Decomposition of ordinary differential polynomials

Gao, Xiao-Shan; Zhang, Mingbo

doi:10.1007/s00200-008-0059-z

Cited by 7 publications

(7 citation statements)

References 32 publications

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“…We note that our definition of generalized divisors is in the frames of differential ideals [9], rather than the definition of factorization from [13], [1] being in terms of a composition of nonlinear ordinary differential polynomials. In [1] a decomposition algorithm is designed.…”

Section: Introductionmentioning

confidence: 99%

“…One can also introduce a different (from [13], [1]) concept of composition (which yields generalized divisors) as follows. For a differential field K consider an operator A = 0≤i≤n a i • d i dx i acting on the algebra K{y} of differential polynomials in y [9] where the coefficients a i ∈ K{y}, the result of the action we denote by A * z ∈ K{y} for z ∈ K{y}.…”

Section: Introductionmentioning

confidence: 99%

“…Easy examples demonstrate that the two considered compositions differ from each other. In the sense of [13], [1] we have 1 = 1 • z for an arbitrary z ∈ K{y}, while 1 cannot be represented as A * y. On the other hand, y • y = y * y , while one cannot represent y • y as g • y for any g ∈ K{y}.…”

Section: Introductionmentioning

confidence: 99%

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Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations

Grigoriev

Schwarz

2013

Computer Algebra in Scientific Computing

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If solutions of a non-linear differential equation are contained in solutions of another equation we say that the former equation is a generalized divisor of the latter one. We design an algorithm which finds first-order quasi-linear generalized divisors of a second-order quasi-linear ordinary differential equation. If solutions of an equation contain solutions of a pair of equations we say that the equation is a common multiple of the pair. We prove that a quasi-linear common multiple of a pair of quasi-linear equations always exists and design an algorithm which yields a quasilinear common multiple.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations

Grigoriev

Schwarz

2013

Computer Algebra in Scientific Computing

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“…It is obvious that the described decomposition methods may be applied to equations of any order and any degree in the variables, a detailed discussion of these more general cases will be given elsewhere; an algorithm for decomposing equations of any order into rational components has been given by Gao and Zhang [8].…”

Section: Discussionmentioning

confidence: 99%

Decomposition of ordinary differential equations

Schwarz

2017

Bull. Math. Sci.

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Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode's. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only available systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode's and also partial differential equations are suggested.

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“…Actually, Decomposition (1) is in practice obtained by iterating Decomposition (2). Note that Decomposition (2) is related to the decomposition of ordinary differential polynomials [5,6] in the particular case where F is polynomial and W is zero. The additional additivity property is also very interesting since it provides more intrinsic (i.e.…”

Section: Introductionmentioning

confidence: 99%

Additive normal forms and integration of differential fractions

Boulier

Lemaire

Lallemand

et al. 2016

Journal of Symbolic Computation

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This paper presents two new algorithms for integrating fractions of differential polynomials. In a previous work, the authors presented a method for decomposing a fraction as a fraction (the "non integrable part") plus the derivative of another fraction. In this paper, we rigorously formalize this notion of "non integrable part" and introduce a new normal form for decomposing a fraction as a sum of iterated derivations of fractions.

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Decomposition of ordinary differential polynomials

Cited by 7 publications

References 32 publications

Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations

Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations

Decomposition of ordinary differential equations

Additive normal forms and integration of differential fractions

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