scholarly journals Exact Solutions for the Generalized BBM Equation with Variable Coefficients

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Cesar A. Gómez ◽  
Alvaro H. Salas
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Iteration Algorithm ◽  
Variational Iteration ◽  
Generalized Bbm Equation ◽  
Bbm Equation

The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation (BBM) with variable coefficients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered.

scholarly journals Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Alvaro H. Salas S ◽  
Cesar A. Gómez S
Keyword(s):  
Exact Solutions ◽  
Periodic Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Third Order ◽  
Forcing Term ◽  
Projective Riccati Equation Method

The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

scholarly journals New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Alvaro H. Salas ◽  
Lorenzo J. Martinez H ◽  
David L. Ocampo R
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Series Method ◽  
Power Series Method ◽  
Weierstrass Elliptic Function ◽  
Generalized Bbm Equation ◽  
Bbm Equation

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

New Exact Solutions for Two Nonlinear Equations

2006 ◽  
Vol 61 (1-2) ◽  
pp. 1-6 ◽  
Author(s):  
Zonghang Yang
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Exact Solutions ◽  
Water Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
New Exact Solutions ◽  
Complex Phenomena ◽  
Sivashinsky Equation

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutions for the second were just N-soliton solutions. In this paper we present kinds of new exact solutions by using the extended tanh-function method.

Study on soliton solutions of the longitudinal wave equation and magneto-electro-elastic circular rod dynamical model

2021 ◽  
Vol 35 (13) ◽  
pp. 2150168
Author(s):  
Adel Darwish ◽  
Aly R. Seadawy ◽  
Hamdy M. Ahmed ◽  
A. L. Elbably ◽  
Mohammed F. Shehab ◽  
...  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Longitudinal Wave ◽  
Physical Interpretation ◽  
Elliptic Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Jacobi Elliptic Function ◽  
Two Dimensional

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation.

Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method

2019 ◽  
Vol 33 (09) ◽  
pp. 1950106 ◽  
Author(s):  
Behzad Ghanbari
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Graphical Interpretation ◽  
Wave Solutions ◽  
Complex Models ◽  
Exponential Rational Function Method

In this paper, some new traveling wave solutions to the Hirota–Maccari equation are constructed with the help of the newly introduced method called generalized exponential rational function method. Several families of exact solutions are found corresponding to the equation. To the best of our knowledge, these solutions are new, and have never been addressed in the literature. The graphical interpretation of the solutions is also depicted. Moreover, it is contemplated that the proposed technique can also be employed to another sort of complex models.

The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations

2016 ◽  
Vol 71 (2) ◽  
pp. 103-112 ◽  
Author(s):  
E.M.E. Zayed ◽  
Abdul-Ghani Al-Nowehy
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Short Wave ◽  
Simple Equation ◽  
Equation Method ◽  
Wave Resonance ◽  
Modified Simple Equation Method

AbstractThe modified simple equation method, the exp-function method, and the method of soliton ansatz for solving nonlinear partial differential equations are presented. Based on these three different methods, we obtain the exact solutions and the bright–dark soliton solutions with parameters of the long-short wave resonance equations which describe the resonance interaction between the long wave and the short wave. When these parameters take special values, the solitary wave solutions are derived from the exact solutions. We compare the results obtained using the three methods. Also, a comparison between our results and the well-known results is given.

scholarly journals Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Bo Tang ◽  
Xuemin Wang ◽  
Yingzhe Fan ◽  
Junfeng Qu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Mathematical Tool ◽  
Auxiliary Equation Method

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.

Kink-Type Solutions of the MKdV Lattice Equation with an Arbitrary Function

2014 ◽  
Vol 989-994 ◽  
pp. 1716-1719 ◽  
Author(s):  
Sheng Zhang ◽  
Ying Ying Zhou ◽  
Bin Cai
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Arbitrary Function ◽  
Function Method ◽  
Variable Coefficients ◽  
Spatial Structures ◽  
Improved Method ◽  
Lattice Equation

In this paper, the exp-function method is improved for constructing exact solutions of nonlinear differential-difference equations with variable coefficients. To illustrate the validity and advantages of the improved method, the mKdV lattice equation with an arbitrary function is considered. As a result, kink-type solutions are obtained which possess rich spatial structures. It is shown that the improved exp-function method can be applied to some other nonlinear differential-difference equations with variable coefficients.

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