Two Reliable Methods for Solving the (3 + 1)-Dimensional Space-Time Fractional Jimbo-Miwa Equation

Sekson Sirisubtawee,Wanassanun Porka,Sanoe Koonprasert,Chaowanee Khaopant

doi:10.1155/2017/9257019

Sekson Sirisubtawee, Wanassanun Porka + Show 2 more

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https://doi.org/10.1155/2017/9257019

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Abstract

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the G′/G,1/G-expansion method and the novel G′/G-expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel G′/G-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the G′/G-expansion method and the exp-Φ(ξ)-expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions.

Highlights

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders
Study of traveling wave solutions of NPDEs plays a significant role in the investigation of behaviors of nonlinear phenomena
We have found that our exact solutions in (34), (39), and (46), obtained by the (G󸀠/G, 1/G)-expansion method, have the same mathematical structures as their results

Summary

Introduction

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders. It has been found that the above-mentioned methods with their improvements (see, e.g., [30,31,32,33]) are widely applicable to solve FDEs. Searching for exact explicit solutions to nonlinear fractional partial differential equations (NFPDEs) is a research field of active interest. Many approaches with the help of symbolic software packages have been developed to efficiently provide exact solutions of NFPDEs, for example, the improved extended tanhcoth method [34], the improved generalized exp-function method [35], the fractional Riccati expansion method [36], the (G󸀠/G, 1/G)-expansion method [37,38,39,40,41], and the novel (G󸀠/G)-expansion method [42,43,44,45]. The aim of this article is to apply the (G󸀠/G, 1/G)-expansion method and the novel (G󸀠/G)-expansion method to solve the (3 + 1)dimensional space-time fractional Jimbo-Miwa equation in the sense of Jumarie’s modified Riemann-Liouville derivative.

Mathematical Preliminaries

Some Graphical Representations of Some

Conclusions