The Maple package Janet implements the involutive basis technique of V. P. Gerdt and Y. A. Blinkov for computing Janet bases and Janet-like Gröbner bases for linear systems of partial differential equations. It works with left modules over differential algebras defined over differential fields of characteristic zero which exist in Maple. Janet also provides a number of tools for dealing with differential expressions and differential operators. A generic linearization for a non-linear system of partial differential equations can be computed. Some procedures translate differential expressions into jet notation and vice versa. For the Weyl algebra representing ordinary differential operators in characteristic zero whose coefficients are rational functions, an elementary divisor algorithm [Rehm 2001/2002], [Cohn 1985] is implemented to compute the Jacobson normal form of a matrix with entries in this Weyl algebra. Among the orderings for differential monomials which are available in Janet are the degree reverse lexicographical one, the pure lexicographical one, block orderings and their extensions to the case of more than one dependent variable which correspond to ”term over position” and ”position over term” orderings in the polynomial case. Four involutive criteria are implemented to avoid unnecessary reductions during involutive basis computations.

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  1. Al-Omari, Shadi; Zaman, Fiazuddin; Azad, Hassan: Lie symmetries, optimal system and invariant reductions to a nonlinear Timoshenko system (2017)
  2. Lisle, Ian G.; Huang, S.-L.Tracy: Algorithmic calculus for Lie determining systems (2017)
  3. Blinkova, A.Yu.; Blinkov, Yu.A.; Ivanov, S.V.; Mogilevich, L.I.: Nonlinear deformation waves in a geometrically and physically nonlinear viscoelastic cylindrical shell containing viscous incompressible fluid and surrounded by an elastic medium (2015)
  4. Casati, Matteo: On deformations of multidimensional Poisson brackets of hydrodynamic type (2015)
  5. La Scala, Roberto: Gröbner bases and gradings for partial difference ideals (2015)
  6. Robertz, Daniel: Recent progress in an algebraic analysis approach to linear systems (2015)
  7. Lisle, Ian G.; Huang, S.-L.Tracy; Reid, Greg J.: Structure of symmetry of PDE: exploiting partially integrated systems (2014)
  8. Robertz, Daniel: Formal algorithmic elimination for PDEs (2014)
  9. Gerdt, Vladimir P.; Hashemi, Amir; M.-Alizadeh, Benyamin: Involutive bases algorithm incorporating F$_5$ criterion (2013)
  10. Bächler, Thomas; Gerdt, Vladimir; Lange-Hegermann, Markus; Robertz, Daniel: Algorithmic Thomas decomposition of algebraic and differential systems (2012)
  11. Cluzeau, Thomas; Quadrat, Alban: Serre’s reduction of linear partial differential systems with holonomic adjoints (2012)
  12. Levandovskyy, Viktor; Schindelar, Kristina: Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gröbner bases (2012)
  13. Barakat, Mohamed; Lange-Hegermann, Markus: An axiomatic setup for algorithmic homological algebra and an alternative approach to localization (2011)
  14. Jambor, Sebastian: Computing minimal associated primes in polynomial rings over the integers (2011)
  15. Levandovskyy, Viktor; Schindelar, Kristina: Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases (2011)
  16. Lisle, Ian; Huang, S.-L.Tracy: Algorithmic symmetry classification with invariance (2010)
  17. Plesken, Wilhelm; Robertz, Daniel: Linear differential elimination for analytic functions (2010)
  18. Plesken, W.; Fabiańska, A.: An $L_2$-quotient algorithm for finitely presented groups. (2009)
  19. Robertz, Daniel: Noether normalization guided by monomial cone decompositions (2009)
  20. Barakat, Mohamed; Robertz, Daniel: homalg: a meta-package for homological algebra (2008)

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