AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/901540 901540 Research Article Generalized Kudryashov Method for Time-Fractional Differential Equations Tuluce Demiray Seyma 1 Pandir Yusuf 2 Bulut Hasan 1 Baleanu Dumitru 1 Department of Mathematics Firat University, 23119 Elazig Turkey firat.edu.tr 2 Department of Mathematics Bozok University, 66100 Yozgat Turkey bozok.edu.tr 2014 1572014 2014 25 03 2014 09 06 2014 09 06 2014 16 7 2014 2014 Copyright © 2014 Seyma Tuluce Demiray et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

1. Introduction

Partial differential equations are prevalently used as models to identify numerous physical occurrences and have a very crucial role in many sciences. Burgers equation, which is one type of partial differential equations, was first presented by Burgers in 1948 as a model for turbulent phenomena of viscous fluids . The Burgers equation defines the far field of wave propagation in nonlinear dissipative systems. It is well known that this equation is linearizable to the heat equation by using the Cole-Hopf transform. This equation has been considered in a number of fields of implementation such as traffic flows and formation of large clusters in the universe.

The Cahn-Hilliard equation, which is one type of partial differential equations, was first introduced in 1958 as a model for process of phase seperation of a binary alloy under the critical temperature . This equation has also arisen as the modelling equation in numerous other contexts with very disparate length scales. For example, models have been improved in which the Cahn-Hilliard equation is used to represent the evolution of two components of intergalactic material or in ecology in the modeling of the dynamics of two populations or in biomathematics in modeling the dynamics of the biomass and the solvent components of a bacterial film .

Korteweg-de Vries (KdV) equation, which is one type of partial differential equations, has been utilized to define a wide range of physical phenomena as a model for the evolution and interaction of nonlinear waves. It was derived as an evolution equation that conducting one-dimensional, small amplitude, long surface gravity waves propagating in a shallow channel of water . Subsequently, the KdV equation has occurred in a lot of other physical sciences such as collision-free hydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics, and lattice dynamics. Some theoretical physical occurrences in the quantum mechanics domain are expressed by means of a KdV model. It is utilized in fluid dynamics, aerodynamics, and continuum mechanics as a model for shock wave formation, solitons, turbulence, boundary layer behaviour, and mass transport .

The enquiry of exact solutions to nonlinear fractional differential equations has a very crucial role in several sciences such as physics, viscoelasticity, signal processing, probability and statistics, finance, optical fibers, mechanical engineering, hydrodynamics, chemistry, solid state physics, biology, system identification, fluid mechanics, electric control theory, thermodynamics, heat transfer, and fractional dynamics . In recent years, most authors have improved a lot of methods to find solutions of fractional differential equations such as local fractional variational iteration method [9, 10], cantor-type cylindrical-coordinate method , fractional complex transform method , and homotopy decomposition method . Also, exact solutions of fractional differential equations have been considered by using many methods such as the extended trial equation method [14, 15], the modified trial equation method [16, 17], a multiple extended trial equation method , and the modified Kudryashov method .

Our goal in this work is to introduce the exact solutions of time-fractional Burgers equation [16, 22], time-fractional Cahn-Hilliard equation , and time-fractional generalized third-order KdV equation [14, 22, 2629]. In Section 2, we give the description of proposed method. In Section 3, as illustrations, we gain exact solutions of time-fractional Burgers equation [16, 22]: (1)αu(x,t)tα-uxx-βupux=0,t>0,p>0,0<α1, time-fractional Cahn-Hilliard equation : (2)αu(x,t)tα=ux+6u(ux)2+(3u2-1)uxx-uxxxx,xxxxxxxxxxxxxxxxxxxxxixt>0,0<α1, and time-fractional generalized third-order KdV equation [14, 22]: (3)αu(x,t)tα-uxxx-γupux=0,t>0,p>0,0<α1, where α is a parameter describing the order of the fractional derivative.

2. The Generalized Kudryashov Method

Recently, some authors have investigated Kudryashov method . But, in this work, we try to constitute generalized form of Kudryashov method.

