Painlevé property and approximate solutions using Adomian decomposition for a nonlinear KdV-like wave equation

Sohail, Ayesha; Maqbool, Khadija; Hayat, Tasawar

doi:10.1016/j.amc.2013.12.056

Cited by 9 publications

(4 citation statements)

References 22 publications

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“…Indeed, there exist methods that only work for a particular set of equations, but even so, the subsequent mathematics is often quite different from linear PDEs, and they generally present an evident difficulty and complexity. In particular, various direct methods have been developed to deal with the determination of exact solutions of nonlinear PDEs, for instance, the extended simplest equation method [10][11][12], the tanh-sech method [13][14][15], the Painlevé analysis [16,17], the variational iteration method [18], the Hirota's method [19], and other special methods.…”

Section: Introductionmentioning

confidence: 99%

Applications of Solvable Lie Algebras to a Class of Third Order Equations

et al. 2022

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A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

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Section: Introductionmentioning

confidence: 99%

Applications of Solvable Lie Algebras to a Class of Third Order Equations

et al. 2022

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“…It was observed by Sohail et al [29] that wave data obtained from an online measurement technique satisfies this evolution equation, when the wave length is just a few multiples of the fluid depth. In [30], the integrability of this new nonlinear partial differential equation was discussed with a focus on the Painlevé property, the compatibility condition, and the Bäcklund transformation.…”

Section: Journal Of Applied Mathematicsmentioning

confidence: 99%

Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

Zhao

2014

Journal of Applied Mathematics

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The simplest equation method presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a new nonlinear KdV-like wave equation by means of the simplest equation method, the modified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin film mass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

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“…The majority of these methods are only valid for the fractional differential equations with linear and constant coefficients. For nonlinear fractional differential equations, there are some techniques for studying the existence and multiplicity of their solution, for example, the theories of a fixed point, the theory of Leray-Shauder, Adomian decomposition and the variational iteration method [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. In [43], the authors studied the impact of the fractional derivatives on the prorogation and the formulation of the solitary waves corresponding to certain type of KdV equation in the form…”

Section: Introductionmentioning

confidence: 99%

Generalized KdV equation involving Riesz time-fractional derivatives: constructing and solution utilizing variational methods

Alshenawy

Al-Alwan

Elmandouh

2020

Journal of Taibah University for Science

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In this work, the time-fractional generalized Korteweg de Vries (TFGKdV) is derived by utilizing the method of a semi-inverse and the variational principles. Based on the initial condition relying on the dispersion and nonlinear coefficients, we can apply the He's variational iteration to construct an approximated solution for the TFGKdV equation. Finally, we study the impact of the fractional derivatives on the propagation and the structure of the solitary waves obtained from the solution of TFGKdV equation.

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Painlevé property and approximate solutions using Adomian decomposition for a nonlinear KdV-like wave equation

Cited by 9 publications

References 22 publications

Applications of Solvable Lie Algebras to a Class of Third Order Equations

Applications of Solvable Lie Algebras to a Class of Third Order Equations

Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

Generalized KdV equation involving Riesz time-fractional derivatives: constructing and solution utilizing variational methods

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