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

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 ( / , 1/ )-expansion method and the novel ( / )-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 ( / )-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the ( / )-expansion method and the exp(−Φ( ))-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.

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Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

“…Useful methods for solving NLEEs numerically are those such as the finite element method [13], the finite volume method [14], and the finite-difference predictor-corrector method [15]. Several efficient and reliable methods which have recently been developed to obtain exact explicit solutions for NLEEs are, for instance, the Jacobi elliptic function method [16], the (G /G)-expansion method and its various modifications [17][18][19][20], the Exp-function method [21], the sub-equation method [22], the first integral method [23], the modified trial equation method [24,25], and the simplest equation method [26].…”

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“…Over the last few decades, exact solutions, analytical approximate solutions, and numerical solutions of many NPDEs have been successfully obtained. The methods for obtaining exact explicit solutions of NPDEs are, for example, the ( / )-expansion method [6][7][8], the ( / , 1/ )-expansion method [9][10][11], the novel ( / )-expansion method [12], the tanh-function method [13], the exp-function method [14,15], the F-expansion method [16], Hirota's direct method [17,18], Kudryashov method [19,20], and the extended auxiliary equation method [21]. Examples of the methods for obtaining analytical approximate solutions to NPDEs are the variational iteration method [22,23] (VIM), the Adomian decomposition method [24,25] (ADM), the homotopy perturbation method [26,27] (HPM), and the reduced differential transform method [28].…”