An extension of the Prelle–Singer method and a Maple implementation

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

doi:10.1016/s0010-4655(01)00462-3

Cited by 20 publications

(20 citation statements)

References 7 publications

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“…In [23,21,22] the authors give an algorithm to compute such integrating factors. The key point is the computation of exponential factors.…”

Section: Liouvillian First Integralsmentioning

confidence: 99%

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Chèze¹

2011

Journal of Complexity

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International audienceIn this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y ), dY/dt= B(X, Y ), where A,B belongs to Z[X, Y ], degA ≤ d, degB ≤ d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y )dX + B(X, Y )dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira's algorithm computes irreducible Darboux polynomials with degree smaller than N, with a polynomial number, relatively to d, log(H) and N, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree

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“…In [23,21,22] the authors give an algorithm to compute such integrating factors. The key point is the computation of exponential factors.…”

Section: Liouvillian First Integralsmentioning

confidence: 99%

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Chèze¹

2011

Journal of Complexity

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“…In other words, apart from the theoretical extension that the present package now makes available for the user, thus making the analytical solving of a new class of rational first order ordinary differential equations possible, it also improves the Maple solving capabilities in the arena where the dsolve Maple command (even with some of its non-default enhancements turned on, e.g. the Lie package) and some previously made implementations of the PS-approach and its extensions [18,15] already act.…”

Section: Discussionmentioning

confidence: 99%

Determining Liouvillian first integrals for dynamical systems in the plane

Avellar

Duarte

Duarte³

et al. 2007

Computer Physics Communications

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Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In [1], we have introduced the basis for the present implementation. The particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE). We present a set of software routines in Maple 10 for solving rational first order ODEs. The package present commands permitting research incursions of some algebraic properties of the system that is being studied.Keyword: Liouvillian functions, first integrals, dynamical systems in the plane, first order ordinary differential equations, computer algebra, Prelle-Singer (PS) PROGRAM SUMMARYTitle of the software package: Lsolver.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 ME, Windows XP. Programming languages used: Maple 10Memory required to execute with typical data: 128 Megabytes.No. of lines in distributed program, including On-Line Help, etc.: 900.Keywords: Liouvillian functions, first integrals, dynamical systems in the plane, first order ordinary differential equations, computer algebra, Prelle-Singer (PS). Nature of mathematical problemSolution of rational first order ordinary differential equations. Methods of solutionThe method of solution is based on a Darboux/PS type approach.Restrictions concerning the complexity of the problem If the integrating factor for the rational first order ODE under consideration presents Darboux Polynomials of high degree ( > 3 ) in the dependent and independent variables, the package may spend an unpractical amount of time to obtain the solution. Typical running timeThis depends strongly on the ODE, but usually under 2 seconds. Unusual features of the programOur implementation not only search for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the Darboux procedure (for example, the algebraic invariants curves and associated cofactors can be calculated). In addition, since our package is based in recent theoretical developments [1], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available.

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“…Having substituted equation (26) into (25) and made the coefficients of the same power of w ″ vanish, one achieves a family of PCDEe (PCDE stands for Partial Complex Differential Equations) with respect to the variables a, b, c, d, e, and f.…”

Section: A: Determination Of Integration Factors and Nullmentioning

confidence: 99%

“…As a result, (32) forms compatible solutions for systems (12)-(17) with ϕ given in (21). Finally, we note that, in the above, we considered only a trivial solution S � 0 for equation (25) and derived the corresponding forms of F and R. However, in the choice S ≠ 0, systems (22) and (23) become a coupled equation in the unknowns F and S. To solve this system, as we did previously, let us make an ansatz:…”

Section: A: Determination Of Integration Factors and Nullmentioning

confidence: 99%

An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

Joohy

Al-Juaifri

Mechee

2020

International Journal of Differential Equations

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A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

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An extension of the Prelle–Singer method and a Maple implementation

Cited by 20 publications

References 7 publications

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Determining Liouvillian first integrals for dynamical systems in the plane

An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

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