3,315 research outputs found
Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations
In this paper we present an algorithmic approach to the generation of fully
conservative difference schemes for linear partial differential equations. The
approach is based on enlargement of the equations in their integral
conservation law form by extra integral relations between unknown functions and
their derivatives, and on discretization of the obtained system. The structure
of the discrete system depends on numerical approximation methods for the
integrals occurring in the enlarged system. As a result of the discretization,
a system of linear polynomial difference equations is derived for the unknown
functions and their partial derivatives. A difference scheme is constructed by
elimination of all the partial derivatives. The elimination can be achieved by
selecting a proper elimination ranking and by computing a Gr\"obner basis of
the linear difference ideal generated by the polynomials in the discrete
system. For these purposes we use the difference form of Janet-like Gr\"obner
bases and their implementation in Maple. As illustration of the described
methods and algorithms, we construct a number of difference schemes for Burgers
and Falkowich-Karman equations and discuss their numerical properties.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Involutive Division Technique: Some Generalizations and Optimizations
In this paper, in addition to the earlier introduced involutive divisions, we
consider a new class of divisions induced by admissible monomial orderings. We
prove that these divisions are noetherian and constructive. Thereby each of
them allows one to compute an involutive Groebner basis of a polynomial ideal
by sequentially examining multiplicative reductions of nonmultiplicative
prolongations. We study dependence of involutive algorithms on the completion
ordering. Based on properties of particular involutive divisions two
computational optimizations are suggested. One of them consists in a special
choice of the completion ordering. Another optimization is related to
recomputing multiplicative and nonmultiplicative variables in the course of the
algorithm.Comment: 19 page
Transitive Lie algebras of vector fields---an overview
This overview paper is intended as a quick introduction to Lie algebras of
vector fields. Originally introduced in the late 19th century by Sophus Lie to
capture symmetries of ordinary differential equations, these algebras, or
infinitesimal groups, are a recurring theme in 20th-century research on Lie
algebras. I will focus on so-called transitive or even primitive Lie algebras,
and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg,
Blattner, and others. This paper gives just one, subjective overview of the
subject, without trying to be exhaustive.Comment: 20 pages, written after the Oberwolfach mini-workshop "Algebraic and
Analytic Techniques for Polynomial Vector Fields", December 2010 2nd version,
some minor typo's corrected and some references adde
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
A point symmetry based method for transforming ODEs with three-dimensional symmetry algebras to their canonical forms
We provide an algorithmic approach to the construction of point
transformations for scalar ordinary differential equations that admit
three-dimensional symmetry algebras which lead to their respective canonical
forms
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases
This paper is a sequel to "Computing diagonal form and Jacobson normal form
of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We
present a new fraction-free algorithm for the computation of a diagonal form of
a matrix over a certain non-commutative Euclidean domain over a computable
field with the help of Gr\"obner bases. This algorithm is formulated in a
general constructive framework of non-commutative Ore localizations of
-algebras (OLGAs). We split the computation of a normal form of a matrix
into the diagonalization and the normalization processes. Both of them can be
made fraction-free. For a matrix over an OLGA we provide a diagonalization
algorithm to compute and with fraction-free entries such that
holds and is diagonal. The fraction-free approach gives us more information
on the system of linear functional equations and its solutions, than the
classical setup of an operator algebra with rational functions coefficients. In
particular, one can handle distributional solutions together with, say,
meromorphic ones. We investigate Ore localizations of common operator algebras
over and use them in the unimodularity analysis of transformation
matrices . In turn, this allows to lift the isomorphism of modules over an
OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of
this lifting with the solutions of the original system of equations. Moreover,
we prove some new results concerning normal forms of matrices over non-simple
domains. Our implementation in the computer algebra system {\sc
Singular:Plural} follows the fraction-free strategy and shows impressive
performance, compared with methods which directly use fractions. Since we
experience moderate swell of coefficients and obtain simple transformation
matrices, the method we propose is well suited for solving nontrivial practical
problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio
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