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.

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.

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.

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.

On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

scholarly journals Second Order Scheme For Korteweg-De Vries (KDV) Equation

2019 ◽  
Vol 43 (1) ◽  
pp. 85-93
Author(s):  
Khandaker Md Eusha Bin Hafiz ◽  
Laek Sazzad Andallah
Keyword(s):  
Stability Condition ◽  
Kdv Equation ◽  
Convex Combination ◽  
Second Order ◽  
Combination Method ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
The Kdv Equation ◽  
De Vries ◽  
Order Scheme

The kinematics of the solitary waves is formed by Korteweg-de Vries (KdV) equation. In this paper, a third order general form of the KdV equation with convection and dispersion terms is considered. Explicit finite difference schemes for the numerical solution of the KdV equation is investigated and stability condition for a first-order scheme using convex combination method is determined. Von Neumann stability analysis is performed to determine the stability condition for a second order scheme. The well-known qualitative behavior of the KdV equation is verified and error estimation for comparisons is performed. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 85-93, 2019

scholarly journals Dispersive dynamics in the characteristic moving frame

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180784 ◽  
Author(s):  
D. J. Ratliff
Keyword(s):  
Shallow Water ◽  
Lagrangian Density ◽  
Kdv Equation ◽  
Moving Frame ◽  
Modulation Equations ◽  
Water Flows ◽  
Shallow Water Flows ◽  
The Kdv Equation ◽  
De Vries ◽  
Klein Gordon

A mechanism for dispersion to automatically arise from the dispersionless Whitham Modulation equations (WMEs) is presented, relying on the use of a moving frame. The speed of this is chosen to be one of the characteristics which emerge from the linearization of the Whitham system, and assuming these are real (and thus the WMEs are hyperbolic) morphs the WMEs into the Korteweg-de Vries (KdV) equation in the boosted coordinate. Strikingly, the coefficients of the KdV equation are universal, in the sense that they are determined by abstract properties of the original Lagrangian density. Two illustrative examples of the theory are given to illustrate how the KdV may be constructed in practice. The first being a revisitation of the derivation of the KdV equation from shallow water flows, to highlight how the theory of this paper fits into the existing literature. The second is a complex Klein–Gordon system, providing a case where the KdV equation may only arise with the use of a moving frame.

The growth of periodic waves in gas-fluidized beds

1996 ◽  
Vol 325 ◽  
pp. 261-282 ◽  
Author(s):  
S. E. Harris
Keyword(s):  
Nonlinear Waves ◽  
Fluidized Beds ◽  
Kdv Equation ◽  
Periodic Waves ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Kdv Equation ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds ◽  
The One

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham & Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham & Merkin (1986) and Anderson, Sundaresan & Jackson (1995) is then carried out.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

scholarly journals Fractional System of Korteweg-De Vries Equations via Elzaki Transform

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 673
Author(s):  
Wenfeng He ◽  
Nana Chen ◽  
Ioannis Dassios ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Fractional Order ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Hybrid Technique ◽  
Transform Method ◽  
De Vries ◽  
In Series ◽  
Analytical Result ◽  
Approximate Result ◽  
Good Agreement

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

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