scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

scholarly journals SOME DEFINABLE GALOIS THEORY AND EXAMPLES

2017 ◽  
Vol 23 (2) ◽  
pp. 145-159 ◽  
Author(s):  
OMAR LEÓN SÁNCHEZ ◽  
ANAND PILLAY
Keyword(s):  
Differential Equations ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Differential Galois Theory ◽  
Algebraic Differential Equations ◽  
Galois Correspondence

AbstractWe make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

scholarly journals Differential Galois theory of infinite dimension

1996 ◽  
Vol 144 ◽  
pp. 59-135 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
19Th Century ◽  
Galois Theory ◽  
Infinite Dimension ◽  
Differential Galois Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The 19Th Century

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

scholarly journals Galois theory of aigebraic and differential equations

1996 ◽  
Vol 144 ◽  
pp. 1-58 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galois Theory ◽  
Linear Partial Differential Equation ◽  
Linear Algebraic Group ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Infinite Dimensional ◽  
Linear Partial Differential Equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

2003 ◽  
Vol 70 (5) ◽  
pp. 732-738
Author(s):  
K. Yagasaki
Keyword(s):  
Differential Equation ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Degree Of Freedom ◽  
Differential Galois Theory ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Infinite Degree ◽  
Simulation Results

We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

scholarly journals Existence of Nontrivial Solutions for Sixth-Order Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1852
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Nontrivial Solution ◽  
Critical Point Theory ◽  
Point Theory ◽  
Sixth Order ◽  
Nontrivial Solutions ◽  
Order Ordinary Differential Equation

We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory.

scholarly journals Variational analysis to fourth-order impulsive differential equations

2018 ◽  
Vol 38 (1) ◽  
pp. 151-163
Author(s):  
Saeid Shokooh
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Fourth Order ◽  
Variational Analysis ◽  
Impulsive Differential Equation ◽  
Impulsive Differential Equations ◽  
Three Solutions ◽  
One Dimensional ◽  
Critical Point Theorems

Applying two critical point theorems, we prove the existence of atleast three solutions for a one-dimensional fourth-order impulsive differential equation with two real parameters.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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