scholarly journals Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 871-880
Author(s):  
Bo Tang ◽  
Jiajia Tao ◽  
Shijun Chen ◽  
Junfeng Qu ◽  
Qian Wang ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Space Time ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Physical Features

Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.

scholarly journals Solutions of Fractional Konopelchenko-Dubrovsky and Nizhnik-Novikov-Veselov Equations Using a Generalized Fractional Subequation Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yanqin Liu ◽  
Limei Yan
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Coupled Equations ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Relationship Of ◽  
The Relationship

A new generalized fractional subequation method based on the relationship of fractional coupled equations is proposed. This method is applied to the space-time fractional coupled Konopelchenko-Dubrovsky equations and Nizhnik-Novikov-Veselov equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. It is observed that the proposed approach provides a simple and reliable tool for solving many other fractional coupled differential equations.

scholarly journals Exact Solutions of the Time Fractional BBM-Burger Equation by Novel(G′/G)-Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Muhammad Shakeel ◽  
Qazi Mahmood Ul-Hassan ◽  
Jamshad Ahmad ◽  
Tauseef Naqvi
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Burger Equation ◽  
Graphical Representation ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Methods Numerical

The fractional derivatives are used in the sense modified Riemann-Liouville to obtain exact solutions for BBM-Burger equation of fractional order. This equation can be converted into an ordinary differential equation by using a persistent fractional complex transform and, as a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions are attained. The performance of the method is reliable, useful, and gives newer general exact solutions with more free parameters than the existing methods. Numerical results coupled with the graphical representation completely reveal the trustworthiness of the method.

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 A Novel (G'/G)-Expansion Method and its Application to the Space-Time Fractional Symmetric Regularized Long Wave (SRLW) Equation

2015 ◽  
Vol 2 ◽  
pp. 1-16 ◽  
Author(s):  
Muhammad Shakeel ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Differential Equation ◽  
Expansion Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Space Time ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Long Wave ◽  
Complex Transformation ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

In this work, we use the fractional complex transformation which converts nonlinear fractional partial differential equation to nonlinear ordinary differential equation. A fractional novel (G`/G) - expansion method is used to look for exact solutions of nonlinear evolution equation with the aid of symbolic computation. To check the validity of the method we choose the space-time fractional symmetric regularized long wave (SRLW) equation and as a result, many exact analytical solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. The performance of the method is reliable, useful and gives more new general exactsolutions than the existing methods.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

2009 ◽  
Vol 64 (9-10) ◽  
pp. 540-552
Author(s):  
Mamdouh M. Hassan
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Wave System ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Long Wave ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

Exact Solutions for Coupled mKdV Equations by a New Symbolic Computation Method

2010 ◽  
Vol 20-23 ◽  
pp. 184-189 ◽  
Author(s):  
Bang Qing Li ◽  
Yu Lan Ma
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Computation Method ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

By introducing (G′/G)-expansion method and symbolic computation software MAPLE, two types of new exact solutions are constructed for coupled mKdV equations. The solutions included trigonometric function solutions and hyperbolic function solutions. The procedure is concise and straightforward, and the method is also helpful to find exact solutions for other nonlinear evolution equations.

The Modified (G'/G)-Expansion Method for Nonlinear Evolution Equations

2011 ◽  
Vol 66 (1-2) ◽  
pp. 33-39 ◽  
Author(s):  
Sheng Zhang ◽  
Ying-Na Sun ◽  
Jin-Mei Ba ◽  
Ling Dong
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

A modified (Gʹ/G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

scholarly journals Various Exact Solutions for the Conformable Time-Fractional Generalized Fitzhugh–Nagumo Equation with Time-Dependent Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sami Injrou ◽  
Ramez Karroum ◽  
Nadia Deeb
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Nagumo Equation

In this paper, the subequation method and the sine-cosine method are improved to give a set of traveling wave solutions for the time-fractional generalized Fitzhugh–Nagumo equation with time-dependent coefficients involving the conformable fractional derivative. Various structures of solutions such as the hyperbolic function solutions, the trigonometric function solutions, and the rational solutions are constructed. These solutions may be useful to describe several physical applications. The results show that these methods are shown to be affective and easy to apply for this type of nonlinear fractional partial differential equations (NFPDEs) with time-dependent coefficients.

scholarly journals Travelling Wave Solutions of the Schrödinger-Boussinesq System

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Adem Kılıcman ◽  
Reza Abazari
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Boussinesq System ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Physical Problems ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

We establish exact solutions for the Schrödinger-Boussinesq Systemiut+uxx−auv=0,vtt−vxx+vxxxx−b(|u|2)xx=0, whereaandbare real constants. The (G′/G)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.

scholarly journals Exact Solutions of the Generalized KP-BBM Equation by the G′ / G-expansion Method and the First Integral Method

2018 ◽  
pp. 1-16
Author(s):  
Huaji Cheng ◽  
Yanxia Hu
Keyword(s):  
Exact Solutions ◽  
Periodic Solutions ◽  
Integral Method ◽  
Expansion Method ◽  
First Integral ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
First Integral Method ◽  
Hyperbolic Function Solutions ◽  
Bbm Equation

In this paper, the generalized KP-BBM equation is considered. The G′ / G-expansion method and the first integral method are applied to integrate the equation. By means of the two methods, the rational solutions, the periodic solutions and the hyperbolic function solutions are thus obtained under some parametric conditions.

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