516 research outputs found
Classification of algebraic ODEs with respect to rational solvability
This is the author’s version of a work that was accepted for publication in
Computational Algebraic and Analytic Geometry, AMS series Contemporary
Mathematics.
Changes resulting from the publishing process, such as peer review,
editing, corrections,
structural formatting, and other quality control mechanisms may not be
reflected in this document.
Changes may have been made to this work since it was submitted for
publication. A definitive version was subsequently published
in Computational Algebraic and Analytic Geometry vol. 572 pp. 193-210,
AMS series Contemporary Mathematics DOI 10.1090/conm/572/11361In this paper, we introduce a group of affine linear transformations
and consider its action on the set of parametrizable algebraic ODEs. In
this way the set of parametrizable ODEs is partitioned into classes with an invariant
associated system, and hence of equal complexity in terms of rational
solvability. We study some special parametrizable ODEs: some well-known
and obviously parametrizable classses of ODEs, and some classes of ODEs
with special geometric shapes, whose associated systems are characterized by
classical ODEs such as separable or homogeneous ones
Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One
The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. NgĂ´ and F. Winkler\ud
(2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-\ud
Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI:\ud
10.5486/PMD.2015.6032 and it is available at http://dx.doi.org/10.5486/PMD.2015.6032An algebro-geometric method for determining the rational solvability\ud
of autonomous algebraic ordinary differential equations is extended from single equations\ud
of order 1 to systems of equations of arbitrary order but dimension 1 in the algebrogeometric\ud
sense. We provide necessary conditions, for the existence of rational solutions,\ud
on the degree and on the structure at infinity of the associated algebraic curve. Furthermore,\ud
from a rational parametrization of a planar projection of the corresponding\ud
space curve one deduces, either by derivation or by lifting the planar parametrization,\ud
the existence and actual computation of all rational solutions if they exist. Moreover, if\ud
the differential polynomials are defined over the rational numbers, we can express the\ud
rational solutions over the same field of coefficients.Vietnam Institute for Advanced Study in Mathematics (VIASM
Birational transformations preserving rational solutions of algebraic ordinary differential equations
We characterize the set of all rational transformations with the property of pre-
serving the existence of rational solutions of algebraic ordinary di erential equations
(AODEs). This set is a group under composition and, by its action, partitions the set
of AODEs into equivalence classes for which the existence of rational solutions is an
invariant property. Moreover, we describe how the rational solutions, if any, of two
different AODEs in the same class are related.Ministerio de EconomĂa y CompetitividadVietnam Institute for Advanced Study in Mathematics (VIASM)Austrian Science Fund (FWF)Research Group ASYNAC
On the complete integrability and linearization of nonlinear ordinary differential equations - Part III: Coupled first order equations
Continuing our study on the complete integrability of nonlinear ordinary
differential equations, in this paper we consider the integrability of a system
of coupled first order nonlinear ordinary differential equations (ODEs) of both
autonomous and non-autonomous types. For this purpose, we modify the original
Prelle-Singer procedure so as to apply it to both autonomous and non-autonomous
systems of coupled first order ODEs. We briefly explain the method of finding
integrals of motion (time independent as well as time dependent integrals) for
two and three coupled first order ODEs by extending the Prelle-Singer(PS)
method. From this we try to answer some of the open questions in the original
PS method. We also identify integrable cases for the two dimensional
Lotka-Volterra system and three-dimensional Rssler system as well as
other examples including non-autonomous systems in a straightforward way using
this procedure. Finally, we develop a linearization procedure for coupled first
order ODEs.Comment: added a new section (section 8) and minor revision. Proc. R. Soc.
London A (accepted
Solving 1ODEs with functions
Here we present a new approach to deal with first order ordinary differential
equations (1ODEs), presenting functions. This method is an alternative to the
one we have presented in [1]. In [2], we have establish the theoretical
background to deal, in the extended Prelle-Singer approach context, with
systems of 1ODEs. In this present paper, we will apply these results in order
to produce a method that is more efficient in a great number of cases.
Directly, the solving of 1ODEs is applicable to any problem presenting
parameters to which the rate of change is related to the parameter itself.
Apart from that, the solving of 1ODEs can be a part of larger mathematical
processes vital to dealing with many problems.Comment: 31 page
Duffing-van der Pol type oscillator
The nonlinear Duffing-van der Pol oscillator system is studied by means of the Lie symmetry reduction method and the Preller-Singer method. With the particular case of coefficients, this system has physical relevance as a simple model in certain flow-induced structural vibration problems. Under certain parametric conditions, we are concerned with the first integrals of the Duffing-van der Pol oscillator system. After making a series of variable transformations, we apply the Preller-Singer method and the Lie symmetry reduction method to obtain the first integrals of the simplified equations without complicated calculations
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe
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