scholarly journals Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Lie Algebras ◽  
First Integral ◽  
First Integrals ◽  
The Other ◽  
Symmetry Class ◽  
Third Order ◽  
Symmetry Properties ◽  
The Relationship

The relationship between first integrals of submaximal linearizable third-order ordinary differential equations (ODEs) and their symmetries is investigated. We obtain the classifying relations between the symmetries and the first integral for submaximal cases of linear third-order ODEs. It is known that the maximum Lie algebra of the first integral is achieved for the simplest equation and is four-dimensional. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the first integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2, and 3 symmetries, and for the 4 symmetry class of linear third-order ODEs, they are 0, 1, and 2 symmetries, respectively. In the case of submaximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals.

scholarly journals Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Lie Algebra ◽  
First Integral ◽  
First Integrals ◽  
Order Equation ◽  
Third Order ◽  
Maximal Symmetry ◽  
Symmetry Properties ◽  
Group Methods ◽  
Algebraic Properties ◽  
The Relationship

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equationy′′′=0is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

scholarly journals Boundedness and oscillation of third order neutral differential equations

2009 ◽  
Vol 43 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Božena Mihalíková ◽  
Eva Kostiková
Keyword(s):  
Differential Equations ◽  
Third Order ◽  
Neutral Differential Equations ◽  
The Third ◽  
The Relationship ◽  
Oscillation Of Solutions

Abstract The relationship between boundedness and oscillation of solutions of the third order neutral differential equations are presented.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

EQUIVALENCE OF THE TWO METHODS IN CALCULATING THE EQUILIBRIUM SHAPES OF CYLINDRICAL VESICLES

1999 ◽  
Vol 13 (16) ◽  
pp. 547-553
Author(s):  
SHAOGUANG ZHANG ◽  
ZHONGCAN OUYANG ◽  
JIXING LIU
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
The Other ◽  
General Shape ◽  
Calculus Of Variation ◽  
And Topology ◽  
Shape Equation ◽  
Equilibrium Shapes ◽  
Comparison Of The Results ◽  
The Relationship

So far, two methods are often used in solving the equilibrium shapes of vesicles. One method is by starting with the general shape equation and restricting it to the shapes with particular symmetry. The other method is by assuming the symmetry and topology of the vesicle first and treating it with the calculus of variation to get a set of ordinary differential equations. The relationship between these two methods in the case of cylindrical vesicles, and a comparison of the results are given.

On first integrals of a class of third-order differential equations

2009 ◽  
Vol 45 (10) ◽  
pp. 1536-1538
Author(s):  
N. S. Berezkina ◽  
I. P. Martynov ◽  
V. A. Pron’ko
Keyword(s):  
Differential Equations ◽  
First Integrals ◽  
Third Order

scholarly journals A relação entre História e Sociologia no horizonte da conceitualização e da explicação dos objetos históricos: reflexões sobre o pensamento de Max Weber * The relationship between History and Sociology over the horizon of conceptualization...

2014 ◽  
Vol 3 (3) ◽  
pp. 28
Author(s):  
ULISSES DO VALLE
Keyword(s):  
Max Weber ◽  
Historical Narrative ◽  
The Other ◽  
Historical Knowledge ◽  
Empirical Reality ◽  
Ideal Types ◽  
Historical Objects ◽  
The One ◽  
The Relationship

<p class="Default"><strong>Resumo</strong>: Este artigo procura refletir sobre as relações entre a disciplina da história e a sociologia a partir do pensamento de Max Weber. Procuramos mostrar como a sociologia exerce uma participação fundamental na constituição do conhecimento histórico com relação a dois procedimentos específicos: a caracterização adequada das entidades históricas individuais, por um lado, e a lógica explicativa que preside a narrativa histórica, por outro. Veremos como Weber, então, introduz a sociologia como uma forma de resolver o intricado problema da interpenetração entre o geral e o particular na representação e na explicação dos objetos históricos, de modo a esclarecer os vínculos formais e metodológicos entre as duas disciplinas assim entendidas.</p><p class="Default"><strong>Palavras-chave</strong>: História; Sociologia; realidade empírica; tipos ideais.</p><p class="Default"><strong><br /></strong></p><p class="Default"><strong>Abstract</strong>: This paper discuss the relationship between the discipline of history and sociology from the thought of Max Weber. We intend to show how sociology plays a key role in the constitution of historical knowledge regarding two specific procedures: the appropriate characterization of individual historical entities, on the one hand, and the explanatory logic of the historical narrative, on the other. We will see how Weber then introduces sociology as a way to solve the intricate problem of interpenetration between the general and the particular in the representation and explanation of historical objects, in order to clarify the formal and methodological links between the two disciplines well understood.</p><p class="Default"><strong>Keywords</strong>: History; Sociology; empirical reality; ideal types.</p>

scholarly journals On Calculation of the First Integrals for Three-dimensional ODE Systems

2019 ◽  
pp. 11-21
Author(s):  
A. V. Kavinov
Keyword(s):  
Vector Field ◽  
General Solution ◽  
Nonlinear Dynamical Systems ◽  
First Integral ◽  
First Integrals ◽  
Third Order ◽  
The Third ◽  
Ode System ◽  
Special Cases ◽  
Ode Systems

The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.

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