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 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 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 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 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 Bifurcation and Solitary-Like Solutions for Compound KdV-Burgers-Type Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yin Li ◽  
Ruiying Wei ◽  
Guoming Jian
Keyword(s):  
Dynamical Systems ◽  
Type Equation ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Bifurcation Method ◽  
Dynamical Characteristics ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Ode Method ◽  
Parameter Condition

Firstly, based on the improved sub-ODE method and the bifurcation method of dynamical systems, we investigate the bifurcation of solitary waves in the compound KdV-Burgers-type equation. Secondly, numbers of solitary patterns solutions are given for each parameter condition and numerical simulations are used to display the dynamical characteristics. Finally, we obtain twelve solitary patterns solutions under some parameter conditions, such as the trigonometric function solutions and the hyperbolic function solutions.

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.

scholarly journals A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fanwei Meng
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Lattice Equation ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Toda Lattice Equation

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

Symmetry reduction, exact solutions and conservation laws of the Bogoyavlenskii equation

2019 ◽  
Vol 35 (01) ◽  
pp. 1950339
Author(s):  
Zhenli Wang ◽  
Chuan Zhong Li ◽  
Lihua Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Power Series Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Group Invariant Solutions ◽  
Symmetry Method

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

Exact solutions of Wick-type stochastic Korteweg – de Vries equation

2012 ◽  
Vol 90 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Sheng Zhang
Keyword(s):  
Vries Equation ◽  
Similarity Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
De Vries ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Hermite Transformation ◽  
Similarity Reductions

An algorithm is devised for using the compatibility method to find symmetries and similarity reductions of Wick-type stochastic partial differential equations. To illustrate the validity and advantages of the devised algorithm, the Wick-type stochastic Korteweg – de Vries equation is considered. With the aid of Hermite transformation and white noise theory, we obtain a symmetry and corresponding reduction of the Wick-type stochastic Korteweg – de Vries equation. Furthermore, some new similarity solutions are obtained including Jacobi elliptic function 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.

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