Kamruzzaman Khan

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.

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.

A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations.

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations

The modified simple equation (MSE) method is executed to find the traveling wave solutions for the coupled Konno-Oono equations and the variant Boussinesq equations. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It has been shown that the proposed method is direct, effective, and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.

The exp(−Φ(ξ))–expansion method for finding travelling wave solutions of Vakhnenko–Parkes equation

The paper employs the exp(−Φ(ξ))–expansion method for finding exact solutions of the Vakhnenko–Parkes (VP) equation. Each of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, exponential function solutions and rational function solutions, contain an explicit function of the variables in the considered equation. It has been shown that the method provides a powerful mathematical tool for solving non–linear wave equations in mathematical physics and engineering problems.

Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations

In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions–hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters.PACS numbers02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg

Study of analytical method to seek for exact solutions of variant Boussinesq equations

In this paper, we have been acquired the soliton solutions of the Variant Boussinesq equations. Primarily, we have used the enhanced (G′/G)-expansion method to find exact solutions of Variant Boussinesq equations. Then, we attain some exact solutions including soliton solutions, hyperbolic and trigonometric function solutions of this equation.Mathematics subject classification35 K99; 35P05; 35P99

The improved F-expansion method and its application to the MEE circular rod equation and the ZKBBM equation

Abstract In the present article, we implement the improved F-expansion method combined with Riccati equation to attain traveling wave solutions to nonlinear evolution equations (NLEEs) via the Magneto-electro-elastic circular rod equation and the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. We establish three classes of explicit solutions-hyperbolic, trigonometric and rational solutions containing several free parameters. For particular values of the parameters, solitary wave solutions are emanated from the traveling wave solutions. It turns out that, the method is straightforward, concise and it can be applied to many other NLEEs in mathematical physics and engineering.

Solitary wave solutions of the (2+1)-dimensional Zakharov-Kuznetsevmodified equal-width equation

Abstract In this letter, the enhanced (G′/G)-expansion method is applied to seek exact solitary wave solutions of the nonlinear evolution equations (NLEEs). Here we derive exact solutions for the (2+1)-dimensional Zakharov-Kuznetsev-Modified Equal-Width (ZK-MEW) equation. The obtained results show that the applied equation reveal richness of explicit solitons and periodic solutions. It has been shown that the method is effective and can be used for many other NLEEs in mathematical physics.