The DifferentialAlgebra package implements algebraic and differential elimination algorithms which are key for simplifying systems of polynomial differential equations and computing formal power series solutions for them. The underlying theory is the differential algebra of Joseph Fels Ritt and Ellis Robert Kolchin [R50, K73] (see References).

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The main functionality of the package is provided by the RosenfeldGroebner function, which permits triangularizing a differential equation system so that it can be solved eliminating one variable at a time, simplifying the system with respect to its integrability conditions, or determining its singular cases. Commands are also provided to solve related problems, such as BelongsTo for deciding membership to a differential ideal, and ReducedForm for reducing a system with respect to another one. The command for computing formal power series solutions to differential equation systems is PowerSeriesSolution. Other commands for analyzing mathematical properties of differential systems or performing algebraic manipulation and related programming are listed below.

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For examples illustrating the use of the package's commands, see the Examples section of the RosenfeldGroebner command and the DifferentialAlgebra Examples page.

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For more information about the mathematical terminology used in the help pages of this package, see the Glossary page.

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The DifferentialAlgebra package is based on the Bibliotheques Lilloises d'Algebre Differentielle (BLAD) software developed by F. Boulier. The redesign of the interface of BLAD to DifferentialAlgebra was done by F. Boulier and E. S. Cheb-Terrab in Maple 14.

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Each command in the DifferentialAlgebra package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

Description of the DifferentialAlgebra package commands

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BelongsTo decides membership in differential ideals.

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DifferentialRing constructs a computational representation of a differential polynomial ring embedding the ranking of the dependent and independent variables.

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Equations returns the equations representing the chain of a differential ideal

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Get returns varied type of information regarding mathematical objects related to a differential ring or ideal

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Inequations returns the inequations of a differential idea, that is the initials and separants.

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Is returns true or false regarding properties of differential rings and ideals

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NormalForm computes normal forms modulo regular differential chains.

ReducedForm computes reduced forms modulo a differential ideal.

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RosenfeldGroebner returns a representation of the radical of the differential ideal generated by a system of equations as an intersection of radical differential ideals with respect to a given ranking (differential ring) and rewrites a prime differential ideal using a different ranking.

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Tools is a subpackage with tool commands for performing diverse operations that constitute the main blocks of differential algebra computations.

Description of the DifferentialAlgebra[Tools] package commands

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Coeffs extracts the coefficients of a rational differential fraction.

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DeltaPolynomial returns the Delta-polynomial generated by two differential polynomials.

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DifferentialPrem is an implementation of the Ritt reduction algorithm.

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Differentiate differentiates differential polynomial or rational fractions using different notations (jet, function, etc.).

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Display displays the relevant information related to a given differential ring.

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FactorDerivative extracts the derivation operator (product of differentiation variables) of a derivative.

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FieldElement decides membership in differential base fields.

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FromJet rewrites an expression in function notation.

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Initial returns the initial of a differential polynomial.

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LeadingCoefficient returns the leading coefficient of a differential polynomial.

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LeadingDerivative returns the leading derivative of a differential polynomial.

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LeadingRank returns the leading rank of a differential polynomial.

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PreparationEquation returns the preparation equation of a differential polynomial.

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