scholarly journals Symmetries and Conservation Laws for Some Compacton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Gandarias ◽  
M. S. Bruzón ◽  
M. Rosa
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Dynamical Behaviour ◽  
Point Of View ◽  
Nonlinear Dispersion ◽  
Multipliers Method

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

scholarly journals Symmetry Reductions, Exact Solutions, and Conservation Laws of a Modified Hunter-Saxton Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Andrew Gratien Johnpillai ◽  
Chaudry Masood Khalique
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Lie Algebra ◽  
Point Of View ◽  
Third Order ◽  
Theoretic Point ◽  
Group Invariant Solutions ◽  
Nonlocal Conservation Laws ◽  
Symmetry Generators

We study a modified Hunter-Saxton equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the underlying equation are derived. We utilize the Lie algebra admitted by the equation to obtain the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras are then used to reduce the underlying equation to nonlinear third-order ordinary differential equations. Exact traveling wave group-invariant solutions for the equation are constructed by integrating the reduced ordinary differential equations. Moreover, using the variational method, we construct infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation.

The Poincaré-Cartan forms of one-dimensional variational integrals

2020 ◽  
Vol 70 (6) ◽  
pp. 1381-1412
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Differential Equations ◽  
Variational Problem ◽  
Ordinary Differential Equations ◽  
Differential Forms ◽  
Point Of View ◽  
Full Generality ◽  
One Dimensional ◽  
Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

scholarly journals Control of Conservation Laws – An Application

2018 ◽  
Vol 71 (1) ◽  
pp. 155-174
Author(s):  
Vladimir Răsvan
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Conservation Laws ◽  
Power Plants ◽  
Main Tool ◽  
Nonlinear Boundary Conditions ◽  
Point Of View ◽  
Hyperbolic Partial Differential Equations ◽  
Hydroelectric Power ◽  
And Control

Abstract We present here three types of controlled boundary value problems for conservation laws arising from energy co-generation, hydraulic flows and water hammer for hydroelectric power plants and control of the open channel flows (shallow water). The novelty of these models, from the mathematical point of view, is that they are described by nonlinear hyperbolic partial differential equations of the conservation laws with (possibly) nonlinear boundary conditions. At their turn these boundary conditions are controlled by some systems of ordinary differential equations. The engineering requirements for such systems are asymptotic stability and disturbance rejection: these properties have to be achieved by feedback control. In our setting the main tool for tackling these problems is a suitable Lyapunov functional arising from the energy identity. The hints for “guessing” this functional are to be found in the linearized version of the aforementioned mathematical objects.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

scholarly journals New dynamical behaviour of the coronavirus (COVID-19) infection system with nonlocal operator from reservoirs to people

2020 ◽  
Author(s):  
P. Veeresha ◽  
D. G. Prakasha ◽  
Naveen Sanju Malagi ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Order ◽  
Dynamical Behaviour ◽  
Nonlocal Operator ◽  
Nonlinear Ordinary Differential Equations ◽  
Transform Method ◽  
Homotopy Analysis ◽  
The Fixed Point Theorem ◽  
The Mathematical Model

Abstract The fundamental aim of the present study is to analyse and find the solution for the system of nonlinear ordinary differential equations describing the deadly and most dangerous virus from the lost three months called coronavirus. The mathematical model consisting of six nonlinear ordinary differential equations are exemplified and the corresponding solution is studied within the frame of 𝑞-homotopy analysis transform method (𝑞-HATM). Moreover, a newly defined fractional operator is employed in order to understand more effectively, known as Atangana-Baleanu (AB) operator. For the obtained results, the fixed point theorem is hired to present the exactness as well as uniqueness. For diverse arbitrary order, the behaviour of the outcomes is presented in terms of plots. Finally, the present study may help to examine the wild class of real-world models and also aid to predict their behaviour with respect to parameters considered in the models.

scholarly journals Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2083
Author(s):  
María S. Bruzón ◽  
Tamara M. Garrido-Letrán ◽  
Rafael de la Rosa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Soliton Solutions ◽  
Third Order ◽  
Partial Differential ◽  
Symmetry Reductions ◽  
Multipliers Method ◽  
The Family

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

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