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.

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.

Extension of the Homotopy Perturbation Method for Solving Nonlinear Differential-Difference Equations

2010 ◽  
Vol 65 (12) ◽  
pp. 1060-1064 ◽  
Author(s):  
Mohamed Medhat Mousa ◽  
Aidarkan Kaltayev ◽  
Hasan Bulut
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Lattice Equation ◽  
Convenient Tool ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
De Vries

In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schr¨odinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations.

scholarly journals Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Khaled A. Gepreel ◽  
A. R. Shehata
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Elliptic Functions ◽  
Mathematical Physics ◽  
Jacobi Elliptic Function ◽  
Lattice Equation ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Saturable Nonlinearity ◽  
New Exact Solutions ◽  
Elliptic Solutions

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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.

On exact solutions of fractional differential-difference equations with Ψ-Riemann–Liouville derivative

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Rajagopalan Ramaswamy ◽  
Mohamed S. Abdel Latif ◽  
Amr Elsonbaty ◽  
Abas H. Abdel Kader
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Three Dimensional ◽  
Generalized Solutions ◽  
Lattice Equation ◽  
Closed Form Solutions ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
First Time ◽  
Rl Fractional Derivative

Abstract The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods.

Application of the Exp-Function Method for Solving Some Evolution Equations with Nonlinear Terms of any Orders

2010 ◽  
Vol 65 (12) ◽  
pp. 1039-1044 ◽  
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Evolution Equations ◽  
Burgers Equation ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Full Agreement ◽  
Different Types

In this paper, suitable transformations and a so-called exp-function method are used to obtain different types of exact solutions for some nonlinear evolution equations with variable coefficients and nonlinear terms of any orders. The Korteweg-de Vries equation and the Burgers equation with nonlinear terms of any orders are chosen to show how to apply the exp-function method for these kinds of nonlinear equations. These exact solutions are in full agreement with the previous results obtained by Ebaid and by Zhu.

Homotopy Perturbation Method for Solving Nonlinear Differential- Difference Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 511-517 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Schrödinger Equation ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Schrodinger Equation ◽  
Lattice Equation ◽  
Homotopy Perturbation

In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons between the results of the presented method and exact solutions are made. The results reveal that the HPM is very effective and convenient for solving such kind of equations.

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