Exact Solutions Expressible in Rational Formal Hyperbolic and Elliptic Functions for Nonlinear Differential-Difference Equation

2008 ◽  
Vol 50 (2) ◽  
pp. 303-308
Author(s):  
Bai Cheng-Jie ◽  
Zhao Hong ◽  
Han Ji-Guang
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Elliptic Functions ◽  
Differential Difference Equation

Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative

2016 ◽  
Vol 8 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Ahmet Bekir ◽  
Ozkan Guner ◽  
Burcu Ayhan ◽  
Adem C. Cevikel
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Difference Equations ◽  
Applied Mathematics ◽  
Expansion Method ◽  
New Exact Solutions ◽  
Liouville Derivative ◽  
Nonlinear Lattice ◽  
Fractional Differential ◽  
Differential Difference Equation

AbstractIn this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

Differential-Difference Equation Arising in Nanotechnology and it's Exact Solutions

2013 ◽  
Vol 23 ◽  
pp. 113-116 ◽  
Author(s):  
Sheng Zhang ◽  
Qian An Zong ◽  
Qun Cao ◽  
Dong Liu
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Continuum Hypothesis ◽  
Nanoporous Materials ◽  
Difference Model ◽  
Electron Conduction ◽  
New Exact Solutions ◽  
Alternative Approach ◽  
Differential Difference Equation ◽  
Model Equations

Differential-difference model equations are often considered as an alternative approach to describing some phenomena arising in heat/electron conduction and flow in carbon nanotubes and nanoporous materials, this is due to the fact that continuum hypothesis is no longer valid. A (2+1)-dimensional nonlinear differential-difference equation with an arbitrary function is introduced and new exact solutions are obtained.

N-soliton-like and double Casoratian solutions of a nonisospectral Ablowitz–Ladik equation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640008
Author(s):  
Shou-Ting Chen ◽  
Qi Li ◽  
Deng-Yuan Chen ◽  
Jun-Wei Cheng
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Spectral Problem ◽  
Bilinear Form ◽  
Direct Method ◽  
Dependent Variables ◽  
Differential Difference Equation ◽  
Bilinear Equation

A bilinear form of a nonisospectral differential-difference equation related to the Ablowitz–Ladik (AL) spectral problem is derived by a transformation of dependent variables. Exact solutions to the resulting bilinear equation are found. The [Formula: see text]-soliton-like solutions and the double Casoratian solutions are derived by means of Hirota’s direct method and the double Casoratian technique, respectively. Moreover, the connection between those two classes of solutions is explored.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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