On the Origins of Symmetries of Partial Differential Equations: the Example of the Korteweg—de Vries Equation

2008 ◽  
Vol 15 (sup1) ◽  
pp. 60-68 ◽  
Author(s):  
Keshlan S Govinder ◽  
Barbara Abraham-Shrauner
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vries Equation ◽  
Partial Differential ◽  
De Vries

scholarly journals Asymptotic analysis of the singularly perturbed Korteweg-de Vries equation

2019 ◽  
pp. 194-197
Author(s):  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
V. S. Vovk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Vries Equation ◽  
Singular Part ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
De Vries

The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

scholarly journals New exact solutions of fractional Hirota-Satsuma coupled Korteweg-de Vries equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
Lian-Xiang Cui ◽  
Li-Mei Yan ◽  
Yan-Qin Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

An improved extended tg-function method, which combines the fractional complex transform and the extended tanh-function method, is applied to find exact solutions of non-linear fractional partial differential equations. Generalized Hirota-Satsuma coupled Korteweg-de Vries equations are used as an example to elucidate the effectiveness and simplicity of the method.

scholarly journals Extension of Optimal Homotopy Asymptotic Method with Use of Daftardar–Jeffery Polynomials to Coupled Nonlinear-Korteweg-De-Vries System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Rashid Nawaz ◽  
Zawar Hussain ◽  
Abraiz Khattak ◽  
Adam Khan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Method ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Higher Order ◽  
Partial Differential ◽  
De Vries ◽  
Optimal Homotopy Asymptotic Method

In this paper, Daftardar–Jeffery Polynomials are introduced in the Optimal Homotopy Asymptotic Method for solution of a coupled system of nonlinear partial differential equations. The coupled nonlinear KdV system is taken as test example. The results obtained by the proposed method are compared with the multistage Optimal Homotopy Asymptotic Method. The results show the efficiency and consistency of the proposed method over the Optimal Homotopy Asymptotic Method. In addition, accuracy of the proposed method can be improved by taking higher order approximations.

scholarly journals Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 884
Author(s):  
Linyu Peng
Keyword(s):  
Partial Differential Equations ◽  
General Theory ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Vries Equation ◽  
Lie Point Symmetries ◽  
Nonlocal Equations ◽  
De Vries ◽  
Nonlocal Nonlinear

In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.

scholarly journals An Analytical Technique to Solve the System of Nonlinear Fractional Partial Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 505 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Analytical Technique ◽  
Full Support ◽  
Research Article ◽  
De Vries ◽  
Physical Phenomena

The Kortweg–de Vries equations play an important role to model different physical phenomena in nature. In this research article, we have investigated the analytical solution to system of nonlinear fractional Kortweg–de Vries, partial differential equations. The Caputo operator is used to define fractional derivatives. Some illustrative examples are considered to check the validity and accuracy of the proposed method. The obtained results have shown the best agreement with the exact solution for the problems. The solution graphs are in full support to confirm the authenticity of the present method.

Riccati Equation Solutions to Higher Order Korteweg-de Vries Equation

2014 ◽  
Vol 926-930 ◽  
pp. 3240-3244
Author(s):  
Hong Lei Wang ◽  
Chun Huan Xiang
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Riccati Equation ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

The traveling wave solutions to the heigher order Korteweg-de Vries equation is obtained by using Riccati equation. The method is straightforward and concise, the applications are promising to obtain traveling wave solutions of various partial differential equations. It is shown that the Riccati equation method, with the symbolic computation, provide an effective and powerful mathematical tools for solving such systems. The numerical simulation of the solutions are given for completeness.

Approximate Solutions of Generalized Fifth-Order Korteweg-De Vries (KdV) Equation by the Standard Truncated Expansion Method

2012 ◽  
Vol 201-202 ◽  
pp. 246-250
Author(s):  
Jiang Long Wu ◽  
Wei Rong Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Special Cases ◽  
De Vries ◽  
Fifth Order

It is difficult to obtain exact solutions of the nonlinear partial differential equations (PDEs) due to the complexity and nonlinearity, especially for non-integrable systems. In this case, some reasonable approximations of real physics are considered, by means of the standard truncated expansion approach to solve real nonlinear system is proposed. In this paper, a simple standard truncated expansion approach with a quite universal pseudopotential is used for generalized fifth-order Korteweg-de Vries (KdV) equation, we can get two kinds of approximate solutions of the above equation, in some special cases, the approximate solutions may become exact. The same idea can also used to find approximate solutions of other well known nonlinear equations. We find a quite universal expansion approach which is valid for various nonlinear partial differential equations (PDEs).

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