ACM Communications in Computer Algebra, Vol. 41, No. 3, September 2007 Dissertation Abstracts Abstracts of Recent Doctoral Dissertations in Computer Algebra Communicated by Mark Giesbrecht In this issue we are pleased to present a new section of the ACM Communications in Computer Algebra on the abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates (and their supervisors), who have defended in the past year, to submit their abstracts for publication here. We hope you agree that this is a great way to bring attention to the young researchers in our eld, and see the new directions of their research. Please send abstracts to the CCA editors editors SIGSAM@acm.org for consideration. Symbolic-Algebraic Methods for Linear Partial Differential Operators Dr. Ekaterina Shemyakova Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, AUSTRIA (kath@risc.uni-linz.ac.at) Supervisor: Professor Franz Winkler, External examiner: Professor Elizabeth Mans eld. Defence date: July 20, 2007 This thesis is devoted to the study of symbolic-algebraic factorization, classi cation, and integration methods for Linear Partial Differential Operators (LPDOs). A new theoretical notion, an obstacle to factorizations of LPDOs of general form, that simpli es the considerations of factorization algorithms is introduced. A full system of invariants for third-order bivariate hyperbolic LPDOs is found. The factorizations of LPDOs of orders two, three, and four with completely factorable symbols and without any additional requirement are studied. We prove that âirreducibleâ parametric factorizations can exist only for a few certain types of factorizations. For these cases explicit examples are given. For operators of orders two and three, it is shown that a family may be parameterized by at most one function in one variable. New transformations (Generalized Laplace Transformations) of bivariate hyperbolic second order LPDOs are introduced. The important application is the possibility to extend the class of analytically solvable partial differential equations. Examples are given. The results have been obtained with the help of a specially created M APLE-package. Also the procedures for computing the obstacles to factorizations and invariants are implemented in the package.
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Symbolic-algebraic methods for linear partial differential operators
Shemyakova, Ekaterina
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, Volume 41 (3)
Association for Computing Machinery – Sep 1, 2007
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