scholarly journals Solution of Moving Boundary Space-Time Fractional Burger’s Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
E. A.-B. Abdel-Salam ◽  
E. A. Yousif ◽  
Y. A. S. Arko ◽  
E. A. E. Gumma
Keyword(s):  
Moving Boundary ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Space Time ◽  
Hyperbolic Function ◽  
Burger's Equation ◽  
Burger’S Equation ◽  
Hyperbolic Function Solutions ◽  
First Time ◽  
Boundary Space

The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.

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 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.

scholarly journals New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method

2019 ◽  
Vol 4 (1) ◽  
pp. 129-138 ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut ◽  
Tukur Abdulkadir Sulaiman
Keyword(s):  
Wave Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Structures ◽  
Computational Program ◽  
Hyperbolic Function Solutions

AbstractIn this paper, a powerful sine-Gordon expansion method (SGEM) with aid of a computational program is used in constructing a new hyperbolic function solutions to one of the popular nonlinear evolution equations that arises in the field of mathematical physics, namely; longren-wave equation. We also give the 3D and 2D graphics of all the obtained solutions which are explaining new properties of model considered in this paper. Finally, we submit a comprehensive conclusion at the end of this paper.

scholarly journals The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Guiying Chen ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method

2021 ◽  
pp. 2150312
Author(s):  
Rodica Cimpoiasu
Keyword(s):  
Riccati Equation ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Time Variable ◽  
Auxiliary Equation ◽  
Natural Phenomena ◽  
Generalized Riccati Equation ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.

Exact Soliton Solutions To An Averaged Dispersion-Managed Fiber System Equation

2004 ◽  
Vol 59 (12) ◽  
pp. 919-926
Author(s):  
Biao Li
Keyword(s):  
Evolution Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Fiber System ◽  
System Equation ◽  
Generalized Riccati Equation ◽  
Hyperbolic Function Solutions ◽  
Hyperbolic Function Method

By introducing a set of ordinary differential equations which possess q-deformed hyperbolic function solutions, and a new ansatz, a method is developed for constructing a series of exact analytical solutions of some nonlinear evolution equations. The proposed method is more powerful than various tanh methods, the secq-tanhq-method, generalized hyperbolic-function method, generalized Riccati equation expansion method, generalized projective Riccati equations method and other sophisticated methods. As an application of the method, an averaged dispersion-managed (DM) fiber system equation, which governs the dynamics of the core of the DM soliton, is chosen to illustrate the method. With the help of symbolic computation, rich new soliton solutions are obtained. From these solutions, some previously known solutions obtained by some authors can be recovered by means of some suitable choices of the arbitrary functions and arbitrary constants. Further, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

scholarly journals New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures

2016 ◽  
Vol 6 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut
Keyword(s):  
Complex Structure ◽  
Three Dimensional ◽  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Zakharov Equation ◽  
Wolfram Mathematica ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

scholarly journals Applications of an Extended -Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
E. M. E. Zayed ◽  
Shorog Al-Joudi
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Nonlinear Pdes ◽  
Hyperbolic Function ◽  
Soliton Equation ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

We construct the traveling wave solutions of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, the (2+1)-dimensional typical breaking soliton equation, the (1+1)-dimensional classical Boussinesq equations, and the (2+1)-dimensional Broer-Kaup-Kuperschmidt equations by using an extended -expansion method, whereGsatisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic 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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