Traveling wave solutions of Benny Luke equation via the enhanced (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"><mml:mrow><mml:mi>G</mml:mi><mml:mtext>'</mml:mtext><mml:mo stretchy="false">/</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:math>)-expansion method

Hossain, Altaf; Akbar, M. Ali

doi:10.1016/j.asej.2017.03.018

Cited by 15 publications

(5 citation statements)

References 43 publications

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“…where C, D are arbitrary nonzero constants. The exact traveling wave solutions of Equation ( 9) can be obtained by inserting the obtained values of a 0 , a j , b j (j = 1, 2, ..., N), c 1 , c 2 , ..., c n , k and the solutions ( 15)-( 17) into Equation (12) with the transformation (10).…”

Section: Algorithm Of the G /G 2 -Expansion Methodsmentioning

confidence: 99%

“…Numerous kinds of exact solutions such as solitons, positons, complexitons, dromions, cuspon, rational, kink, periodic, and quasiperiodic solutions have been obtained via solving integrable NLEEs. In the past couple of decades, many robust, efficient, and powerful methods exist that have been developed for finding exact solutions of NLEEs, including the (G /G, 1/G)-expansion method [11], the enhanced (G /G)-expansion method [12], the exp-function method [13], the Jacobi elliptic equation method [14], the generalized Kudryashov's method [15], the sine-Gordon expansion method [16], the sub-equation method [17], the improved tan(φ/2)-expansion method [18], and the extended direct algebraic method [19,20]. More recently, the (G /G 2 )-expansion method [4,[21][22][23][24][25][26][27][28][29] has attracted a remarkable amount of attention of many researchers who employed the method to construct exact solutions of certain NPDEs.…”

Section: Introductionmentioning

confidence: 99%

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Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

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The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

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Section: Algorithm Of the G /G 2 -Expansion Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

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“…There is no effective method for solving all varieties of nonlinear equations till now. As a result many powerful and efficient methods have been developed to find exact traveling wave solutions such as, the tanh-method [1][2], the extended tanhfunction method [3], the improved Bernoulli sub-equation function (IBSEF) method [4], the Global Multi-Trace Method [5][6], the extended direct algebraic method [7][8], the (G′/G)expansion method [9][10][11], the extended (G′/G)-expansion method [12], the enhanced (G′/G)expansion method [13][14][15][16], The FD-BPM algorithm [17], the modified F-expansion method [18], the Hirota bilinear method [19][20], the modified simple equation method [21][22][23][24][25][26][27], the simple equation method [28], the complex method [29], the free space radiation mode (FSRM) method [30], the homogeneous balance method [31], the Exp(-Φ(η))-expansion method [32][33][34], the new generalized (G′/G)-expansion method [35][36][37][38], the Adomian decomposition method [39], the sine-cosine method [40] etc. The new generalized 𝐺 ′ /𝐺)eexpansion method is a very impressive, powerful and effective mathematical tool for searc...…”

Section: Introductionmentioning

confidence: 99%

Soliton solutions of DSW and Burgers equations by generalized (G′/G)-expansion method

Hossain,

Akter,

Akbar

2024

Opt Quant Electron

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Nonlinear evolution equations play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences such as condensed matter physics, biophysics, atomic chains, optical fiber, chemical kinematics, molecular crystals and mathematical biology. The new generalized (G′/G)-expansion method provides an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics for their easy calculation procedure. In this paper, two nonlinear evolution equations named the Drinfeld-Sokolov-Wilson equation and the Burgers equation are considered to find more new exact solutions by executing the new generalized (G′/G)-expansion method. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations. We provide some 3D plots to realize characteristics of the solutions.

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“…However, in the existing literature, there is no unique method that has been developed to effectively analyze all types of NLEEs. Consequently, various techniques have been established to obtain analytical and numerical solutions to NLEEs, such as the Painlevé approach [13], the Jacobi elliptic function scheme [14], the homotopy perturbation technique [15], the exponential function process [16], the modified simple equation technique [17], the improved F-expansion approach [18], the functional variable scheme [19], the enhanced MSE method [20], the improved Kudryashov procedure [21], the enhanced ( / ¢ G G)-expansion technique [22], the Bilinear residual network method [23], the new auxiliary equation scheme [24], the modified exp-function process [25], etc and put into use with the assistance of computer algebra like Mathematica. One of the most effective and direct techniques for extracting wave solutions to NLEEs is the first integral approach.…”

Section: Introductionmentioning

confidence: 99%

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

Hussain,

Akbar,

İlhan

2024

Phys. Scr.

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In this article, we investigate the (2+1)-dimensional dispersive long water wave equation and the (1+1)-dimensional PHI-four equation, which describe the behavior of long gravity waves with small amplitudes, long wave propagation in oceans and seas, coastal structures and harbor design, effects of wave motion on sediment transport, quantum field theory, phase transitions of matter, ferromagnetic systems, liquid-gas transitions, and the structure of optical solitons. We use the first integral technique and obtain new and generic solutions for the models under consideration. By setting definite values for the associated parameters, various types of richly structured solitons are generated. The solitons include kink, flat kink, bell-shaped, anti-bell-shaped, and singular kink formations. These solutions allow for a profound understanding of the behavior and properties of the phenomena, offering new insights and potential applications in the associated field. The first integral technique is simpler, directly integrates the models, and the solutions offer clear insights into the underlying phenomena without requiring intermediate steps, making it widely applicable to various other models, including nonlinear equations and those that are challenging to solve using other standard techniques.

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Traveling wave solutions of Benny Luke equation via the enhanced (G'/G)-expansion method

Cited by 15 publications

References 43 publications

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Soliton solutions of DSW and Burgers equations by generalized (G′/G)-expansion method

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

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