7 research outputs found
Differential equations with natural matrices coefficients
In this paper we present the main results of the master thesis in applied mathematics
of the second named author, which was supervised by the first named author. Such results, and
for instance this paper, concerns to some differential and algebraic results involving natural
matrices. The problem of solving differential equations is very ancient and is very important to
get explicit solutions of differential equations to be applied in physics and other areas. In this
paper, as well in the master thesis, we study the differential and algebraic structure of linear
differential equations with natural matrix coefficients and generalizations. These results are
original and important for researchers interested in differential algebra and applications of
differential equations
Variations for Some Painlev\'e Equations
This paper first discusses irreducibility of a Painlev\'e equation . We
explain how the Painlev\'e property is helpful for the computation of special
classical and algebraic solutions. As in a paper of Morales-Ruiz we associate
an autonomous Hamiltonian to a Painlev\'e equation . Complete
integrability of is shown to imply that all solutions to are
classical (which includes algebraic), so in particular is solvable by
''quadratures''. Next, we show that the variational equation of at a given
algebraic solution coincides with the normal variational equation of
at the corresponding solution. Finally, we test the Morales-Ramis
theorem in all cases to where algebraic solutions are present,
by showing how our results lead to a quick computation of the component of the
identity of the differential Galois group for the first two variational
equations. As expected there are no cases where this group is commutative
A model of anaerobic digestion for biogas production using Abel equations
Some time ago has been studied mathematical models for biogas production due
to its importance in the use of control and optimization of re\-new\-able
resources and clean energy. In this paper we combine two algebraic methods to
obtain solutions of Abel equation of first kind that arise from a mathematical
model to biogas production formulated in France on 2001. The aim of this paper
is obtain Liouvillian solutions of Abel's equations through Hamiltonian
Algebrization. As an illustration, we present graphics of solutions for Abel
equations and solutions for algebrized Abel equations.Comment: 12 pages, 3 figure
Nonautonomous Hamiltonian Systems and Morales-Ramis Theory I. The Case
In this paper we present an approach towards the comprehensive analysis of
the non-integrability of differential equations in the form
which is analogous to Hamiltonian systems with 1+1/2 degree of freedom. In
particular, we analyze the non-integrability of some important families of
differential equations such as Painlev\'e II, Sitnikov and Hill-Schr\"odinger
equation.
We emphasize in Painlev\'e II, showing its non-integrability through three
different Hamiltonian systems, and also in Sitnikov in which two different
version including numerical results are shown. The main tool to study the
non-integrability of these kind of Hamiltonian systems is Morales-Ramis theory.
This paper is a very slight improvement of the talk with the almost-same title
delivered by the author in SIAM Conference on Applications of Dynamical Systems
2007.Comment: 15 pages without figures (19 pages and 6 figures in the published
version