scholarly journals Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families

1986 ◽  
Vol 114 (2) ◽  
pp. 340-359 ◽  
Author(s):  
Kurt Bernardo Wolf ◽  
Ladislav Hlavatý ◽  
Stanly Steinberg
Keyword(s):  
Differential Equations ◽  
Group Action ◽  
Vries Equation ◽  
Nonlinear Differential Equations ◽  
De Vries

scholarly journals Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

2021 ◽  
Vol 2 (2) ◽  
pp. 62-77
Author(s):  
Rajeev Kumar ◽  
Sanjeev Kumar ◽  
Sukhneet Kaur ◽  
Shrishty Jain
Keyword(s):  
Differential Equations ◽  
Series Solution ◽  
Vries Equation ◽  
Convergent Series ◽  
De Vries ◽  
Conformable Derivatives ◽  
Graphical View ◽  
Conditionally Convergent Series ◽  
The Given ◽  
Explicit Series Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

scholarly journals A Note on the Fractional Generalized Higher Order KdV Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Yongyi Gu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Differential Equations ◽  
Mathematical Physics ◽  
Higher Order ◽  
Complex Method ◽  
Applied Method ◽  
De Vries

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.

scholarly journals SOME LINEAR AND NONLINEAR EQUATIONS OF HIGHER ORDER

2016 ◽  
pp. 46-52
Author(s):  
Nadezda Chuesheva ◽  
Nadezda Chuesheva ◽  
Aleksandr Chueshev ◽  
Aleksandr Chueshev
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Uniqueness Of Solutions ◽  
Partial Derivatives ◽  
Petviashvili Equation ◽  
De Vries ◽  
Linear And Nonlinear

In this article we consider boundary value problems for some linear and nonlinear differential equations with partial derivatives of the sixth, fifth, fourth and third orders. We write out conditions on equation coefficients for which existence and uniqueness of solutions from Sobolev's space occur. If these conditions on equation coefficients are not valid, then there are given examples when solution is not unique, or is not unstable, or does not belong to Sobolev's space from existence and uniqueness theorem even for analytical coefficients and analytical right side of differential equation. After S.P. Novikov’s fundamental study in 1974 the interest to the nonlinear Korteweg-de Vries equation, Kadomtsev-Petviashvili equation and other nonlinear equations significantly grew. In this study of such equations we used methods of algebraic geometry integration and expansion method. In these studies exact solutions of special nonlinear equations series in partial derivatives play a big role. Solvability of similar equations was also studied in articles of A.I. Kozhanov, N.A. Larkin and other authors. The aim of this article is to find some exact solutions for special series partial differential equations. Solution graphs of such problems for linear equations and for the Korteweg-de Vries, Burgers-Korteweg-deVries, and Kadomtsev-Petviashvili equations are constructed.

The Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation

2008 ◽  
Vol 63 (10-11) ◽  
pp. 621-626 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Vries Equation ◽  
Nonlinear Differential Equations ◽  
Convergent Series ◽  
Science And Engineering ◽  
Weak Nonlinearity ◽  
Homotopy Perturbation ◽  
Engineering Problems ◽  
De Vries

The homotopy perturbation method (HPM) is employed successfully for solving the modified Korteweg-de Vries equation. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions or perturbation theory. The results show applicability, accuracy and efficiency of the HPM in solving nonlinear differential equations. It is predicted that the HPM can be widely applied in science and engineering problems.

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 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.

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