We consider the following nonlinear partial differential equation with fractional order for a function u of two real variables, space x and time t: (4)P(u,Dtαu,ux,uxx,uxxx,)=0.

The basic phases of the generalized Kudryashov method are explained as follows.

Step 1.

First of all, we must get the travelling wave solution of (4) in the following form: (5)u(x,t)=u(η),η=kx-λtαΓ[1+α], where k and λ are arbitrary constants. Equation (4) was converted into a nonlinear ordinary differential equation of the form (6)N(u,u,u′′,u′′′,)=0, where the prime indicates differentiation with respect to η.

Step 2.

Suggest that the exact solutions of (6) can be written in the following form: (7)u(η)=i=0NaiQi(η)j=0MbjQj(η)=A[Q(η)]B[Q(η)], where Q is 1/(1±eη). We note that the function Q is solution of equation : (8)Qη=Q2-Q. Taking into consideration (7), we obtain (9)u(η)=AQB-ABQB2=Q[AB-ABB2]u(η)=(Q2-Q)[AB-ABB2],(10)u′′(η)=Q2-QB2u′′(η)=×[(2Q-1)(AB-AB)+Q2-QBu′′(η)=ii××[B(A′′B-AB′′)-2BAB+2A(B)2]Q2-QB],(11)u′′′(η)=(Q2-Q)3×[6A(B)3B4((A′′′B-AB′′′-3A′′B-3B′′A)B+6B(AB′′+BA))(B3)-1-6A(B)3B4]+3(Q2-Q)2(2Q-1)×[B(A′′B-AB′′)-2BAB+2A(B)2B3]+(Q2-Q)(6Q2-6Q+1)[AB-ABB2].

Step 3.

Under the terms of proposed method, we suppose that the solution of (6) can be explained in the following form: (12)u(η)=a0+a1Q+a2Q2++aNQN+b0+b1Q+b2Q2++bMQM+. To calculate the values M and N in (12) that is the pole order for the general solution of (6), we progress conformably as in the classical Kudryashov method on balancing the highest-order nonlinear terms in (6) and we can determine a formula of  M and N. We can receive some values of M and N.

Step 4.

Replacing (7) into (6) provides a polynomial R(Ω) of Ω. Establishing the coefficients of R(Ω) to zero, we acquire a system of algebraic equations. Solving this system, we can describe λ and the variable coefficients of a0,a1,a2,,aN,b0,b1,b2,,bM. In this way, we attain the exact solutions to (6).

3. Applications to the Time-Fractional Equations

In this chapter, we search the exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV by using the generalized Kudryashov method.

Example 1.

We take the travelling wave solutions of (1) and we use the transformation u(x,t)=u(η) and η=kx-(λtα/Γ[1+α]), where k and λ are constants. Then, integrating this equation with respect to η and putting the integration constant to zero, we acquire (13)-λu-k2u-βkup+1p+1=0. When we take into consideration the transformation (14)u(η)=v1/p(η), we obtain the following formula: (15)λp(p+1)v-k2(p+1)v-βpkv2=0. Setting (7) and (9) into (15) and balancing the highest-order nonlinear terms of v and v2 in (15), then the following relation is attained: (16)N-M+1=2N-2MN=M+1. If we choose M=1 and N=2, then (17)u(η)=a0+a1Q+a2Q2b0+b1Q,(18)u(η)=(Q2-Q)×[(a1+2a2Q)(b0+b1Q)-b1(a0+a1Q+a2Q2)(b0+b1Q)2],u′′(η)=Q2-Q(b0+b1Q)2(2Q-1)×[(a1+2a2Q)(b0+b1Q)-b1(a0+a1Q+a2Q2)]+(Q2-Q)2(b0+b1Q)3×[2a2(b0+b1Q)2-2b1(a1+2a2Q)(b0+b1Q)+2b12(a0+a1Q+a2Q2)],u′′′(η)=(Q2-Q)(6Q2-6Q+1)×[(a1+2a2Q)(b0+b1Q)-b1(a0+a1Q+a2Q2)(b0+b1Q)2]+3(Q2-Q)2(2Q+1)×[(2a2(b0+b1Q)2-2b1(a1+2a2Q)(b0+b1Q)+2b12(a0+a1Q+a2Q2))((b0+b1Q)3)-1]+(Q2-Q)3×[-6a2b1(b0+b1Q)+6b12(a1+2a2Q)(b0+b1Q)3-6b13(a0+a1Q+a2Q2)(b0+b1Q)4].

