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Seyma Tuluce Demiray 1 and Yusuf Pandir 2 and Hasan Bulut 1
Academic Editor:Dumitru Baleanu
1, Department of Mathematics, Firat University, 23119 Elazig, Turkey
2, Department of Mathematics, Bozok University, 66100 Yozgat, Turkey
Received 25 March 2014; Revised 9 June 2014; Accepted 9 June 2014; 16 July 2014
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.
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 [1]. 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 [2]. 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 [3].
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 [4]. 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 [5].
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 [6-8]. 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 [11], fractional complex transform method [12], and homotopy decomposition method [13]. 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 [18], and the modified Kudryashov method [19-21].
Our goal in this work is to introduce the exact solutions of time-fractional Burgers equation [16, 22], time-fractional Cahn-Hilliard equation [23-25], and time-fractional generalized third-order KdV equation [14, 22, 26-29]. 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]: [figure omitted; refer to PDF] time-fractional Cahn-Hilliard equation [24]: [figure omitted; refer to PDF] and time-fractional generalized third-order KdV equation [14, 22]: [figure omitted; refer to PDF] where α is a parameter describing the order of the fractional derivative.
2. The Generalized Kudryashov Method
Recently, some authors have investigated Kudryashov method [30-32]. 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 : [figure omitted; refer to PDF]
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: [figure omitted; refer to PDF] where k and λ are arbitrary constants. Equation (4) was converted into a nonlinear ordinary differential equation of the form [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] where Q is 1/(1±eη ) . We note that the function Q is solution of equation [30]: [figure omitted; refer to PDF] Taking into consideration (7), we obtain [figure omitted; refer to PDF]
Step 3.
Under the terms of proposed method, we suppose that the solution of (6) can be explained in the following form: [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] When we take into consideration the transformation [figure omitted; refer to PDF] we obtain the following formula: [figure omitted; refer to PDF] Setting (7) and (9) into (15) and balancing the highest-order nonlinear terms of v[variant prime] and v2 in (15), then the following relation is attained: [figure omitted; refer to PDF] If we choose M=1 and N=2 , then [figure omitted; refer to PDF]
The exact solutions of (1) are obtained as follows.
Case 1.
Consider [figure omitted; refer to PDF] When we substitute (19) into (17), we get the following solution of (1): [figure omitted; refer to PDF] Using several simple transformations to this solution, we procure new exact solutions to (1): [figure omitted; refer to PDF] where K=b0 /a1 , L=-βp/k(1+p) , k1 =k/2 , and λ1 =k2 /2pΓ(1+α) .
Case 2.
Consider [figure omitted; refer to PDF] When we set (23) into (17), we attain the following solution of (1): [figure omitted; refer to PDF] Performing several simple transformations to this solution, we obtain new exact solutions to (1): [figure omitted; refer to PDF] where M=k(1+p)b0 /βp and λ2 =-k2 /2pΓ(1+α) .
Case 3.
Consider [figure omitted; refer to PDF] When we set (27) into (17), we obtain the following solution of (1): [figure omitted; refer to PDF] Performing several simple transformations to this solution, we find new exact solutions to (1): [figure omitted; refer to PDF] where E=2k(1+p)/βp and λ3 =k2 /pΓ(1+α) .
Case 4.
Consider [figure omitted; refer to PDF] When we embed (31) into (17), we get the following solution of (1): [figure omitted; refer to PDF] Implementing several simple transformations to this solution, we gain kink solutions to (1): [figure omitted; refer to PDF] where N=-k(1+p)/2βp .
Case 5.
Consider [figure omitted; refer to PDF] When we replace (35) into (17), we reach the following solution of (1): [figure omitted; refer to PDF] Applying several simple transformations to this solution, we attain new exact solutions to (1): [figure omitted; refer to PDF] where λ4 =-k2 /pΓ(1+α) .
Case 6.
Consider [figure omitted; refer to PDF] When we put (39) into (17), we have the following solution of (1): [figure omitted; refer to PDF] Fulfilling several simple transformations to this solution, we acquire kink solutions to (1): [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] Putting (7), (9), and (11) into (43) and balancing the highest-order nonlinear terms of u[variant prime][variant prime][variant prime] and u2u[variant prime] in (43), then the following formula is procured: [figure omitted; refer to PDF] In an attempt to obtain exact solutions of (2), if we take [figure omitted; refer to PDF] and put (45) into (17), we obtain the following solution of (2): [figure omitted; refer to PDF] Fulfilling several simple transformations to this solution, we get new exact solutions to (2): [figure omitted; refer to PDF] 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 [24], 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 [figure omitted; refer to PDF] When we take into consideration the transformation [figure omitted; refer to PDF] we find the following formula: [figure omitted; refer to PDF] Setting (7) and (10) into (51) and balancing the highest-order nonlinear terms of vv[variant prime][variant prime] and v3 in (51), then the following relation is attained: [figure omitted; refer to PDF] If we take M=1 and N=3 , then [figure omitted; refer to PDF] In an attempt to find the exact solution of (3), if we choose [figure omitted; refer to PDF] and embed (55) into (53), we obtain the following solution of (3): [figure omitted; refer to PDF] Performing several simple transformations to this solution, we get kink solutions to (3): [figure omitted; refer to PDF] where S=k2 (1+p)(2+p)/2γ and λ6 =-k3 /2p2 Γ(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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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 .
(a) [figure omitted; refer to PDF]
(b) [figure omitted; refer to PDF]
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.
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Abstract
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.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer