Numerical Solutions and Pole Expansion for Perturbed Korteweg-de Vries Equation

1994 ◽  
Vol 63 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Sho Hosoda ◽  
Takuji Kawahara
Keyword(s):  
Numerical Solutions ◽  
Vries Equation ◽  
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.

scholarly journals New numerical solutions of fractional-order Korteweg-de Vries equation

2020 ◽  
Vol 19 ◽  
pp. 103326
Author(s):  
Mustafa Inc ◽  
Mohammad Parto-Haghighi ◽  
Mehmet Ali Akinlar ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Order ◽  
Numerical Solutions ◽  
Vries Equation ◽  
De Vries

scholarly journals Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yuexing Bai ◽  
Temuer Chaolu ◽  
Sudao Bilige
Keyword(s):  
Deep Learning ◽  
Prediction Error ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Mkdv Equation ◽  
Dynamic State ◽  
Limited Memory ◽  
De Vries ◽  
Physical Phenomena

In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches 1 0 − 6 . The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.

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 Generation of solitary waves by transcritical flow over a step

2007 ◽  
Vol 587 ◽  
pp. 235-254 ◽  
Author(s):  
R. H. J. GRIMSHAW ◽  
D.-H. ZHANG ◽  
K. W. CHOW
Keyword(s):  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Undular Bore ◽  
Downstream Side ◽  
Upstream Side ◽  
Transcritical Flow ◽  
De Vries ◽  
Positive Step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg–de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent

1978 ◽  
Vol 361 (1706) ◽  
pp. 265-275 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Critical Value ◽  
Similarity Solutions ◽  
Amplitude Parameter ◽  
Painlevé Transcendent ◽  
De Vries

The second Painlevé transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equation and the modified Korteweg-de Vries equation. Asymptotic expansions and numerical solutions are presented; the main result is the existence of a critical value for the amplitude parameter above which the relevant solutions become singular.

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.

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