M. Ali Akbar

New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method

This article was published in the Journal of Applied Mathematics [© 2012 Hasibun Naher et al.] and the definite version is available at :http://dx.doi.org/10.1155/2012/575387 The Journals website is at:https://www.hindawi.com/journals/jam/2012/575387/

A Generalized and Improved -Expansion Method for Nonlinear Evolution Equations

A generalized and improved -expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.

The exp-function method for new exact solutions of the nonlinear partial differential equations

This article was published in the International Journal of Physical Sciences [© 2011 Academic Journals] and the definite version is available at : 10.5897/IJPS11.1026

The (′/)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the (𝐺/𝐺)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (𝐺/𝐺)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method

Abstract Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this article, we implemented the modified simple equation (MSE) method for finding the exact solutions of NLEEs via the (2+1)-dimensional cubic Klein–Gordon (cKG) equation and the (3+1)-dimensional Zakharov–Kuznetsov (ZK) equation and achieve exact solutions involving parameters. When the parameters are assigned special values, solitary wave solutions are originated from the exact solutions. It is established that the MSE method offers a further influential mathematical tool for constructing exact solutions of NLEEs in mathematical physics.

Generalized and Improved (G′/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation

The generalized and improved -expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.

Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G′/G)-expansion method

The new approach of the generalized (G′/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G′/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method

Mathematical modeling of many autonomous physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear evolution equations plays a significant role in the study of nonlinear physical phenomena. In this article, the enhanced (G′/G)-expansion method has been applied for finding the exact traveling wave solutions of longitudinal wave motion equation in a nonlinear magneto-electro-elastic circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. It has been shown that the applied method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields.

Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations

AbstractIn this paper, we implement the exp(−Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

New extended (G’/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation

In this article, a new extended (G′/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.