Finding elementary first integrals for rational second order ordinary differential equations

Duarte, L.G.S.; Mota, L.A.C.P. da

doi:10.1063/1.3050281

Cited by 13 publications

(39 citation statements)

References 17 publications

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“…As mentioned in [1], the fact that our method uses the differential operator defines as in Eq. (1) is very advantageous.…”

Section: Summary Of Revisionsmentioning

confidence: 99%

“…We have realized that one can extract information regarding the Darboux polynomials, correspondent to the D-operator related to the 2ODE (please see [1]) being studied in a very straightforward way. This, although a simple procedure, will prove essential to solve (or at least reduce) some 2ODEs.…”

Section: Summary Of Revisionsmentioning

confidence: 99%

“…Considering this D-operator, one can see, by inspection on (1), that the cases that will be of interest to us are the ones listed below:…”

Section: Summary Of Revisionsmentioning

confidence: 99%

“…Unusual features: Since our package is based on our theoretical developments [1], it can successfully reduce some rational 2ODEs that were not solved (or reduced) by some of the best-known methods available. This situation is even enhanced via the improvements that we have made here.…”

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confidence: 99%

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

Avellar

Duarte

Mota

2015

Computer Physics Communications

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a b s t r a c tWe 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 apply a procedure to greatly reduce the time expenditure in determining these needed Darboux polynomials for a class of second order differential equations. Catalogue identifier of previous version: AEQL_v1_0Journal reference of previous version: Comput. Phys. Commun. 185(2014)307 Does the new version supersede the previous version?: YesNature of problem: Determination of first order differential invariants for rational second order ordinary differential equations. Solution method:The method of solution is based on a Darboux type approach. Reasons for the new version:We have been working on analyzing and solving systems of first and second order differential equations (1ODEs and 2ODEs, respectively) from a numerical point of view, using Lie

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“…As mentioned in [1], the fact that our method uses the differential operator defines as in Eq. (1) is very advantageous.…”

Section: Summary Of Revisionsmentioning

confidence: 99%

Section: Summary Of Revisionsmentioning

confidence: 99%

“…Considering this D-operator, one can see, by inspection on (1), that the cases that will be of interest to us are the ones listed below:…”

Section: Summary Of Revisionsmentioning

confidence: 99%

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confidence: 99%

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

Avellar

Duarte

Mota

2015

Computer Physics Communications

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“…• Next, we present the concept of associated 1ODE (introduced in [44]) and show how its solution is related to the solving/reducing of rational 2ODEs presenting Liouvillian first integrals.…”

Section: The S-functions and The Associated 1odes Of A 2odementioning

confidence: 99%

Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The S-function method

Avellar

Cardoso

Duarte

et al. 2019

Computer Physics Communications

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Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package -InSyDE. The package is designed, apart from materializing the algorithms presented, to provide a set of tools to allow the user to analyse the intermediary steps of the process. PROGRAM SUMMARYTitle of the software package: InSyDE -Invariants and Symmetries of (racional second order ordinary) Differential Equations.Catalogue number: (supplied by Elsevier) Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Licensing provisions: noneOperating systems under which the program has been tested: Windows 8. Programming languages used: Maple 17.Memory required to execute with typical data: 200 Megabytes. No. of lines in distributed program, including On-Line Help, etc.: 537Nature of mathematical problem Search for first integrals of rational 2ODEs. Methods of solutionThe method of solution is based on an algorithm described in this paper. Restrictions concerning the complexity of the problemIf the rational 2ODE that is being analysed presents a very high degree in (x, y, z), then the method may not work well. Typical running timeThis depends strongly on the 2ODE that is being analysed. Unusual features of the programOur implementation can find first integrals in many cases where the rational 2ODE under study can not be reduced by other powerful solvers. Besides that, the package presents some useful research commands.

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$$\lambda $$ -Symmetries, isochronicity, and integrating factors of nonlinear ordinary differential equations

Guha

Choudhury²,

Khanra³

2013

J Eng Math

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Finding elementary first integrals for rational second order ordinary differential equations

Cited by 13 publications

References 17 publications

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

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

Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The S-function method

$$\lambda $$ -Symmetries, isochronicity, and integrating factors of nonlinear ordinary differential equations

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