scholarly journals Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Tahir Ayaz ◽  
Farhad Ali ◽  
Wali Khan Mashwani ◽  
Israr Ali Khan ◽  
Zabidin Salleh ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Order Differential Equation ◽  
Kdv Equation ◽  
Third Order ◽  
Lagrange Method ◽  
Approximate Symmetries ◽  
Weakly Nonlinear ◽  
The Kdv Equation ◽  
De Vries ◽  
Perturbed Kdv Equation

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

2021 ◽  
Author(s):  
Murat Polat ◽  
Ömer Oruç
Keyword(s):  
Conservation Laws ◽  
Lie Group ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
High Order ◽  
Basis Functions ◽  
Numerical Integrator ◽  
The Kdv Equation ◽  
De Vries ◽  
Novel Method

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

scholarly journals Solutions of the KdV Equation through Analysis of Regular Symmetries

2021 ◽  
Vol 20 ◽  
pp. 387-398
Author(s):  
S. Y. Jamal ◽  
J. M. Manale
Keyword(s):  
Burgers Equation ◽  
Kdv Equation ◽  
Lie Symmetries ◽  
Third Order ◽  
The Kdv Equation ◽  
De Vries ◽  
Topological Manifolds

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

scholarly journals Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yu-Shan Bai ◽  
Qi Zhang
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Symmetry Algebra ◽  
Optimal System ◽  
Approximate Symmetry ◽  
Invariant Solutions ◽  
Approximate Symmetries ◽  
One Dimensional ◽  
Perturbed Kdv Equation ◽  
Partial Lagrangian

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.

scholarly journals Nonlinear water waves (KdV) equation and Painleve technique

2015 ◽  
Vol 4 (2) ◽  
pp. 216
Author(s):  
Attia Mostafa
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Nonlinear Pde ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
The Third ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

scholarly journals Solutions of the perturbed KdV equation for convecting fluids by factorizations

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Octavio Cornejo-Pérez ◽  
Haret Rosu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Kdv Equation ◽  
Travelling Wave ◽  
Second Order ◽  
Travelling Wave Solutions ◽  
Second Order Differential Equation ◽  
Wave Solutions ◽  
Perturbed Kdv Equation

AbstractIn this paper, we obtain some new explicit travelling wave solutions of the perturbed KdV equation through recent factorization techniques that can be performed when the coefficients of the equation fulfill a certain condition. The solutions are obtained by using a two-step factorization procedure through which the perturbed KdV equation is reduced to a nonlinear second order differential equation, and to some Bernoulli and Abel type differential equations whose solutions are expressed in terms of the exponential andWeierstrass functions.

SPHERICAL KORTEWEG – DE VRIES SOLITONS FROM RELATIVISTIC HYDRODYNAMICS

2007 ◽  
Vol 16 (09) ◽  
pp. 3019-3023 ◽  
Author(s):  
D. A. FOGAÇA ◽  
F. S. NAVARRA
Keyword(s):  
Fluid Dynamics ◽  
Equation Of State ◽  
Kdv Equation ◽  
Relativistic Hydrodynamics ◽  
Spherical Coordinates ◽  
Relativistic Fluid Dynamics ◽  
Continuity Equations ◽  
Relativistic Fluid ◽  
The Kdv Equation ◽  
De Vries

We study the conditions for the formation and propagation of Korteweg – de Vries (KdV) solitons in relativistic fluid dynamics using an appropriate equation of state. It is known that in cartesian coordinates the combination of the Euler and continuity equations leads to the KdV equation. Recently it has been shown that this is true also in relativistic hydrodynamics. Here we show that a KdV-like equation can be obtained in relativistic hydrodynamics in spherical coordinates.

scholarly journals A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1254
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Sobia Sultana ◽  
Zakia Hammouch ◽  
Rasool Shah ◽  
...  
Keyword(s):  
Nonlinear Waves ◽  
Analytical Procedure ◽  
Integral Transform ◽  
Kdv Equation ◽  
Analytical Techniques ◽  
Transform Method ◽  
Fractional Pdes ◽  
Approximate Analytical Solutions ◽  
The Kdv Equation ◽  
De Vries

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

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