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UK funding (99 957 £) : Algèbre informatique pour les problèmes de limites linéaires Ukri24/04/2012 UK Research and Innovation, Royaume Uni
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Algèbre informatique pour les problèmes de limites linéaires
| Abstract | Boundary problems are arguably the most popular models for describing nature and society. They combine a generic part (the differential equation) with a specific part (the boundary conditions). The generic part typically describes a law of nature or similar principle by analysing how small changes in one quantity cause small changes in another (measured by differential quotients). A typical example is given by Fourier's law of heat transfer in a solid material, relating small changes in temperature to small changes in heat energy. Since differential equations apply uniformly to large classes of situations, additional data is needed for mastering any particular situation, for example heat transfer in an iron slab of a particular shape with a particular control mechanism (like cooling) at its boundary. This is why the specific part is called a boundary problem: It is typically determined by the shape of a boundary and the values of certain quantities of interest (like a uniformly cool temperature) on this boundary. Due to their enormous importance in applications, there is a rich arsenal of computational methods for solving boundary problems. However, virtually all of these methods are based on numerical approximation. This is fully acceptable for the applied scientist who is mostly interested in the numerical description (and visualisation) of key quantities. For the mathematical fine analysis it is often more advantageous to have an exact or symbolic solution. Also parameter dependence can be studied most efficiently in this manner. Despite their great importance, boundary problems are rather neglected in symbolic computation, both in theory and in practise. The deeper reason for this is that the algebraic treatment of boundary problems does not fit into the common frameworks (notably a branch called differential algebra and differential Galois theory) used for the symbolic treatment of differential equations. It is the aim of this project to extend and generalise such algebraic frameworks to cover boundary problems, specifically those for linear partial differential equations. Symbolic methods for boundary problems are not meant to compete with numerical methods; in fact, class of the boundary problems amenable to exact solutions is rather restricted. But symbolic methods are not only good for solving boundary problems, they can also effect various other operations. The ultimate goal of our research on boundary problems is thus to achieve a tight interaction with numerical methods: We want to analyse / manipulate / decompose boundary problems symbolically and solve the atomic chunks numerically. In this project we focus on the following symbolic operations: decompose higher-order problems into lower-order ones, transform a given problem to a simpler geometry, explore situations of solvable model problems, and combine exact solutions. |
| Category | Research Grant |
| Reference | EP/I037474/1 |
| Status | Closed |
| Funded period start | 24/04/2012 |
| Funded period end | 23/04/2014 |
| Funded value | £99 957,00 |
| Source | https://gtr.ukri.org/projects?ref=EP%2FI037474%2F1 |
Participating Organisations
| University of Kent | |
| The National Institute for Research in Computer Science and Control (INRIA) | |
| Tennessee Technological University | |
| École Centrale de Lille | |
| University of Lille |
Cette annonce se réfère à une date antérieure et ne reflète pas nécessairement l’état actuel. L’état actuel est présenté à la page suivante : University of Kent, Canterbury, Royaume Uni.