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conduction and flow in carbon nanotubes and nanoporous materials, this is due to the fact that continuum hypothesis is no longer valid.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.

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.

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.

scholarly journals Infinite many conservation laws of discrete system associated with a 3×3 matrix spectral problem

2017 ◽  
Vol 21 (4) ◽  
pp. 1613-1619
Author(s):  
Sheng Zhang ◽  
Dongdong Liu
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Difference Equations ◽  
Discrete System ◽  
Nanoporous Materials ◽  
Electron Conduction ◽  
Alternative Approach ◽  
Non Linear ◽  
Linear Differential ◽  
Matrix Spectral Problem

Differential-difference 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. Infinite many conservation laws play important role in discussing the integrability of non-linear differential equations. In this paper, infinite many conservation laws of the non-linear differential-difference equations associated with a 3?3 matrix spectral problem are obtained.

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

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