An efficient modification of the decomposition method with a convergence parameter for solving Korteweg de Vries equations

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.

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The Time-Fractional Coupled-Korteweg-de-Vries Equations

Abstract and Applied Analysis ◽

10.1155/2013/947986 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 44

Author(s):

Abdon Atangana ◽

Aydin Secer

Keyword(s):

Decomposition Method ◽

Homotopy Perturbation Method ◽

Numerical Solutions ◽

Adomian Decomposition Method ◽

Variational Iteration Method ◽

Analytical Technique ◽

Adomian Decomposition ◽

Homotopy Decomposition ◽

De Vries ◽

Homotopy Analysis

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

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Adomian decomposition method for solving a Generalized Korteweg – De Vries equation with boundary conditions

Journal of King Abdulaziz University-Science ◽

10.4197/sci.23-2.6 ◽

2011 ◽

Vol 23 (2) ◽

pp. 79-90

Author(s):

Bothayna Kashkari

Keyword(s):

Boundary Conditions ◽

Decomposition Method ◽

Vries Equation ◽

Adomian Decomposition Method ◽

Adomian Decomposition ◽

De Vries

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

AIMS Mathematics ◽

10.3934/math.2022092 ◽

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 < \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>

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