scholarly journals Features of numerical solutions of some problems for cnoidal waves as periodic solutions of the Korteweg - de Vries

2021 ◽  
Vol 13 (5) ◽  
pp. 885-901
Author(s):  
Alexander P Chernyaev ◽  
Sofia A Chernyaeva
Keyword(s):  
Periodic Solutions ◽  
Numerical Solutions ◽  
Cnoidal Waves ◽  
De Vries

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics

2020 ◽  
Vol 34 (29) ◽  
pp. 2050282
Author(s):  
Asıf Yokuş ◽  
Doğan Kaya
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Traveling Wave Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Traveling Wave Solution ◽  
Von Neumann ◽  
Wave Solutions ◽  
De Vries ◽  
The Stability

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

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

DYNAMIC ANALYSIS OF A HOLLING I PREDATOR-PREY SYSTEM WITH MUTUAL INTERFERENCE CONCERNING PEST CONTROL

2005 ◽  
Vol 13 (01) ◽  
pp. 45-58 ◽  
Author(s):  
YUJUAN ZHANG ◽  
ZHILONG XU ◽  
BING LIU ◽  
LANSUN CHEN
Keyword(s):  
Biological Control ◽  
Periodic Solutions ◽  
Pest Control ◽  
Chemical Control ◽  
Numerical Solutions ◽  
Dynamical Behavior ◽  
Continuous System ◽  
Mutual Interference ◽  
Predator Prey ◽  
Predator Prey Model

A Holling I predator-prey model with mutual interference concerning pest control is proposed and analyzed. The prey and predator are considered to be a pest and a natural enemy, respectively. The model is forced by the addition of periodic impulsive terms representing predator import (biological control) and pesticide application (chemical control) at different fixed moments. By using Floquet theory and small amplitude perturbations, we show the existence and stability of pest-free periodic solutions. Further, we prove that when the stability of pest-free periodic solutions is lost, the system is permanent by using analytic methods of differential equation theory. Numerical solutions are also given, which show that when stability of pest-free periodic solutions is lost, more exotic behavior can occur, such as quasi-periodic oscillation or chaos. We investigate the effect of impulsive perturbations on the unforced continuous system, and find that the forced system has a different dynamical behavior with a different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system. Finally, we compare the validity of the combination of biological control and chemical control with classical methods and conclude that the synthetical strategy is more effective than classical methods if we take effective chemical control.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
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).

scholarly journals Control and stabilization of the periodic fifth order Korteweg-de Vries equation

2019 ◽  
Vol 25 ◽  
pp. 38
Author(s):  
Cynthia Flores ◽  
Derek L. Smith
Keyword(s):  
Exponential Stability ◽  
Periodic Solutions ◽  
Vries Equation ◽  
Regularity Property ◽  
Contraction Principle ◽  
Smoothing Effect ◽  
Dissipative Term ◽  
De Vries ◽  
Fifth Order

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in Hs(𝕋), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.

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