Exact Solutions for The Generalized Zakharov-Kuznetsov Equation with Variable Coefficients Using The Generalized (GʹG)-expansion Method

2010 ◽  
Author(s):  
Elsayed M. E. Zayed ◽  
Mahmoud A. M. Abdelaziz ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients

scholarly journals Exact Solutions for the Wick-Type Stochastic Schamel-Korteweg-de Vries Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Xueqin Wang ◽  
Yadong Shang ◽  
Huahui Di
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Hermite Transformation

We consider the Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients in this paper. With the aid of symbolic computation and Hermite transformation, by employing the (G′/G,1/G)-expansion method, we derive the new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered equations.

Study of the Solitoff Solution and Fractal Solution for the (2+1)-Dimensional Broek-Kaup System with Variable Coefficients

2014 ◽  
Vol 532 ◽  
pp. 356-361
Author(s):  
Wei Ting Zhu
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Variable Separation ◽  
New Family ◽  
Localized Excitations ◽  
Variable Separation Method

Starting from a (G'/G)-expansion method and a variable separation method, a new family of exact solutions of the (2+1)-dimensional Broek-Kaup system with variable coefficients(VCBK) is obtained. Based on the derived solitary wave solution, we obtain some special localized excitations such as solitoff solutions and fractal solutions.

Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

2021 ◽  
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Interaction Solution ◽  
Group Invariant Solutions ◽  
Dimensional Variable

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Optimal System ◽  
Differential Invariants ◽  
Group Method ◽  
Equivalence Transformations ◽  
3 Dimensional ◽  
Dimensional Variable

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

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