scholarly journals SOLUSI NUMERIK PERSAMAAN GELOMBANG KORTEWIEG DE VRIES (KDV)

2013 ◽  
Vol 7 (2) ◽  
pp. 1-7
Author(s):  
Francis Y. Rumlawang
Keyword(s):  
Numerical Solutions ◽  
Kdv Equation ◽  
Wave Profile ◽  
Wave Form ◽  
De Vries ◽  
Running Wave

One of KdV wave form is 𝑢𝑡 + 6𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0. This paper deals with finding numerical solutions of KdV’s equation which form a running wave 𝑢(𝑥, 𝑡) = 𝑢(𝑥 − 𝜆𝑡), by using Stepeest DescentMethod which is charged on Hamilton 𝐻(𝑢) and Momentum 𝑀(𝑢). By using MAPLE software, we obtain numerical solutions of KdV equation in the form of running wave profile

scholarly journals The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idris Dag
Keyword(s):  
Numerical Experiments ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Algebraic Equations ◽  
Thomas Algorithm ◽  
B Spline ◽  
Suggested Algorithm ◽  
De Vries

The exponential cubic B-spline algorithm is presented to find the numerical solutions of the Korteweg-de Vries (KdV) equation. The problem is reduced to a system of algebraic equations, which is solved by using a variant of Thomas algorithm. Numerical experiments are carried out to demonstrate the efficiency of the suggested algorithm.

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.

Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method

2018 ◽  
Vol 32 (29) ◽  
pp. 1850365 ◽  
Author(s):  
Asıf Yokuş
Keyword(s):  
Difference Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Expansion Method ◽  
Von Neumann ◽  
Forward Difference ◽  
De Vries ◽  
Conformable Derivatives ◽  
Von Neumann Stability Analysis

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G[Formula: see text])-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2and L[Formula: see text] error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables.

scholarly journals Analytical approach to a generalized Hirota-Satsuma coupled Korteweg-de Vries equation by modified variational iteration method

2016 ◽  
Vol 20 (3) ◽  
pp. 885-888 ◽  
Author(s):  
Jun-Feng Lu ◽  
Li Ma
Keyword(s):  
Iteration Method ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
De Vries ◽  
Modified Variational Iteration Method

In this paper, we apply the modified variational iteration method to a generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation. The numerical solutions of the initial value problem of the generalized Hirota-Satsuma coupled KdV equation are provided. Numerical results are given to show the efficiency of the modified variational iteration method.

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

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.

Some numerical solutions for Lamb's problem

1974 ◽  
Vol 64 (2) ◽  
pp. 473-491
Author(s):  
Harold M. Mooney
Keyword(s):  
Rayleigh Wave ◽  
Poisson’S Ratio ◽  
Pulse Width ◽  
Numerical Solutions ◽  
Pulse Height ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Form ◽  
Lamb’S Problem ◽  
Wave Forms

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

scholarly journals On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source

2019 ◽  
Vol 50 (3) ◽  
pp. 281-291 ◽  
Author(s):  
G. U. Urazboev ◽  
A. K. Babadjanova
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Scattering Data ◽  
Self Consistent ◽  
De Vries ◽  
The Matrix

In this work we deduce laws of the evolution of the scattering  data for the matrix Zakharov Shabat system with the potential that is the solution of the matrix modied KdV equation with a self consistent source.

THE SECOND-ORDER KdV EQUATION AND ITS SOLITON-LIKE SOLUTION

2009 ◽  
Vol 23 (14) ◽  
pp. 1771-1780 ◽  
Author(s):  
CHUN-TE LEE ◽  
JINN-LIANG LIU ◽  
CHUN-CHE LEE ◽  
YAW-HONG KANG
Keyword(s):  
Solitary Waves ◽  
Soliton Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Close Resemblance ◽  
De Vries

This paper presents both the theoretical and numerical explanations for the existence of a two-soliton solution for a second-order Korteweg-de Vries (KdV) equation. Our results show that there exists "quasi-soliton" solutions for the equation in which solitary waves almost retain their identities in a suitable physical regime after they interact, and bear a close resemblance to the pure KdV solitons.

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