The Differential Counting Polynomial

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional asumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions

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

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

Algorithms for Mappings and Symmetries of Differential Equations

Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points

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

Counting solutions of differential equations

Systems of differential equations are notoriously hard to solve, and many such systems do not admit closed form solutions in "elementary" functions. Despite this, increasingly good heuristics are implemented in computer algebra systems to find solutions. Given a set of closed form solutions returned by a computer algebra system, the question remains whether this set is the complete solution set. The aim of this thesis is a quantitative analysis of the solution set of a system of differential equations, which decides whether the solutions found by a heuristical solver form a proper subset of the complete solution set. Therefore, this thesis examines three measures of the size of the solution set of a system of differential equations: the differential dimension polynomial, the counting sequence, and the differential counting polynomial. The differential dimension polynomial was originally introduced by Kolchin to describe the size of solution sets of a prime differential ideal in the sense that it generically describes the number of free power series coefficients up to any order. This thesis generalizes the differential dimension polynomial and its invariance conditions under differential birational maps from differential prime ideals to ideals associated to so-called simple differential systems. The differential dimension polynomial carries enough information to reliably answer the question whether two full solution sets of ideals associated to simple differential systems included in each other are equal, and this sufficiently describes most common differential systems. To give a non-generic description of the size of the solution set of a system of differential equations, this thesis introduces the counting sequence. The counting sequence describes the set of Taylor polynomials of solutions of each degree precisely, in the sense that it accounts for finite and countably infinite exceptional sets. If there exists a closed form polynomial that ultimately describes the counting sequence, then this closed form is called the differential counting polynomial. It is well-known that there cannot be an algorithm to compute the counting sequence or the differential counting polynomial. Nevertheless, in this thesis the counting sequence and the differential counting polynomial are computed for many important classes of systems of differential equations, in particular linear systems, most common semilinear systems, and quasilinear first order ordinary differential equations; in particular, for these classes of differential equations the existence of both the counting sequence and the differential counting polynomial is proved. Both these measures can decide whether an inclusion of two solution sets is proper under the condition that no countably infinite exceptional sets appear. The differential dimension polynomial, the counting sequence, and the differential counting polynomial determine classical measures that describe the size of the solution set of a system of differential equations, including the number of free functions, Cartan's characters and index of generality, Einstein's strength, and classical invariants from differential algebra like the differential type, the differential dimension, and the typical dimension. The Thomas decomposition algorithm, which is implemented as part of this thesis, is the algorithmic foundation for these descriptions of the size of solution sets. This algorithm partitions the solution set into solution sets of simple differential systems. It allows to compute the differential dimension polynomial and also certain consequences of differential systems, which are independent of counting