scholarly journals The General Traveling Wave Solutions of the Fisher Equation with Degree Three

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Wenjun Yuan ◽  
Qiuhui Chen ◽  
Jianming Qi ◽  
Yezhou Li
Keyword(s):  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Necessary Condition ◽  
Complex Method ◽  
Mathematical Tool ◽  
Meromorphic Solutions ◽  
Sufficient And Necessary Condition ◽  
Powerful Mathematical Tool

We employ the complex method to research the integrality of the Fisher equations with degree three. We obtain the sufficient and necessary condition of the integrable of the Fisher equations with degree three and the general meromorphic solutions of the integrable Fisher equations with degree three, which improves the corresponding results obtained by Feng and Li (2006), Guo and Chen (1991), and Ağırseven and Öziş (2010). Moreover, allwg,1(z)are new general meromorphic solutions of the Fisher equations with degree three forc=±3/2. Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.

scholarly journals A Note about the General Meromorphic Solutions of the Fisher Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Jian-ming Qi ◽  
Qiu-hui Chen ◽  
Wei-ling Xiong ◽  
Wen-jun Yuan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Physics ◽  
Fisher Equation ◽  
Complex Method ◽  
Mathematical Tool ◽  
Meromorphic Solutions ◽  
Partial Differential ◽  
Powerful Mathematical Tool

We employ the complex method to obtain the general meromorphic solutions of the Fisher equation, which improves the corresponding results obtained by Ablowitz and Zeppetella and other authors (Ablowitz and Zeppetella, 1979; Feng and Li, 2006; Guo and Chen, 1991), andwg,i(z)are new general meromorphic solutions of the Fisher equation forc=±5i/6.Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

scholarly journals Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Libing Zeng ◽  
Keding Qin ◽  
Shengqiang Tang
Keyword(s):  
Traveling Wave ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Physics ◽  
Traveling Wave Solution ◽  
Mathematical Tool ◽  
Extended Tanh Method ◽  
Physical Problems ◽  
Tanh Method ◽  
Powerful Mathematical Tool

The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

scholarly journals New solitary Solution for the Kudryashov-Sinelshchikov (KS) equation by Modern Extension of the Hyperbolic Method

2020 ◽  
Vol 25 (2) ◽  
pp. 124
Author(s):  
Ali H. Hazza1 ◽  
Wafaa M. Taha2 ◽  
Raad A. Hameed1 ◽  
, Israa A. Ibrahim1 ◽  
, Israa A . Ibrahim1
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Applied Mathematics ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Mathematical Tool ◽  
3D Graphics ◽  
Hyperbolic Method ◽  
Powerful Mathematical Tool

In the present paper, we apply the modern extension of the hyperbolic tanh function method of nonlinear partial differential equations (NLPDEs) of Kudryashov - Sinelshchikov (KS) equation for obtaining exact and solitary traveling wave solutions. Through our solutions, we gain various functions, such as, hyperbolic, trigonometric and rational functions. Additionally, we support our results by comparisons with other methods and painting 3D graphics of the exact solutions. It is shown that our method provides a powerful mathematical tool to find exact solutions for many other nonlinear equations in applied mathematics   http://dx.doi.org/10.25130/tjps.25.2020.039

scholarly journals Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1003-1010
Author(s):  
Asıf Yokuş ◽  
Hülya Durur ◽  
Taher A. Nofal ◽  
Hanaa Abu-Zinadah ◽  
Münevver Tuz ◽  
...  
Keyword(s):  
Traveling Wave ◽  
Analytical Methods ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Dark Soliton ◽  
Mathematical Tool ◽  
Symbolic Calculation ◽  
Advantages And Disadvantages ◽  
Wave Solutions

Abstract In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

scholarly journals Exact Traveling Wave Solutions to the (2 + 1)-Dimensional Jaulent-Miodek Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yongyi Gu ◽  
Bingmao Deng ◽  
Jianming Lin
Keyword(s):  
Differential Equations ◽  
Computer Simulations ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Complex Method ◽  
Exact Traveling Wave Solutions ◽  
Applied Method ◽  
Wave Solutions

We derive exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation by means of the complex method, and then we illustrate our main result by some computer simulations. It has presented that the applied method is very efficient and is practically well suited for the nonlinear differential equations that arise in mathematical physics.

scholarly journals Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fu Zhang ◽  
Jian-ming Qi ◽  
Wen-jun Yuan
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

Multiple wave solutions for nonlinear burgers equations using the multiple exp-function method

2021 ◽  
pp. 2150149
Author(s):  
Khaled A. Gepreel ◽  
E. M. E. Zayed
Keyword(s):  
Traveling Wave ◽  
Software Package ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Multiple Wave ◽  
Wave Solutions ◽  
The One

In this paper, we use the multiple exp-function method to explicity present traveling wave solutions, double-traveling wave (DTW) solutions and triple-traveling wave solutions (TWs) which include one-soliton, double-soliton and triple-soliton solutions for nonlinear partial differential equations (NPDEs) via, the (2+1)-dimensional and (3+1)-dimensional nonlinear Burgers PDEs in mathematical physics. In this work, we build some series of straightforward and new solutions successfully with the help of a computerized symbol computational software package like Maple or Mathematica. We will make some drawings in some cases with specific values for the relevant parameters for each obtained solutions such as the one-traveling wave solutions, double-traveling wave solutions and TWs. This method is efficient and powerful in solving a wide class of NPDEs.

scholarly journals Exact Traveling Wave Solutions of the Gardner Equation by the Improved tanΘϑ-Expansion Method and the Wave Ansatz Method

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hatıra Günerhan
Keyword(s):  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Tool ◽  
Ansatz Method ◽  
Gardner Equation ◽  
Wave Solutions ◽  
Analytical Approaches

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

scholarly journals The Two-Variable -Expansion Method for Solving the Nonlinear KdV-mKdV Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
E. M. E. Zayed ◽  
M. A. M. Abdelaziz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mkdv Equation ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Powerful Mathematical Tool

We apply the two-variable (, )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ()-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ()-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

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