The Modified Exp-Function Method and Its Application to the Nonlinear Dispersive Equation with Variable Coefficients

2014 ◽  
Vol 7 (1) ◽  
pp. 63-79
Author(s):  
Mahmoud A. M. Abdelaziz
Keyword(s):  
Function Method ◽  
Variable Coefficients ◽  
Dispersive Equation

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 Multi-wave solutions for a non-isospectral KDV-type equation with variable coefficients

2012 ◽  
Vol 16 (5) ◽  
pp. 1476-1479 ◽  
Author(s):  
Sheng Zhang ◽  
Qun Gao ◽  
Qian-An Zong ◽  
Dong Liu
Keyword(s):  
Variable Coefficient ◽  
Function Method ◽  
Evolution Equations ◽  
Vries Equation ◽  
Type Equation ◽  
Variable Coefficients ◽  
Linear Evolution ◽  
Wave Solutions ◽  
De Vries ◽  
Linear Evolution Equations

As a typical mathematical model in fluids and plasmas, Korteweg-de Vries equation is famous. In this paper, the Exp-function method is extended to a nonisos-pectral Korteweg-de Vries type equation with three variable coefficients, and multi-wave solutions are obtained. It is shown that the Expfunction method combined with appropriate ansatz may provide with a straightforward, effective and alternative method for constructing multi-wave solutions of variable-coefficient non-linear evolution equations.

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 ON THE RELATION BETWEEN CHANGES IN INTEGRAL QUANTITIES OF SHOALING WAVES AND BREAKING INCEPTION

1982 ◽  
Vol 1 (18) ◽  
pp. 2
Author(s):  
Takeshi Yasuda ◽  
Shintaro Goto ◽  
Yoshito Ysuchiya
Keyword(s):  
Conservation Laws ◽  
Stream Function ◽  
Function Method ◽  
Breaking Waves ◽  
Variable Coefficients ◽  
Shoaling Waves ◽  
Wave Profiles ◽  
Internal Properties ◽  
Particle Velocities ◽  
The Waves

This paper describes a mechanism of breaking waves over sloping bottoms in terms of changes in integral quantities of the waves. Systematic computations are made of wave profiles of shoaling waves up to the numerical unstable points by using the K-dV equation with variable coefficients and internal properties such as horizontal and vertical water particle velocities by a stream function method satisfying the conservation laws of mass and energy. Applicability of the numerical results is examined and a relation between numerical unstable points and actual breaker points is found. Characteristics of the integral quantities of shoaling waves are investigated in relation to the existence of the extremum of the energy of the shoaling waves and their breaking inception.

scholarly journals Dirichlet problem for a second-order ordinary differential equation with a distributed differentiation operator

2021 ◽  
Vol 21 (4) ◽  
pp. 37-44
Author(s):  
B.I. Efendiev ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dirichlet Problem ◽  
Function Method ◽  
Order Differential Equation ◽  
Variable Coefficients ◽  
Differentiation Operator ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation

For an ordinary second-order differential equation with an operator of continuously distributed differentiation with variable coefficients, a solution to the Dirichlet problem is constructed using the Green’s function method.

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