A domain theoretic account of Euler's method for solving initial value problems

Abstract

This paper presents a method of solving initial value problems using Euler's method, based on the domain of interval valued functions of a real variable. In contrast to other interval based techniques, the actual computation of enclosures to the solution is not based on the code list (term representation) of the vector field that defines the equation, but assumes instead that the vector field is approximated to an arbitrary degree of accuracy. By using approximations defined over rational or dyadic numbers, we obtain proper data types for approximating both the vector field and the solution. As a consequence, we can guarantee the speed of convergence also for an implementation of the method. Furthermore, we give estimates on the algebraic complexity for computing approximate solutions. © Springer-Verlag Berlin Heidelberg 2006.