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Solving Differential Equations in Exact Real Arithmetic

Marko Krznaric

PhD Thesis
September, 2005
Abstract

In this thesis we provide a method for solving differential equations in a form of extended, domain-theoretic analogue of the classical Picard operator, whose fix-point give the solution of the differential equation. This framework provides the first proper data type for a solution of differential equations, which can be a basis for a computer software in future, and it brings the problem of solving differential equations into the realm of computer science.

Given a Scott continuous, interval-valued and time-dependent scalar field and a Scott continuous initial function consistent with the scalar field, which bounds the required solution of the differential equation, the extended Picard's theorem acts on the domain of continuously differentiable functions by successively updating the information about the solution and the information about its derivative.

The solution of the differential equation is approximated by a sequence of pairs of polynomials (lower and upper bounds), one increasing and the other one decreasing, both tending to the function in the limit. Hence, this enables us to compute the solution of the differential equation up to any desired accuracy. In particular, this technique can be used to solve the classical initial value problem.

We have built on top of earlier work on exact real arithmetic package using linear fractional transformations (LFT), but any properly implemented exact interval arithmetic package can be used as the basis for our implementation of Picard's operator.

Keywords
London e-Science Centre
Theory of Computational Systems
BibTEX file for the publication
 

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