A package to work with linear partial differential operators.

*(English. Russian original)*Zbl 1311.65180
Program. Comput. Softw. 39, No. 4, 212-219 (2013); translation from Programmirovanie 39, No. 4 (2013).

Summary: The paper describes package LPDO, which is designed for work with linear partial differential operators with symbolic coefficients in the computer algebra system MAPLE. In addition to basic procedures (operator creation, determination, modification, and various simplifications of their coefficients, as well as algebraic operations on them), it implements generating systems of gauge invariants for separate operators and operator pairs, the Laplace transformation method (not related to the integral Laplace method), procedures returning necessary and sufficient conditions for factoring third-order operators in the plane into compositions of operators of one or another form in terms of invariants, and several procedures related to the Darboux transformations.

##### MSC:

65Y15 | Packaged methods for numerical algorithms |

68W30 | Symbolic computation and algebraic computation |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

##### Keywords:

package LPDO; linear partial differential operators; symbolic coefficients; computer algebra system MAPLE; operator creation; determination; modification; gauge invariant; operator pair; Laplace transformation method; Darboux transformations##### References:

[1] | Darboux, G., Le ons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, vol. 2, Paris: Gauthier-Villars, 1915. · JFM 45.0881.05 |

[2] | Fels, M; Olver, PJ, Moving coframes. II: regularization and theoretical foundations, Acta Appl. Math., 55, 127-208, (1999) · Zbl 0937.53013 |

[3] | Krichever, I, Algebraic-geometric N-orthogonal curvilinear coordinate systems and solutions of the associativity equations, Functional Analysis Its Applications, 31, 25-39, (1997) · Zbl 1004.37052 |

[4] | Mansfield, E.L., A Practical Guide to the Invariant Calculus, Cambridge Univ. Press, 2010. · Zbl 1203.37041 |

[5] | Matveev, V. B. and Salle, M.A., Darboux Transformations and Solutions, Springer, 1991. · Zbl 0744.35045 |

[6] | Olver, PJ, No article title, Differential invariant algebras, Contemp. Math., 549, 95-121, (2011) |

[7] | Shemyakova, ES; Winkler, F, On the invariant properties of hyperbolic bivariate third-order linear partial differential operators, Lec. Notes Computer Sci., 5081, 199-212, (2007) · Zbl 1166.35346 |

[8] | Shemyakova, ES; Mansfield, EL, Moving frames for Laplace invariants, 295-302, (2008) |

[9] | Shemyakova, ES, Invariant properties of third-order non-hyperbolic linear partial differential operators, Lec. Notes Computer Sci., 5625, 154-169, (2009) · Zbl 1247.68328 |

[10] | Shemyakova, E.S., Gauge Invariants for Pairs of Operators of Arbitrary Order, 2013. · Zbl 1305.70038 |

[11] | Shemyakova, E.S., Proof of the completeness of Darboux Wronskian formulas for order two, Canadian J. Math., Accepted to see electronically at http://arxiv.org/abs/1111.1338, 2012. · Zbl 1275.53088 |

[12] | Tsarev, SP, On Darboux integrable nonlinear partial differential equations, Proc. Steklov Inst. Math., 225, 372-381, (1999) · Zbl 0990.35008 |

[13] | Tsarev, SP, Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations, proc. of the 2005 int, 325-331, (2005), Beijing · Zbl 1360.35116 |

[14] | Tsarev, SP, On factorization and solution of multidimensional linear partial differential equations, 43-66, (2007) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.