Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics

Abstract

In this paper, we employ the modified simple equation method to find the exact traveling wave solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional generalized shallow water-wave equation and the(2+1)-dimensional KdV-Burgers equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the proposed method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations.

Modified extended tanh-function method and its applications to the Bogoyavlenskii equation

“…On comparing our solutions Eq. (21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32) with that obtained in [39,40], we have many new solutions using the proposed method of this paper which are equivalent in some cases and not in other cases and this solutions are spelled out explicitly and we put them in this form to allow reader clearly different forms of solution. The performance of this method is reliable and effective and can be applied to many other nonlinear evolution equations.…”

“…Exact solutions for these equations play an important role in many phenomena in physics such as solid state physics, fluid mechanics, hydrodynamics, Optics, Plasma physics and so on. Recently many new approaches for finding these solutions have been proposed, for example, tanh -sech method [2][3][4], sine -cosine method [5][6][7], homogeneous balance method [8,9], Jacobi elliptic function method [10][11][12][13], F-expansion method [14][15][16], exp-function method [17][18], trigonometric function series method [19], ( G G )− expansion method [20][21][22][23], the modified simple equation method [24][25][26], extended tanh -method [27][28][29]. The extended tanh method, developed by Wazwaz [30,31], is a direct and effectie algebraic method for handling nonlinear equations.…”

Analytical Solutions of Nonlinear Coupled Schrodinger–KdV Equation via Advance Exponential Expansion

“…Analytical solutions allow researchers to design and perform experiments, by creating suitable natural situations, to determine these functions and parameters. There are several types of well-established methods that have been devoted to evaluate analytical solutions of NPDEs, such as the modified simple equation method [1,2], the ( / ) G G ′ expansion method [3,4], the tanh method [5,6], the Homotopy perturbation technique [7], the homogeneous balance method [8,9], the Hirota method [10], the Expfunction method [11,12], the exp ( ( )) ϕ ξ − -expansion method [13][14][15][16][17][18], the modified Kudryashov method [19], the generalized exp ( ( )) ϕ ξ − -expansion method [20,21], and so on. Due to the effectiveness of mathematical approaches, the advance exp(-( )) ξ Φ -expansion method may be easily applicable with the aid of symbolic computational software to find more general solitary and periodic wave solutions of NPDEs in mathematical physics and engineering.…”

An exponential expansion method and its application to the strain wave equation in microstructured solids

“…The wave propagation phenomena are observed in microstructured solids, plasma physics, chemical physics, elastic media, optical fibers, fluid dynamics, quantum mechanics, etc. With the rapid development of nonlinear science based on computer algebraic system, many effective methods have been presented, such as, the tanh method [1], the extended tanh method [2,3], the modified extended tanhfunction method [4,5], the Exp-function method [6][7][8], the improved F-expansion method [9], the exp(ÀU(n))-expansion method [10][11][12][13][14], the sine-cosine method [15], the modified simple equation method [16][17][18][19][20][21][22], the (G 0 /G)-expansion method [23,24], the novel (G 0 /G)-expansion method [25], new approach of the generalized (G 0 /G)-expansion method [26,27], the Jacobi elliptic function method [28,29], the homogeneous balance method [30][31][32], the Hirota's bilinear method [33], the homotopy perturbation technique [34] and others.…”