Related papers: Order one equations with the Painlev\'e property
We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained…
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear…
This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for…
A class of first order linear impulsive differential equation with continuous and piecewise constant arguments is studied. Sufficient conditions for the oscillation of the solutions are obtained.
The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e and A. Denjoy. The long-standing problem of generalising these results for the equations of the order $k>1$ (or for the…
We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a…
We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between…
An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…
We study properties that allow first-order theories to be disjointly combined, including stable infiniteness, shininess, strong politeness, and gentleness. Specifically, we describe a Galois connection between sets of decidable theories,…
In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…
We introduce Galois corings, and give a survey of properties that have been obtained so far. The Definition is motivated using descent theory, and we show that classical Galois theory, Hopf-Galois theory and coalgebra Galois theory can be…
The Obreshkov method is a single-step multi-derivative method used in the numerical solution of differential equations and has been used in recent years in efficient circuit simulation. It has been shown that it can be made of arbitrary…