The exact solutions of (1) are obtained as follows.

Case 1.

Consider (19)a0=0,a2=0,b1=-βpa1k(1+p)-b0,λ=k2p. When we substitute (19) into (17), we get the following solution of (1): (20)v1(x,t)=(a1(11±ekx-(k2tα/pΓ(1+α))))×(b0-(βpa1k(1+p)+b0)(11±ekx-(k2tα/pΓ(1+α))))-1. Using several simple transformations to this solution, we procure new exact solutions to (1): (21)u1(x,t)=[1-tanh(k1x-λ1tα)K[1+tanh(k1x-λ1tα)]+L[1-tanh(k1x-λ1tα)]]1/p,(22)u2(x,t)=[1-coth(k1x-λ1tα)K[1+coth(k1x-λ1tα)]+L[1-coth(k1x-λ1tα)]]1/p, where K=b0/a1, L=-βp/k(1+p), k1=k/2, and λ1=k2/2pΓ(1+α).

Case 2.

Consider (23)a0=k(1+p)b0βp,a1=-k(a+p)b0βp,a2=0,λ=-k2p. When we set (23) into (17), we attain the following solution of (1): (24)v2(x,t)=(k(1+p)b0βp-k(1+p)b0βp(11±ekx+(k2tα/pΓ(1+α))))×(b0+b1(11±ekx+(k2tα/pΓ(1+α))))-1. Performing several simple transformations to this solution, we obtain new exact solutions to (1): (25)u3(x,t)=[M[(1/2)+(1/2)tanh(k1x-λ2tα)]b0+b1[(1/2)-(1/2)tanh(k1x-λ2tα)]]1/p,(26)u4(x,t)=[M[(1/2)+(1/2)coth(k1x-λ2tα)]b0+b1[(1/2)-(1/2)coth(k1x-λ2tα)]]1/p, where M=k(1+p)b0/βp and λ2=-k2/2pΓ(1+α).

Case 3.

Consider (27)a0=0,a1=0,a2=-k(1+p)b1βp,b0=-b12,λ=2k2p. When we set (27) into (17), we obtain the following solution of (1): (28)v3(x,t)=(-k(1+p)b1βp(11±ekx-(2k2tα/pΓ(1+α)))2)×(-b12+b1(11±ekx-(2k2tα/pΓ(1+α))))-1. Performing several simple transformations to this solution, we find new exact solutions to (1): (29)u5(x,t)=[E[(1/2)-(1/2)tanh(k1x-λ3tα)]2tanh(k1x-λ3tα)]1/p,(30)u6(x,t)=[E[(1/2)-(1/2)coth(k1x-λ3tα)]2coth(k1x-λ3tα)]1/p, where E=2k(1+p)/βp and λ3=k2/pΓ(1+α).

Case 4.

Consider (31)a0=0,a1=k(1+p)b12βp,a2=-k(1+p)b1βp,b0=-b12,λ=k2p. When we embed (31) into (17), we get the following solution of (1): (32)v4(x,t)=(k(1+p)b12βp(11±ekx-(k2tα/pΓ(1+α)))-k(1+p)b1βp(11±ekx-(k2tα/pΓ(1+α)))2)×(-b12+b1(11±ekx-(k2tα/pΓ(1+α))))-1. Implementing several simple transformations to this solution, we gain kink solutions to (1): (33)u7(x,t)=[N(1-tanh(k1x-λ1tα))]1/p,(34)u8(x,t)=[N(1-coth(k1x-λ1tα))]1/p, where N=-k(1+p)/2βp.

Case 5.

