J. Avellar

Bio: J. Avellar is an academic researcher from Rio de Janeiro State University. The author has contributed to research in topics: Ordinary differential equation & Symbolic computation. The author has an hindex of 4, co-authored 12 publications receiving 40 citations.

Finding Elementary First Integrals for Rational Second Order Ordinary Differential Equations

Abstract: Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the present work. In \cite{royal}, the authors used these results and developed a method to deal with SOODEs and a classification of those. Our present algorithm is based on a much more solid theoretical basis (many theorems are presented) and covers a much broader family of SOODEs than before since we do not work with restricted ansatz. Furthermore, our present approach allows for an easy integrability analysis of SOODEs and much faster actual calculations.

Finding elementary first integrals for rational second order ordinary differential equations

Abstract: Here we present a semialgorithm to find elementary first integrals of a class of rational second order ordinary differential equations The method is based on a Darboux-type procedure and it is an attempt to construct an analogous of the method built by Prelle and Singer [“Elementary first integral of differential equations,” Trans Am Math Soc 279, 215 (1983)] for rational first order ordinary differential equations

3D polynomial dynamical systems with elementary first integrals

Abstract: Here we present a semi-algorithm to find elementary first integrals of 3D polynomial dynamical systems. It is a Darboux type procedure that extends the method built by Prelle and Singer for 2D systems. Although it cannot deal with the general case, the method presents a direct/simple way to find elementary first integrals.

Symbolic Computations of First Integrals for Polynomial Vector Fields

Abstract: In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $$\tilde{\mathcal {O}}(N^{\omega +1})$$ , where N is the bound on the degree of a representation of the first integral and $$\omega \in [2;3]$$ is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms.

Symbolic Computations of First Integrals for Polynomial Vector Fields.

Abstract: In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{\omega+1})$, where $N$ is the bound on the degree of a representation of the first integral and $\omega \in [2;3]$ is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.