2003-2007

N/A

Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology,financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for along time, and there are both symbolic and numerical approaches to the analysis and solution of differentialequations.In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions ofdifferential problems to algebraic ones and algebraic methods for these derived problems.This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiatedthe investigation of transformations, which leave a given differential equation invariant. Such transformations arecommonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find agroup of symmetries of the differential equations and then use this group to reduce the order or the number ofvariables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the conceptof involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generatorsfor differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to themethod of Gröbner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Anotherapproach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formalintegration, and Liouville's method has been extended to an algorithm for solving linear differential equations.There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able togo the whole way and find a symbolic solution to the given DE;if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation ofsolutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we canderive differential systems in triangular form such that the solutions of the original system are the (non-singular)solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutionsof a given system of differential algebraic equations (DAEs).An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. thetheory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is welldeveloped, and we can answer the main questions such as consistency of ideals, equality of ideals, the membershipproblem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solvingthese problems are resultants, Gröbner bases, and algebraic involutive bases. For differential ideals there are stillmany open problems. For instance, the membership problem or the ideal inclusion problems for finitely generateddifferential ideals are still not solved.The analysis and improvement of existing algorithms as well as the development of new algorithmic methods indifferential elimination theory is the major goal of the project.Additionally, we will also consider the important application area of control theory. More... »

http://pf.fwf.ac.at/en/research-in-practice/project-finder/11495

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