Consider (35)a0=2k(1+p)b0βp,a1=-4k(1+p)b0βp,a2=2k(1+p)b0βp,b1=-2b0,λ=-2k2p. When we replace (35) into (17), we reach the following solution of (1): (36)v5(x,t)=(2k(1+p)b0βp-4k(1+p)b0βp(11±ekx+(2k2tα/pΓ(1+α)))+2k(1+p)b0βp(11±ekx+(2k2tα/pΓ(1+α)))2)×(b0-2b0(11±ekx+(2k2tα/pΓ(1+α))))-1. Applying several simple transformations to this solution, we attain new exact solutions to (1): (37)u9(x,t)=[E[(1/2)+(1/2)tanh(k1x-λ4tα)]2tanh(k1x-λ4tα)]1/p,(38)u10(x,t)=[E[(1/2)+(1/2)coth(k1x-λ4tα)]2coth(k1x-λ4tα)]1/p, where λ4=-k2/pΓ(1+α).

Case 6.

Consider (39)a0=k(1+p)b0βp,a1=-3k(1+p)b0βp,a2=2k(1+p)b0βp,b1=-2b0,λ=-k2p. When we put (39) into (17), we have the following solution of (1): (40)v6(x,t)=(k(1+p)b0βp-3k(1+p)b0βp(11±ekx+(k2tα/pΓ(1+α)))+2k(1+p)b0βp(11±ekx+(k2tα/pΓ(1+α)))2)×(b0-2b0(11±ekx+(k2tα/pΓ(1+α))))-1. Fulfilling several simple transformations to this solution, we acquire kink solutions to (1): (41)u11(x,t)=[D(1+tanh(k1x-λ2tα))]1/p,(42)u12(x,t)=[D(1+coth(k1x-λ2tα))]1/p, where D=k(1+p)/2βp.

Remark 2.

The exact solutions of (1) were found by using generalized Kudryashov method and have been checked by means of Mathematica Release 9. Comparing our results with results in [16, 22], then we can say that exact solutions of (1) that we obtained in this paper were firstly presented to the literature. Also, the advantage of our method compared to other methods in [16, 22] is to give more exact solutions.

Example 3.

We take the travelling wave solutions of (2) and we implement the transformation u(x,t)=u(η) and η=kx-(λtα/Γ(1+α)), where k and λ are constants. Then, integrating this equation with respect to η and embedding the integration constant to zero, we obtain (43)(λ+k)u+3k2u2u-k2u-k4u′′′=0. Putting (7), (9), and (11) into (43) and balancing the highest-order nonlinear terms of u′′′ and u2u in (43), then the following formula is procured: (44)N-M+3=3N-3M+1N=M+1. In an attempt to obtain exact solutions of (2), if we take (45)a1=a0[-(1+k2)b0+(1+k2)(-3a02+(1+k2)b02)](1+k2)b0,a2=0,b1=-b0+(1+k2)(-3a02+(1+k2)b02)(1+k2),λ=-k, and put (45) into (17), we obtain the following solution of (2): (46)u(x,t)=(ggxxga0+((11±ekx+(ktα/Γ(1+α)))fffa0[-(1+k2)b0+(1+k2)(-3a02+(1+k2)b02)]×(11±ekx+(ktα/Γ(1+α)))fff)((1+k2)b0)-1ggxxg)×([(k2)(a02(k2)02)(k2)]b0+[-b0+(1+k2)(-3a02+(1+k2)b02)(1+k2)][(k2)(a02(k2)02)(k2)]×(11±ekx+(ktα/Γ(1+α))))-1. Fulfilling several simple transformations to this solution, we get new exact solutions to (2): (47)u1(x,t)=P[1+1tanh(k1x-λ5tα)1-tanh(k1x-λ5tα)]+R,(48)u2(x,t)=P[1+1coth(k1x-λ5tα)1-coth(k1x-λ5tα)]+R, where P=a0(1+k2)/(-3a02+(1+k2)b02), R=a0/b0, and λ5=-k/2Γ(1+α).

Remark 4.

The solutions given by (47) and (48) of (2) were attained by using the generalized Kudryashov method and have been controlled by means of Mathematica Release 9. If we compare our results with results in , then it is clear that the exact solutions of (2) that we obtained in this paper were firstly introduced to the literature.

