27 research outputs found
The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs
We present the Maple package TDDS (Thomas Decomposition of Differential Systems) for decomposition of polynomially nonlinear differential systems, which in addition to equations may contain inequations, into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given
Thomas Decomposition and Nonlinear Control Systems
This paper applies the Thomas decomposition technique to nonlinear control
systems, in particular to the study of the dependence of the system behavior on
parameters. Thomas' algorithm is a symbolic method which splits a given system
of nonlinear partial differential equations into a finite family of so-called
simple systems which are formally integrable and define a partition of the
solution set of the original differential system. Different simple systems of a
Thomas decomposition describe different structural behavior of the control
system in general. The paper gives an introduction to the Thomas decomposition
method and shows how notions such as invertibility, observability and flat
outputs can be studied. A Maple implementation of Thomas' algorithm is used to
illustrate the techniques on explicit examples
Singularities of Algebraic Differential Equations
We combine algebraic and geometric approaches to general systems of algebraic
ordinary or partial differential equations to provide a unified framework for
the definition and detection of singularities of a given system at a fixed
order. Our three main results are firstly a proof that even in the case of
partial differential equations regular points are generic. Secondly, we present
an algorithm for the effective detection of all singularities at a given order
or, more precisely, for the determination of a regularity decomposition.
Finally, we give a rigorous definition of a regular differential equation, a
notion that is ubiquitous in the geometric theory of differential equations,
and show that our algorithm extracts from each prime component a regular
differential equation. Our main algorithmic tools are on the one hand the
algebraic resp. differential Thomas decomposition and on the other hand the
Vessiot theory of differential equations.Comment: 45 pages, 5 figure
Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables
In this paper we study systems of autonomous algebraic ODEs
in several differential indeterminates. We develop a notion of
algebraic dimension of such systems by considering them as
algebraic systems. Afterwards we apply differential elimination
and analyze the behavior of the dimension in the resulting
Thomas decomposition. For such systems of algebraic dimension
one, we show that all formal Puiseux series solutions can be
approximated up to an arbitrary order by convergent solutions. We
show that the existence of Puiseux series and algebraic solutions
can be decided algorithmically. Moreover, we present a symbolic
algorithm to compute all algebraic solutions. The output can
either be represented by triangular systems or by their minimal
polynomials.Agencia Estatal de InvestigaciónAustrian Science Fun
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs
We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given
The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs
We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given. Program summary: Program Title: TDDS Program Files doi: http://dx.doi.org/10.17632/twk8zjxgbz.1 Licensing provisions: GNU LGPL Programming language: MAPLE 11 to MAPLE 2017, available independently in MAPLE 2018 Nature of problem: Systems of polynomially nonlinear partial differential equations are not given in a formally integrable form in general. In order to determine analytic solutions in terms of power series, symbolic manipulations are necessary to find a complete set of conditions for the unknown Taylor coefficients. A particular case of that problem is deciding consistency of a system of PDEs. Nonlinear PDEs require splitting into different cases in general. Deciding whether another PDE is a consequence of a given system depends on similar symbolic manipulations. Computing all consequences of a given system which involve only a subset of the unknown functions or a certain subset of their derivatives are instances of differential elimination problems, which arise, e.g., in detection of hidden constraints in singular dynamical systems and field theoretical models. Solution method: The solution method consists, in principle, of pseudo-division of differential polynomials, as in Euclid's algorithm, with case distinctions according to vanishing or non-vanishing leading coefficients and discriminants, combined with completion to involution for partial differential equations. Since an enormous growth of expressions can be expected in general, efficient versions of these techniques need to be used, e.g., subresultants, Janet division, and need to be applied in an appropriate order. Factorization of polynomials, while not strictly necessary for the method, should be utilized to reduce the size of expressions whenever possible. © 2018 Elsevier B.V