Ekaterina Shemyakova Franz Winkler
Abstract
We investigate the problem of the approximate factorization of a linear partial differential operator of arbitrary order and in arbitrarily many variables. For such an operator, we define common obstacles and the associated ring of obstacles to factorizations of the operator, which extends some specified factorization of its symbol. We derive some facts about obstacles (for instance, they are stable, they cannot be of certain order) and give an exhaustive enumeration of obstacles for bivariate operators of order two and three (also a short form was found for these huge formulas).We notice that for a second-order strictly hyperbolic operator, obstacles are exactly the well-known invariants of Laplace. For operators of order three a generalization (not straightforward) is possible. It allows us to find a full system of invariants (with respect to operation conjugation) for operators of order three.
- D.Grigoriev, F. Schwarz Factoring and Solving Linear Partial Differential Equationse, In J. Computing 73, pp. 179--197 (2004) Google Scholar
Digital Library
- N. Ibragimov English Translation in Journal of Applied Mechanics and Technical Physics, Vol. 45, No. 2, pp. 158--166, (2004).Google ScholarCross Ref
- E. Kartashova Hierarchy of general invariants for bivariate LPDOs, J. Theoretical and Mathematical Physics, pp. 1--8. to appear in May 2006.Google Scholar
- www.maple.com.Google Scholar
- E. Shemyakova, F. Winkler, Obstacle to Factorization of LPDOs, Proc. Transgressive Computing 2006, Granada, Spain, 2006.Google Scholar
- S. P. Tsarev. Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations, proc. ISSAC'05. 2005. Google ScholarDigital Library
Index Terms
- Approximate factorization of linear partial differential operators: full system of invariants for order three
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