Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis

Bilal, Muhammad; Seadawy, Aly R.; Younis, Muhammad; Rizvi, Syed T. R.; Zahed, Hanadi

doi:10.1002/mma.7013

Cited by 116 publications

(28 citation statements)

References 42 publications

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“…In this section, we briefly describe the G /G 2 -expansion method, which is discussed in [21][22][23][24][25][26][27]29,63]. Consider the nonlinear conformable partial differential equation of the unknown function u = u(x 1 , x 2 , ..., x n , t) consisting of the independent variables x 1 , x 2 , ..., x n , and t as follows:…”

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

confidence: 99%

“…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. In 2018, Arshed and Sadia [23] used the (G /G 2 )-expansion method to obtain some new traveling wave solutions for the time-fractional Burgers equation, the fractional biological population model, and the space-time fractional Whitham-Broer-Kaup equations.…”

Section: Introductionmentioning

confidence: 99%

“…In 2020, the (G /G 2 )-expansion approach was employed to construct some novel exact traveling wave solutions of the (2 + 1)-dimensional Boiti-Leon-Pempinelli system [28]. In 2021, Bilal et al [29] proposed new exact solutions, which consist of shock, singular, shock-singular, and singular periodic wave solutions obtained by the (G /G 2 )expansion method, to unidirectional Dullin-Gottwald-Holm (DGH) system describing the prorogation of waves in shallow water.…”

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%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

et al. 2021

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“…In 2020, the (G /G 2 )-expansion method was used to build numerous novel accurate travelling wave solutions for the Boiti-Leon-Pempinelli system in two dimensions [31]. The unidirectional Dullin-Gottwald-Holm (DGH) system describing wave prorogation in shallow water, which include singular periodic wave solutions, shock-singular, shock, and singular solutions investigated in 2021 by Bilal et al [32] to find novel exact solutions via the (G /G 2 )-expansion approach.…”

Section: Introductionmentioning

confidence: 99%

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

Shakeel

Chung

2022

Fractal Fract

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In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

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“…Researchers are looking for new and general solitons from various NLPDEs, such as the generalized Burgers, Broer-Kaup-Kupershmidt, modified Kawahara, Dullin-Gottwald-Holm, modified Zakharov-Kuznetsov, fractional wazwazbenjamin-bona-mahony, Camassa-Holm, (3 + 1)-dimensional extended Jimbo-Miva equation, and nonlinear Schrödinger dynamical models. Furthermore, they have recently proposed and implemented integration strategies to analyze NLPDEs to find the soliton solutions [19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Pattern Formation of a Bubbly Fluid Mixture under the Effect of Thermodynamics via Kudryashov‐Sinelshchikov Model

Abdel‐Aty

Raza

Batool

et al. 2022

Journal of Mathematics

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In this paper, new explicit wave solutions via liquid-gas bubbles are obtained for the fractional Kudryashov-Sinelshchikov (KS) equation under thermodynamic assumptions. A new fractional de nition is applied to get these solutions that are utilized to represent the phenomenon of pressure waves under thermodynamic conditions. Two analytical techniques are used to explore the model which is sinh-Gorden equation expansion and Riccati-Bernoulli Sub-ODE methods. ese approaches provide complex hyperbolic, hyperbolic, complex trigonometric, and trigonometric solutions for the fractional KS equation, particularly singular, combined singular, dark, bright, combined dark-bright, and other soliton solutions. Furthermore, acquired results are illustrated by 3D graphs for suitable parametric values that highlight the physical importance and dynamical behaviors of the equation. It is also demonstrated that the purposed approaches are powerful strategies for developing exact traveling wave solutions for a wide range of problems found in mathematical sciences.

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Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis

Cited by 116 publications

References 42 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

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

Pattern Formation of a Bubbly Fluid Mixture under the Effect of Thermodynamics via Kudryashov‐Sinelshchikov Model

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