Modified method of simplest equation for obtaining exact solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms

Yu, Jun; Wang, Deng‐Shan; Sun, Yan; Wu, Suping

doi:10.1007/s11071-016-2837-7

Cited by 22 publications

(6 citation statements)

References 59 publications

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“…It is interesting to know that other simplest equations can also be rearranged into the Riccati-type equations. The famous examples are Jacobi [15] and Weierstrass Equations [16], which can solve a large class of nonlinear differential equations. Let us consider Jacobi type equation with variable coefficients,…”

Section: The Jacobi and Weierstrass Equationsmentioning

confidence: 99%

On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

Nugroho¹,

Darwito²,

Wahyuono³

et al. 2021

Recent Developments in the Solution of Nonlinear Differential Equations

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The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.

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Section: The Jacobi and Weierstrass Equationsmentioning

confidence: 99%

On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

Nugroho¹,

Darwito²,

Wahyuono³

et al. 2021

Recent Developments in the Solution of Nonlinear Differential Equations

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show abstract

“…There are many known powerful methods that can be used to find the exact solutions of nonlinear partial differential equations, such as Hirota's bilinear method [2], the inverse scattering transform method [1], the ( G G )-expansion method [15,16], the Riccati-Bernoulli sub-ODE method [17,18], the homogeneous balance method [19], and the generalized Riccati equation mapping method [20,21]. Other recent meritorious work on finding exact solutions include [22][23][24][25][26][27][28]. Further, readers interested in the solutions of fractional differential equations or fractional forms of the KdV equation should consult [29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation

2022

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In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature.

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“…In order to better understand the mechanism behind the phenomena described by nonlinear fractional partial differential equations, it is necessary to obtain the exact solution, which also provides a reference for the accuracy and stability of the numerical solution. With the rapid development of computer algebraic system-based nonlinear sciences like Mathematica or Maple, divers' effective methods have been pulled out to acquire precise solutions to nonlinear fractional-order partial differential equations, such as the fractional first integral method [5,6], the fractional simplest equation method [7,8], the improved fractional subequation method [9], the Kudryashov method [10], the fractional subequation method [11,12], the generalised Kudryashov method [13], the fractional exp-function method [14][15][16][17][18][19], the sech-tanh function expansion method [20,21], the fractional (G′/G)-expansion method [22][23][24][25][26][27][28][29], the generalized Sinh-Gorden expansion method [30], the fractional functional variable method [31], the rational (G ′ /G)-expansion method [32], the modified Khater method [33][34][35][36], and the fractional modified trial equation method [37,38]. Many of these methods are constructed by fractional complex transform [39,40] and use of the solutions of some solvable differential equations.…”

Section: Introductionmentioning

confidence: 99%

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

Fan

Zhou

2020

Advances in Mathematical Physics

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With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

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Modified method of simplest equation for obtaining exact solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms

Cited by 22 publications

References 59 publications

On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

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