Standard bases of differential ideals

Ollivier, François

doi:10.1007/3-540-54195-0_60

Cited by 42 publications

(49 citation statements)

References 9 publications

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“…Carra-Ferro [38] defined a differential analogue of Grobner bases, and showed that such differential Grobner bases can be infinite (unlike the case for polynomial equations). Ollivier [39] has given methods which construct the differential Grobner bases of Carra-Ferro up to a finite order. Mansfield [40,41] obtained an algorithm which always terminates but not necessarily with a differential Grobner basis.…”

Section: Differential-algebraic Approaches To Nonlinear Systemsmentioning

confidence: 99%

“…A major advance in polynomial ideal theory was given by Buchberger's algorithm, which converts systems of polynomially nonlinear algebraic equations to the form of a Grobner basis. One of the current objectives of differential-algebraic approaches is to obtain polynomially nonlinear pde systems in the form of a differential analogue of a Grobner basis [38][39][40][41]. Carra-Ferro [38] defined a differential analogue of Grobner bases, and showed that such differential Grobner bases can be infinite (unlike the case for polynomial equations).…”

Section: Differential-algebraic Approaches To Nonlinear Systemsmentioning

confidence: 99%

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Reduction of systems of nonlinear partial differential equations to simplified involutive forms

1996

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We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

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Section: Differential-algebraic Approaches To Nonlinear Systemsmentioning

confidence: 99%

Section: Differential-algebraic Approaches To Nonlinear Systemsmentioning

confidence: 99%

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

1996

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“…We shall refer to this as the monomial order induced by the pseudo-ranking. Both Carrá-Ferro (1989) and Ollivier (1990) have used such monomial orderings to define the concept of a differential Gröbner basis.…”

Section: Polynomial Rings and Pseudo-rankingmentioning

confidence: 99%

The Differential Ideal [ P ] : M∞

Morrison

1999

Journal of Symbolic Computation

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“…The classical notation {Σ} leads to too many confusions in an algorithmic context. Classical references are the books by Ritt and Kolchin [14,22], recent research literature include [6,30,17,18,2,20,3,24,10,4] while reviews and tutorials are to be found in [13,9].…”

Section: Nonlinear Differential Systems From An Algebraic Viewpointmentioning

confidence: 99%

Computing power series solutions of a nonlinear PDE system

Hubert

Roux

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

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This paper presents a new algorithm to compute the power series solutions of a significant class of nonlinear systems of partial differential equations. The algorithm is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equations to get coefficients of the power series, one at a time. The algorithm presented here relies on using the linearisation of the system and the associated recurrences. At each step the order up to which the power series solution is known is doubled. The algorithm can be seen as belonging to the family of Newton iteration methods.

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Standard bases of differential ideals

Cited by 42 publications

References 9 publications

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

The Differential Ideal [ P ] : M∞

Computing power series solutions of a nonlinear PDE system

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