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Computer Algebra Tools for Handling Ordinary Differential Equations

Objective

The objectives of the CATHODE Working Group are the study of the specification of a fundamental description of solutions of differential equations and the development, for the first time, of an integrated set of tools for handling ordinary differential equations (including a large library of programs) using computer algebra.

The eventual aim of the partners is to develop computer algebra software that accepts a single ordinary differential equation or a system of such equations, and returns all analytic and quantitative information on its solutions which is available, utilising the complete existing literature of the last two centuries.

Within this framework, a number of particular topics will be studied and new methods developed, in the areas of: the closed form solution of single linear and non-linear equations; the normal forms of systems of linear and non-linear equations; methods for studying bifurcation and other qualitative properties; the equivalance of apparently different equations and systems under transfrmations of variables; numerical evaluation; and graphical representation of the solutions.
A study has been made of the specification of fundamental descriptions of solutions of equations and primitive functions to develop, for the first time, an integrated set of tools, using computer algebra, for handling ordinary differential equations.

An international workshop was organized and a book containing the main lectures has been published by Cambridge University Press. Four workshops have been organized by the WG, with published proceedings each time. We intend to go on with this collaboration in aspects of research, software development and applications.
ACTIVITIES

The central activities of the group will be a series of workshops. These will be of two main types: the CADE series, open to a wider audience and presenting the latest developments in the theory and algorithms for differential equations; and a series of internal meetings aimed at the development of the specifications of solutions and algorithms. In association with the internal workshops, a series of 1 or 2 day meetings for contacts in industry and academic research will be organised. The aim of these meetings is not only to disseminate our results but also to clarify which of our lines of development offers the most immediate benefit to practical users. To advance work on specific topics within the main theme, a programme of short visits between group members will also be organised.

The results of our workshops will be published in the CADE proceedings and as a series of technical reports, and programs will be disseminated by electronic transfer. Work on individual topics will appear as research papers. We will maintain contacts with the ESPRIT projects ALCOM, POSSO and NEURON by interchange of visits to workshops and meetings.

POTENTIAL

The long-term all-inclusive objective of this work is to provide a computerised toolkit (CATHODE) for analytically studying ordinary differential equations that scientists and engineers throughout industry and academia would use on their workstations as a complement to or instead of the numerical solvers and simulators currently running on supercomputers. The activities of the group will provide an essential basis for such a development.

We believe this software could be exploited in the wide range of applications fields in which modelling and simulation by differential equations is used, for example, analysis of algorithmic complexity, non-linear circuit theory, non-linear optics, chemical kinetics, wave propagation, ecological processes, economics and traffic flow.

Coordinator

Université de Grenoble I (Université Joseph Fourier)
Address
385 Avenue De La Bibliothèque
38041 Grenoble
France

Participants (5)

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE
Switzerland
Address
Gloriastrasse 35
8092 Zurich
Gesellschaft für Mathematik und Datenverarbeitung mbH
Germany
Address
Schloß Birlinghoven
53754 Sankt Augustin
KATHOLIEKE UNIVERSITEIT NIJMEGEN
Netherlands
Address
, 9108
6500 HK Nijmegen
Queen Mary and Westfield College
United Kingdom
Address
Mile End Road
E1 4NS London
VRIJE UNIVERSITEIT BRUSSEL
Belgium
Address
Avenue F.d. Roosevelt, 50, 165
1050 Bruxelles