sinh ( Δ 1 ξ) ic 1 sinh ( Δ 1 ξ) c 2 cosh ( Δ 1 ξ) ], (39) where ξ = /Γ(β + 1) 4K 3 Δ 1 ( /Γ(α+1)). Eqution (36) cnberewrittentc 1 = c 2 ;soweget the other hyperbolic function solution of (2): u (x, t) = 3K2 (Δ 2C) 6CK2 Δ tnh [ Δ 2 ( Kxβ Γ(β+1) K3 Δ Γ (α+1) )] + 6CK2 Δ Eqution (36)becomes tnh 2 [ Δ 2 ( Kxβ Γ(β+1) K3 Δ Γ (α+1) )]. u (x, t) = 3K2 (Δ 2C) t c 1 =c 2. 6CK2 Δ coth [ Δ 2 ( Kxβ Γ(β+1) K3 Δ Γ (α+1) )] + 6CK2 Δ (4) coth 2 [ Δ 2 ( Kxβ Γ(β+1) K3 Δ Γ (α+1) )] (41) Remrk 5. Kudryshov et l. [41 44]hve showed tht every solution, which ws obtined when soliton solutions hve been found by some nlytic methods, is not new solution. They lso showed tht these methods re very similr. Furthermore, they mentioned tht uthors who used these methods should check the obtined results very crefully. The reson for using improved (G /G)-expnsion method in this work is to use nonliner eqution (14) insted of liner eqution G λg μg=, (42) which ws used in stndrd (G /G) method nd to obtin lots of different solutions. 5. Conclusion In this pper, we introduced n improved (G /G)-expnsion method nd crried it out to obtin new trvelling wve solutions of the spce-time frctionl fom dringe eqution nd the spce-time frctionl KdV eqution. This method gives new exct solutions for nonliner FPDEs. These solutions include the hyperbolic function solution, the exponentil function solution, the tringulr function solution, nd the trigonometric function solution. These solutions re useful to understnd the mechnisms of the complicted nonliner physicl phenomen. References [1] I. Podlubny, Frctionl Differentil Equtions,vol.198ofMthemtics in Science nd Engineering, Acdemic Press, Sn Diego, Clif, USA, 1999, An introduction to frctionl derivtives, frctionl differentil equtions, to methods of their solution nd some of their pplictions. [2] C.Li,A.Chen,ndJ.Ye, Numericlpprochestofrctionl clculus nd frctionl ordinry differentil eqution, Journl of Computtionl Physics,vol.23,no.9,pp ,211. [3] X. Yng, Locl frctionl integrl trnsforms, Progress in Nonliner Science, no. 4, pp , 211. [4] S. Momni nd Z. Odibt, A novel method for nonliner frctionl prtil differentil equtions: combintion of DTM nd generlized Tylor s formul, Computtionl nd Applied Mthemtics,vol.22,no.1-2,pp.85 95,28. [5] J. Sbtier, A. Oustloup, J. C. Trigesson, nd N. Mmri, A Lypunov pproch to the stbility of frctionl differentil equtions, Signl Processing, vol. 91, no. 3, pp , 211. [6] J.-H. He, Homotopy perturbtion technique, Computer Methods in Applied Mechnics nd Engineering,vol.178,no.3-4,pp , [7] J.-H. He, Homotopy perturbtion method with n uxiliry term, Abstrct nd Applied Anlysis, vol. 212, Article ID , 7 pges, 212. [8] V. Dftrdr-Gejji nd S. Bhlekr, Solving multi-term liner nd non-liner diffusion-wve equtions of frctionl order by Adomin decomposition method, Applied Mthemtics nd Computtion,vol.22,no.1,pp ,28. [9] V. Dftrdr-Gejji nd H. Jfri, Solving multi-order frctionl differentil eqution using Adomin decomposition, Applied Mthemtics nd Computtion, vol.189,no.1,pp , 27. [1] J. He, Vritionl itertion method for dely differentil equtions, Communictions in Nonliner Science nd Numericl Simultion,vol.2,no.4,pp ,1997. [11] J. H. He, Vritionl itertion method for dely differentil equtions, Computtionl nd Applied Mthemtics, vol.25,pp.3 17,27. [12] M. Inc, The pproximte nd exct solutions of the spcend time-frctionl Burgers equtions with initil conditions by vritionl itertion method, Mthemticl Anlysis nd Applictions,vol.345,no.1,pp ,28. [13] S. Lio, On the homotopy nlysis method for nonliner problems, Applied Mthemtics nd Computtion, vol. 147, no. 2, pp , 24. [14] M. Gnjini, Solution of nonliner frctionl differentil equtions using homotopy nlysis method, Applied Mthemticl Modelling,vol.34,no.6,pp ,21.

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