scholarly journals On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative

2021 ◽  
Vol 10 (1) ◽  
pp. 46-65
Author(s):  
Yeşim Sağlam Özkan ◽  
Emrullah Yaşar ◽  
Nisa Çelik
Keyword(s):  
Exact Solutions ◽  
Wave Solution ◽  
Expansion Method ◽  
Wave Transformation ◽  
Equation Modeling ◽  
Periodic Wave Solution ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Electrostatic Wave ◽  
Function Solution

Abstract The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) - dimensional conformable KdV equation modeling the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. By using the improved tan(φ/2)-expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.

scholarly journals New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yinghui He ◽  
Yun-Mei Zhao ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Wave Model ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

scholarly journals Exact solutions with external linear functions for the potential Yu-Toda-Sasa-Fukuyama equation

2018 ◽  
Vol 22 (4) ◽  
pp. 1621-1628 ◽  
Author(s):  
Sheng Zhang ◽  
Qianan Zong
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Dynamical Property ◽  
Linear Functions ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Modified Method ◽  
Linear Pde ◽  
Non Linear ◽  
Hyperbolic Function Solutions

Constructing exact solutions of non-linear PDE is of both theoretical and practical values. In this paper, a modified F-expansion method is proposed to construct exact solutions of non-linear PDE. To illustrate the validity and advantages of the proposed method, a (3+1)-D potential Yu-Toda-Sasa-Fukuyama equation is considered and more general exact solutions with external linear functions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. It is shown that the original F-expansion method can not construct exact solutions of the potential Yu-Toda-Sasa-Fukuyama equation but the modified method is valid. The modified F-expansion method can lead to such exact solutions with external linear functions which possess a remarkable dynamical property, which is the solitary wave does not propagate in the horizontal direction as the traditional waves. The modified F-expansion method can be used for exactly solving some other non-linear PDE.

New Exact Solutions of (2+1)-Dimensional Generalization of Shallow Water Wave Equation by (G′/G)-Expansion Method

2010 ◽  
Vol 20-23 ◽  
pp. 1516-1521 ◽  
Author(s):  
Bang Qing Li ◽  
Mei Ping Xu ◽  
Yu Lan Ma
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Function Solution ◽  
Shallow Water Wave

Extending a symbolic computation algorithm, namely, (G′/G)-expansion method, a new series of exact solutions are constructed for (2+1)-dimensional generalization of shallow water wave equation. These solutions included hyperbolic function solution, trigonometric function solution and rational function solution. The procedure can illustrate that the new algorithm is concise, powerful and straightforward, and it can also be applied to find exact solutions for other high dimensional nonlinear evolution equations.

scholarly journals Exact Solutions for(4+1)-Dimensional Nonlinear Fokas Equation Using ExtendedF-Expansion Method and Its Variant

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Weierstrass Elliptic Function ◽  
New Exact Solutions ◽  
Higher Dimensional

The construction of exact solution for higher-dimensional nonlinear equation plays an important role in knowing some facts that are not simply understood through common observations. In our work,(4+1)-dimensional nonlinear Fokas equation, which is an important physical model, is discussed by using the extendedF-expansion method and its variant. And some new exact solutions expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

scholarly journals Exact solutions with arbitrary functions of the (4+1)-dimensional fokas equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2403-2411 ◽  
Author(s):  
Bo Xu ◽  
Sheng Zhang
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Trigonometric Function ◽  
High Dimensional ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

In this paper, the (4+1)-dimensional Fokas equation is solved by the generalized F-expansion method, and new exact solutions with arbitrary functions are obtained. The obtained solutions include Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the generalized F-expansion method can be used for constructing exact solutions with arbitrary functions of some other high dimensional partial differential equations in fluids.

scholarly journals F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Trigonometric Functions ◽  
Extended Version ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions

Based on theF-expansion method, and the extended version ofF-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained. When the modulus m of Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.

scholarly journals New Jacobi Elliptic Function Solutions for the Kudryashov-Sinelshchikov Equation Using Improved F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Based on the F-expansion method with a new subequation, an improved F-expansion method is introduced. As illustrative examples, some new exact solutions expressed by the Jacobi elliptic function of the Kudryashov-Sinelshchikov equation are obtained. When the modulusmof the Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function and trigonometric function can also be obtained. The method is straightforward and concise and is promising and powerful for other nonlinear evolution equations in mathematical physics.

A Class of Exact Solutions of (3+1)-Dimensional Generalized B-Type Kadomtsev–Petviashvili Equation

2017 ◽  
Vol 18 (2) ◽  
pp. 137-143 ◽  
Author(s):  
Shuang Liu ◽  
Yao Ding ◽  
Jian-Guo Liu
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
New Exact Solutions ◽  
Petviashvili Equation

AbstractBy employing the generalized$(G'/G)$-expansion method and symbolic computation, we obtain new exact solutions of the (3 + 1) dimensional generalized B-type Kadomtsev–Petviashvili equation, which include the traveling wave exact solutions and the non-traveling wave exact solutions showed by the hyperbolic function and the trigonometric function. Meanwhile, some interesting physics structure are discussed.

scholarly journals New Jacobi Elliptic Function Solutions for the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function

A generalized(G′/G)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

scholarly journals Generalized Kudryashov Method for Time-Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Seyma Tuluce Demiray ◽  
Yusuf Pandir ◽  
Hasan Bulut
Keyword(s):  
Exact Solutions ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Wave Transformation ◽  
Hyperbolic Function ◽  
Third Order ◽  
Hilliard Equation ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Cahn Hilliard Equation

In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

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