Example 5.

We get the travelling wave solutions of (3) and we apply the transformation u(x,t)=u(η) and η=kx-(λtα/Γ(1+α)), where k and λ are constants. Then, integrating this equation with respect to η and setting the integration constant to zero, we attain (49)λ(p+1)u+k3(p+1)u′′+γkup+1p+1=0. When we take into consideration the transformation (50)u(η)=v1/p(η), we find the following formula: (51)λp2(p+1)v2+k3(1-p2)(v)2+k3p(p+1)vv′′+γkv3=0. Setting (7) and (10) into (51) and balancing the highest-order nonlinear terms of vv′′ and v3 in (51), then the following relation is attained: (52)2N-2M+2=3N-3M+1N=M+2. If we take M=1 and N=3, then (53)u(η)=a0+a1Q+a2Q2+a3Q3b0+b1Q,(54)u(η)=(Q2-Q)×[VV((a1+2a2Q+3a3Q2)(b0+b1Q)-b1(a0+a1Q+a2Q2+a3Q3))×((b0+b1Q)2)-1VV],u′′(η)=Q2-Q(b0+b1Q)2(2Q-1)×[(a1+2a2Q+3a3Q2)(b0+b1Q)-b1(a0+a1Q+a2Q2+a3Q3)]+(Q2-Q)2(b0+b1Q)3×[(b0+b1Q)2(2a2+6a3Q)VV-2b1(b0+b1Q)(a1+2a2Q+3a3Q2)VV+2b12(a0+a1Q+a2Q2+a3Q3)]. In an attempt to find the exact solution of (3), if we choose (55)a0=0,a1=2k2(1+p)(2+p)b0γ,a2=-2k2(1+p)(2+p)(b0-b1)γ,a3=-2k2(1+p)(2+p)b1γ,λ=-k3p2, and embed (55) into (53), we obtain the following solution of (3): (56)v(x,t)=(2k2(1+p)(2+p)b0γ(11±ekx+(k3tα/p2Γ(1+α)))-2k2(1+p)(2+p)(b0-b1)γ(11±ekx+(k3tα/p2Γ(1+α)))2-2k2(1+p)(2+p)b1γ(11±ekx+(k3tα/p2Γ(1+α)))3)×(b0+b1(11±ekx+(k3tα/p2Γ(1+α))))-1. Performing several simple transformations to this solution, we get kink solutions to (3): (57)u1(x,t)=[S(1-[tanh(k1x-λ6tα)]2)]1/p,(58)u2(x,t)=[S(1-[coth(k1x-λ6tα)]2)]1/p, where S=k2(1+p)(2+p)/2γ and λ6=-k3/2p2Γ(1+α).

Remark 6.

The solutions given by (57) and (58) of (3) were gained by using the generalized Kudryashov method and have been checked by means of Mathematica Release 9. Comparing our results with results in [14, 22], it can be seen that the exact solutions of (3) that we obtained in this paper were firstly submitted to the literature.

We plot solution (21) of (1) in Figures 1-2, solution (25) of (1) in Figures 3-4, solution (29) of (1) in Figures 5-6, solution (33) of (1) in Figures 7-8, solution (37) of (1) in Figures 9-10, solution (41) of (1) in Figures 11-12, which show the dynamics of solutions with suitable parametric choices. Then we plot solution (47) of (2) in Figures 13-14, which show the dynamics of solutions with suitable parametric choices. Finally we plot solution (57) of (3) in Figures 15-16, which show the dynamics of solutions with suitable parametric choices.

Graph of the solution (21) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, a1=1, b0=2, β=3, 0<x<10, and 0<t<1.

Two-dimensional graph of the solution (21) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, a1=1, b0=2, β=3, t=1, and 0<x<10.

Graph of the solution (25) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, b0=3, b1=1, β=2, -15<x<15, and 0<t<1.

Two-dimensional graph of the solution (25) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, b0=3, b1=1, β=2, t=1, and -15<x<15.

Graph of the solution (29) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=1, β=4, -15<x<15, and 0<t<1.

