On the solution of the non‐linear Korteweg–de Vries equation by the decomposition method

Kybernetes ◽  
2002 ◽  
Vol 31 (5) ◽  
pp. 766-772 ◽  
Author(s):  
Y. Cherruault ◽  
M. Inc ◽  
K. Abbaoui
Keyword(s):  
Decomposition Method ◽  
Vries Equation ◽  
Non Linear ◽  
De Vries

scholarly journals New Cubic B-spline Approximations for Solving Non-linear Third-order Korteweg-de Vries Equation

2019 ◽  
Vol 12 (6) ◽  
pp. 1-9
Author(s):  
Muhammad Abbas ◽  
Muhammad Kashif Iqbal ◽  
Bushra Zafar ◽  
Shazalina Binti Mat Zin ◽  
◽  
...  
Keyword(s):  
Vries Equation ◽  
Third Order ◽  
B Spline ◽  
Non Linear ◽  
De Vries

scholarly journals Computational and Numerical Solutions for 2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Raghda A. M. Attia ◽  
S. H. Alfalqi ◽  
J. F. Alzaidi ◽  
Mostafa M. A. Khater ◽  
Dianchen Lu
Keyword(s):  
Solitary Waves ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Adomian Decomposition Method ◽  
Simplest Equation Method ◽  
Adomian Decomposition ◽  
Tanh Method ◽  
De Vries ◽  
Parameter Values

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation

2017 ◽  
Vol 72 (3) ◽  
pp. 217-222 ◽  
Author(s):  
Jin-Xi Fei ◽  
Wei-Ping Cao ◽  
Zheng-Yi Ma
Keyword(s):  
Vries Equation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
Non Linear ◽  
De Vries ◽  
Finite Transformation ◽  
Non Local

AbstractThe non-local residual symmetry for the classical Korteweg-de Vries equation is derived by the truncated Painlevé analysis. This symmetry is first localised to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we then obtain the finite transformation for the localised residual symmetry. Based on the consistent tanh expansion method, some exact interaction solutions among different non-linear excitations are explicitly presented finally. Some special interaction solutions are investigated both in analytical and graphical ways at the same time.

scholarly journals Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

2021 ◽  
Vol 7 (2) ◽  
pp. 1580-1602
Author(s):  
Khalid Khan ◽  
◽  
Amir Ali ◽  
Manuel De la Sen ◽  
Muhammad Irfan ◽  
...  
Keyword(s):  
Decomposition Method ◽  
Vries Equation ◽  
Coupled System ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Double Laplace Transform ◽  
Proposed Model ◽  
De Vries ◽  
Singular Kernels ◽  
The Waves

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 &lt; \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

Non-linear Dynamics and Exact Solutions for the Variable-Coefficient Modified Korteweg–de Vries Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 143-149 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Rational Solutions ◽  
Transformation Technique ◽  
Linear Dynamics ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

AbstractThis paper presents some new exact solutions which contain soliton solutions, breather solutions and two types of rational solutions for the variable-coefficient-modified Korteweg–de Vries equation, with the help of the multivariate transformation technique. Furthermore, based on these new soliton solutions, breather solutions and rational solutions, we discuss their non-linear dynamics properties. We also show the graphic illustrations of these solutions which can help us better understand the evolution of solution waves.

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