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. Aksteiner, Steffen; Andersson, Lars; Bäckdahl, Thomas; Khavkine, Igor; Whiting, Bernard: Compatibility complex for black hole spacetimes (2021)
  2. Brakhage, Karl-Heinz; Niemeyer, Alice C.; Plesken, Wilhelm; Robertz, Daniel; Strzelczyk, Ansgar: The icosahedra of edge length 1 (2020)
  3. Lange-Hegermann, Markus; Robertz, Daniel: Thomas decomposition and nonlinear control systems (2020)
  4. Quadrat, Alban (ed.); Zerz, Eva (ed.): Algebraic and symbolic computation methods in dynamical systems. Based on articles written for the invited sessions of the 5th symposium on system structure and control, IFAC, Grenoble, France, February 4--6, 2013 and of the 21st international symposium on mathematical theory of networks and systems (MTNS 2014), Groningen, the Netherlands, July 7--11, 2014 (2020)
  5. Khavkine, Igor: Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation (2019)
  6. Michels, D. L.; Gerdt, V. P.; Blinkov, Yu. A.; Lyakhov, D. A.: On the consistency analysis of finite difference approximations (2019)
  7. Zhang, Xiaojing; Gerdt, Vladimir P.; Blinkov, Yury A.: Algebraic construction of a strongly consistent, permutationally symmetric and conservative difference scheme for 3D steady Stokes flow (2019)
  8. Blinkov, Yury A.; Gerdt, Vladimir P.; Lyakhov, Dmitry A.; Michels, Dominik L.: A strongly consistent finite difference scheme for steady Stokes flow and its modified equations (2018)
  9. Al-Omari, Shadi; Zaman, Fiazuddin; Azad, Hassan: Lie symmetries, optimal system and invariant reductions to a nonlinear Timoshenko system (2017)
  10. Lisle, Ian G.; Huang, S.-L. Tracy: Algorithmic calculus for Lie determining systems (2017)
  11. Michels, Dominik L.; Lyakhov, Dmitry A.; Gerdt, Vladimir P.; Hossain, Zahid; Riedel-Kruse, Ingmar H.; Weber, Andreas G.: On the general analytical solution of the kinematic Cosserat equations (2016)
  12. 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)
  13. Casati, Matteo: On deformations of multidimensional Poisson brackets of hydrodynamic type (2015)
  14. La Scala, Roberto: Gröbner bases and gradings for partial difference ideals (2015)
  15. Robertz, Daniel: Recent progress in an algebraic analysis approach to linear systems (2015)
  16. Lisle, Ian G.; Huang, S.-L. Tracy; Reid, Greg J.: Structure of symmetry of PDE: exploiting partially integrated systems (2014)
  17. Robertz, Daniel: Formal algorithmic elimination for PDEs (2014)
  18. Gerdt, Vladimir P.; Hashemi, Amir; M.-Alizadeh, Benyamin: Involutive bases algorithm incorporating F(_5) criterion (2013)
  19. Bächler, Thomas; Gerdt, Vladimir; Lange-Hegermann, Markus; Robertz, Daniel: Algorithmic Thomas decomposition of algebraic and differential systems (2012)
  20. Cluzeau, Thomas; Quadrat, Alban: Serre’s reduction of linear partial differential systems with holonomic adjoints (2012)

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