Improving a family of Darboux methods for rational second order ordinary differential equations

DOI10.1016/J.CPC.2015.04.028zbMATH Open1344.34003arXiv1104.4838OpenAlexW2592566167MaRDI QIDQ311874

J. Avellar, L. A. C. P. da Mota, L. G. S. Duarte

Publication date: 13 September 2016

Published in: Computer Physics Communications (Search for Journal in Brave)

Abstract: We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most costly step of our methods and algorithms of solution is the determination of Darboux polynomials for the associated differential operators. Here, we are going to present some algorithms to greatly reduce the time expenditure in determining these needed Darboux polynomials. Some of them are based on a detailed analysis of the general structure of second order differential equations regarding the associated differential invariants. In order to perform this analysis, we produce a theorem concerning the general form for the differential invariants in terms of the Darboux polynomials.

Full work available at URL: https://arxiv.org/abs/1104.4838

zbMATH Keywords

computer algebra Darboux polynomials second order ordinary differential equations

Mathematics Subject Classification ID

General theory for ordinary differential equations (34A99) Software, source code, etc. for problems pertaining to ordinary differential equations (34-04)