Abstract
In this paper, the generalized exp(-Φ(ξ))-expansion method along with the Jumarie’s modified Riemann-Liouville derivatives is proposed to solve the nonlinear fractional Sharma-Tasso-Olever (STO) equation. Consequently, the exact solutions are obtained in terms of the trigonometric, exponential, hyperbolic, and rational functions, which confirm the proposed technique is very effectual and easily applicable.
Keywords: The fractional STO equation,Traveling wave solutions,Modified Riemann-Liouville derivatives,Generalized Exp (-Φ(ξ))-expansion method ,Fractional differential equation,Exact solutions.
Received: 24 March 2017 / Revised: 12 May 2017 / Accepted:15 June 2017/ Published: 29 June 2017
Contribution/ Originality
This study contributes in the existing literature on the use of the exp(-Φ(ξ))-expansion method. The method is applied to find the exact solutions to the fractional STO equation for the first time. Consequently, we get some new forms of exact solutions.
1. INTRODUCTION
With the development of nonlinear Science, increasing scholars regard the world around us as a nonlinear system and thus a plenty of nonlinear PDEs are widely used as models in various fields of natural sciences [1, 2]. A particular category of nonlinear PDEs are nonlinear fractional PDEs that have continually appeared in physics, chemistry, biology, polymeric materials, electromagnetic, acoustics, neutron point kinetic model, vibration and control, signal and image processing, fluid dynamics and so on [3-6]. Due to its practicability and complexity, it is important to seek the solutions of nonlinear fractional PDEs and researchers [7-24] have put considerable effort into it. For the purpose of solving problems in the practical application fields, more exact traveling wave solutions to the fractional PDEs seem to be useful. Up to now, a large number of methods have been applied to seek the solutions to nonlinear fractional PDEs, such as the fractional first integral method [11, 12] the fractional sub-equation method [13, 14] the (G'/G)-expansion method [15, 16] the improved (G'/G)-expansion method [17] the functional variable method [18] the fractional modified trial equation method [19] the extended spectral method [20] the variational iteration method [21-24] and so on. It is worth mentioning that Li and He [25, 26] have proposed a fractional complex transformation to convert fractional differential equations into ordinary differential equations (ODEs), which makes the problem simple. It means that the analytical methods devoted to advance calculus can also be applied to the fractional differential equations easily [27].
In recent years, the exp(-Φ(ξ))-expansion method have been implemented by many authors [28-30] to search the exact solutions of the nonlinear PDEs appeared in various fields as mentioned earlier. In these articles [28-30] the exp(-Φ(ξ))-expansion method along with the nonlinear ordinary differential equation
comprehensive solutions to nonlinear PDEs, so-called the generalized exp(-Φ(ξ))-expansion method [27]. Thus, the objective of this paper is to present the generalized exp(-Φ(ξ))-expansion method and implement it to find the exact traveling wave solutions of the fractional STO equation [11, 32, 33]. The proposed exp(-Φ(ξ))-expansion method along with the auxiliary nonlinear ODE provides much more comprehensive solutions and easily applicable to solve the nonlinear PDEs. Moreover, we have tried to generalize this method for finding more comprehensive exact traveling wave solutions to the nonlinear fractional PDEs in this paper. Sometimes this method can give solutions in disguised versions of known solutions that already be obtained by other methods. The superiority of this method over the existing methods is that it provides some new exact traveling wave solutions together with additional free parameters [27]. The algebraic computation of this method in this article is realized with the help of algebraic software, i.e., Mathematica.
The rest of the paper is prepared as follows: In section 2, the definitions of the fractional derivative is introduced concisely and the proposed generalized exp(-Ф(ξ))-expansion method is presented in details. Section 3 presents the application of this method to construct the exact traveling wave solutions of the nonlinear fractional STO equation. In comparison with other methods, the advantage of the proposed method is given in section 4. Conclusions have been drawn in Section 5.
2. DESCRIPTION OF THE METHOD
Follow the above steps, we are obtained the multiple explicit solutions of nonlinear fractional PDE (3) by combining the equations (4), (6) and (9).
3. APPLICATION TO NONLINEAR FRACTIONAL STO EQUATION
To illustrate the feasibility of the proposed method, we employ the steps in section 2 to solve a nonlinear fractional PDE.
Let us consider the space-time fractional STO equation as follows [11, 32, 33]
4. DISCUSSION
In Lu [11] the first integral method is applied to seek the exact solutions of fractional STO equation, as a result, the solutions are appeared in the form of hyperbolic functions and trigonometric functions. Considering the solutions (22)~(25), (31)~(34) in this article, if the parameters are set to particular values, the results are in accordance with the solutions (53)~(56) in Lu [11] . In addition, solutions (18), (19), (20), (21), (26), (27), (28), (29), (30), (35) are new exact traveling wave solutions to the nonlinear fractional STO equation.
In Zheng [33] the fractional STO equation is solved by the Exp-function method, the solutions in Zheng [33] are obtained in terms of exponential, hyperbolic and trigonometric functions. While the solutions denoted by (18)~(20), (22)~(25), (27)~(29), (31)~(34) in this paper are obviously different from the results in Zheng [33]. In addition, the solutions denoted by (21), (26), (30), (35) in this paper are in the form of rational functions, which can be seen as new solutions obtained by the generalized exp(-Ф(ξ))-expansion method.
Through the above comparative analysis, it is a remarkable fact that the proposed method is a practical mathematical method to search for the exact solutions of nonlinear fractional PDEs.
5. CONCLUSION
In this paper, the generalized exp(-Φ(ξ))-expansion method along with the Jumarie’s modified Riemann-Liouville derivatives has been successfully applied to the nonlinear fractional STO equation for the first time. The Discussion Section shows that the proposed method can give more solutions in general forms compared with other methods. Besides, this method is straightforward and easily applicable. Based on this advantage, we can also apply it to many other nonlinear fractional PDEs. The exact solutions obtained via this method have its great potential in the further analysis, such as stability analysis and compare with numerical solutions arises in various fields of applied mathematics and mathematical physics [27].
Funding: This study received no specific financial support. |
Competing Interests: The authors declare that they have no competing interests. |
Contributors/Acknowledgement: Both authors would like to express sincere thanks to the referees for their valuable suggestions and comments. |
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