Two-dimensional graph of the solution (29) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=1, β=4, t=1, and -15<x<15.

Graph of the solution (33) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, β=-2, -15<x<15, and 0<t<1.

Two dimensional graph of the solution (33) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, β=-2, t=1, and -15<x<15.

Graph of the solution (37) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, β=3, 0<x<15, and 0<t<1.

Two-dimensional graph of the solution (37) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, β=3, t=1, and 0<x<15.

Graph of the solution (41) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, β=2, -15<x<15, and 0<t<1.

Two-dimensional graph of the solution (41) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=3, β=2, t=1, and -15<x<15.

Graph of the solution (47) corresponding to the values α=0.05, α=0.85, respectively, when k=1, a0=1, b0=2, -5<x<5, and 0<t<1.

Two-dimensional graph of the solution (47) corresponding to the values α=0.05, α=0.85, respectively, when k=1, a0=1, b0=2, t=1, and -5<x<5.

Graph of the solution (57) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, γ=3, -15<x<15, and 0<t<1.

Two-dimensional graph of the solution (57) corresponding to the values α=0.05, α=0.85, respectively, when k=1, p=2, γ=3, t=1, and -15<x<15.

4. Conclusion

The Kudryashov method provides us with the evidential manner to constitute solitary wave solutions for a large category of nonlinear partial differential equations. Previously, many authors have tackled Kudryashov method. But, in this paper, we construct generalized form of Kudryashov method. This type of method will be newly considered in the literature to generate exact solutions of nonlinear fractional differential equations.

According to this information, we can conclude that GKM has an important role to find analytical solutions of nonlinear fractional differential equations. Also, we emphasize that this method is substantially influential and reliable in terms of finding new hyperbolic function solutions. We think that this method can also be implemented in other nonlinear fractional differential equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Burgers J. M. A mathematical model illustrating the theory of turbulence Advances in Applied Mechanics 1948 1 171 199 Cahn J. W. Hilliard J. E. Free energy of a nonuniform system. I. Interfacial free energy The Journal of Chemical Physics 1958 28 2 258 267 10.1063/1.1744102 2-s2.0-33746012315 Novick-Cohen A. The Cahn-Hilliard equation Handbook of Differential Equations, Evolutionary Equations 2008 4 Elsevier Haifa, Israel Korteweg D. J. de Vires G. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wawes Philosophical Magazine 1895 39 422 443 Fung M. K. KdV equation as an Euler-Poincare' equation Chinese Journal of Physics 1997 35 6 789 796 MR1613273 Miller K. S. Ross B. An Introduction to the Fractional Calculus and Fractional Differantial Equations 1993 New York, NY, USA Wiley Kilbas A. A. Srivastava H. M. Trujillo J. Theory and Applications of Fractional Differential Equations 2006 New York, NY, USA Elsevier Science MR2218073 Podlubny I. Fractional Differantial Equations 1999 San Diego, Calif, USA Academic Press Yang X.-J. Baleanu D. Khan Y. Mohyud-din S. T. Local fractional variational iteration method for diffusion and wave equations on cantor sets Romanian Journal of Physics 2014 59 1-2 36 48 Yang X.-J. Baleanu D. Fractal heat conduction problem solved by local fractional variation iteration method Thermal Science 2013 17 2 625 628 10.2298/TSCI121124216Y 2-s2.0-84879324154 Yang X. J. Srivastava H. M. He J. H. Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives Physics Letters A 2013 377 28–30 1696 1700 10.1016/j.physleta.2013.04.012 MR3062238 2-s2.0-84878016367 Yang X.-J. Baleanu D. He J. Transport equations in fractal porous media within fractional complex transform method Proceedings of the Romanian Academy A: Mathematics, Physics, Technical Sciences, Information Science 2013 14 4 287 292 MR3147127 Atangana A. Demiray S. T. Bulut H. Modelling the nonlinear wave motion within the scope of the fractional calculus Abstract and Applied Analysis 2014 2014 7 481657 10.1155/2014/481657 MR3214433 Pandir Y. Gurefe Y. Misirli E. The extended trial equation method for some time fractional differential equations Discrete Dynamics in Nature and Society 2013 13 491359 MR3071603 10.1155/2013/491359 Pandir Y. New exact solutions of the generalized Zakharov-Kuznetsov modified equal width equation Pramana 2014 82 6 949 964 Bulut H. Baskonus H. M. Pandir Y. The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation Abstract and Applied Analysis 2013 2013 8 636802 10.1155/2013/636802 MR3096833 Pandir Y. Tandogan Y. A. Exact solutions of the time-fractional Fitzhugh-Nagumo equation 1558 Proceedings of the 11th International Conference on Numerical Analysis and Applied Mathematics 2013 1919 1922 AIP Conference Proceedings Pandir Y. Gurefe Y. Misirli E. A multiple extended trial equation method for the fractional Sharma-Tasso-Olver equation 1558 Proceedings of the 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM '13) 2013 1927 1930 AIP Conference Proceedings 10.1063/1.4825910 Bulut H. Pandir Y. Baskonus H. M. Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order AIP Conference Proceedings 2013 1558 1914 1918 Tandogan Y. A. Pandir Y. Gurefe Y. Solutions of the nonlinear differential equations by use of modified Kudryashov method Turkish Journal of Mathematics and Computer Science 2013 7 20130021 Pandir Y. Symmetric fibonacci function solutions of some nonlinear partial differantial equations Applied Mathematics Information Sciences 2014 8 5 2237 2241 10.12785/amis/080518 Sahadevan R. Bakkyaraj T. Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations Journal of Mathematical Analysis and Applications 2012 393 2 341 347 10.1016/j.jmaa.2012.04.006 MR2921677 ZBLl1245.35142 2-s2.0-84860915405 Bekir A. Güner Ö. Cevikel A. C. Fractional complex transform and exp-function methods for fractional differential equations Abstract and Applied Analysis 2013 2013 8 426462 MR3064543 10.1155/2013/426462 Dahmani Z. Benbachir M. Solutions of the Cahn-Hilliard equation with time- and space-fractional derivatives International Journal of Nonlinear Science 2009 8 1 19 26 MR2557974 ZBLl1179.35263 Jafari H. Tajadodi H. Kadkhoda N. Baleanu D. Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations Abstract and Applied Analysis 2013 2013 5 587179 MR3034884 10.1155/2013/587179 Hu J. Ye Y. Shen S. Zhang J. Lie symmetry analysis of the time fractional KdV-type equation Applied Mathematics and Computation 2014 233 439 444 10.1016/j.amc.2014.02.010 MR3214998 Zhang Y. Formulation and solution to time-fractional generalized Korteweg-de Vries equation via variational methods Advances in Difference Equations 2014 2014 article 65 12 10.1186/1687-1847-2014-65 El-Wakil S. A. Abulwafa E. M. Zahran M. A. Time-fractional KdV equation: formulation and solution using variational methods Nonlinear Dynamics 2011 65 1-2 55 63 10.1007/s11071-010-9873-5 MR2812008 2-s2.0-79959502347 Wang G. W. Xu T. Z. Feng T. Lie symmetry analysis and explicit solutions of the time fractional fifth-order KdV equation PLoS ONE 2014 9 2 e88336 10.1371/journal.pone.0088336 Kudryashov N. A. One method for finding exact solutions of nonlinear differential equations Communications in Nonlinear Science and Numerical Simulation 2012 17 6 2248 2253 10.1016/j.cnsns.2011.10.016 MR2877672 ZBLl1250.35055 2-s2.0-84855223308 Lee J. Sakthivel R. Exact travelling wave solutions for some important nonlinear physical models Pramana—Journal of Physics 2013 80 5 757 769 10.1007/s12043-013-0520-9 2-s2.0-84877691703 Ryabov P. N. Sinelshchikov D. I. Kochanov M. B. Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations Applied Mathematics and Computation 2011 218 7 3965 3972 10.1016/j.amc.2011.09.027 MR2851496 ZBLl1246.35015 2-s2.0-80054985830