scholarly journals An unconditionally stable space–time FE method for the Korteweg–de Vries equation

2020 ◽  
Vol 371 ◽  
pp. 113297
Author(s):  
Eirik Valseth ◽  
Clint Dawson
Keyword(s):  
Vries Equation ◽  
Space Time ◽  
Fe Method ◽  
Unconditionally Stable ◽  
Stable Space ◽  
De Vries

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

2018 ◽  
Vol 32 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Strong Basis ◽  
De Vries

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

scholarly journals Generation and Analysis of a New Implicit Difference Scheme for the Korteweg-de Vries Equation

2018 ◽  
Vol 173 ◽  
pp. 03006
Author(s):  
Yuri Blinkov ◽  
Vladimir Gerdt ◽  
Konstantin Marinov
Keyword(s):  
Difference Scheme ◽  
Vries Equation ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
Algorithmic Approach ◽  
De Vries ◽  
Volume Method ◽  
Modified Equation ◽  
Two Parameter

In this paper we apply our computer algebra based algorithmic approach to construct a new finite difference scheme for the two-parameter form of the Korteweg-de Vries equation. The approach combines the finite volume method, numerical integration and difference elimination. Being implicit, the obtained scheme is consistent and unconditionally stable. The modified equation for the scheme shows that its accuracy is of the second order in each of the mesh sizes. Using exact one-soliton solution, we compare the numerical behavior of the scheme with that of the other two schemes known in the literature and having the same order of accuracy. The comparison reveals numerical superiority of our scheme.

The Korteweg–de-Vries equation on a noncommutative space-time

2000 ◽  
Vol 278 (3) ◽  
pp. 139-145 ◽  
Author(s):  
A. Dimakis ◽  
F. Müller-Hoissen
Keyword(s):  
Vries Equation ◽  
Space Time ◽  
Noncommutative Space ◽  
De Vries

scholarly journals Hyperbolic Tangent Ansatz Method to Space Time Fractional Modified KdV, Modified EW and Benney–Luke Equations

2018 ◽  
Author(s):  
Ozlem Ersoy Hepson
Keyword(s):  
Exact Solutions ◽  
Vries Equation ◽  
Space Time ◽  
Wave Transformation ◽  
One Dimension ◽  
Governing Equations ◽  
Hyperbolic Tangent ◽  
Ansatz Method ◽  
De Vries

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

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