Details of Grant 

EPSRC Reference:	GR/S31396/01
Title:	Domain-theoretic Solution of Differential Equations
Principal Investigator:	Edalat, Professor A
Other Investigators:
Researcher Co-Investigators:
Project Partners:	Dassault Group
Department:	Computing
Organisation:	Imperial College London
Scheme:	Standard Research (Pre-FEC)
Starts:	01 May 2003	Ends:	31 August 2006	Value (£):	153,983
EPSRC Research Topic Classifications:	Fundamentals of Computing Numerical Analysis
EPSRC Industrial Sector Classifications:	Information Technologies
Related Grants:
Panel History:
Summary on Grant Application Form
The major problem with existing numerical methods for solving differential equations is that it is extremely hard in general to have any reasonable estimate for the error bound. We have recently developed a domain-theoretic model for differential calculus: a domain for continuously differentiable functions. We propose to use this domain to develop a computable framework for solving differential equations such that any computable differential equation with a computable initial value has a computable solution. We will achieve this by constructing a computable, domain-theoretic extension of Picard's theorem. In this framework, we will obtain algorithms for finding approximations to the solution of an initial value problem given by a differential equation, which is specified by elementary functions. At each stage of computation a lower piecewise polynomial function and an upper piecewise polynomial function are determined which give lower and upper bounds to the solution. Using packages in interval anaylysis we produce a library for generating elementary functions with domain-theoretic step functions, which will enable us to reduce the piecewise polynomial functions to piecewise linear functions. By developing a rounding scheme within a fixed size arithmetic, we intend to make these algorithms efficent in time and space complexity.This work will also enable us to solve differential equations whose vector fields are interval-valued and/or the initial value is uncertain, giving rise to new classes of solutions of differential equations. We will produce prototype implementations of our algorithms in Haskell, Java and C++,
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Organisation Website:	http://www.imperial.ac.uk