Domain Theoretic Solutions of Initial Value Problems for Unbounded Vector Fields

This paper extends the domain theoretic method for solving initial value problems, described in [A. Edalat and D. Pattinson. A domain theoretic account of picard's theorem. In Proc. ICALP 2004, number 3142 in Lect. Notes in Comp. Sci., pages 494–505, 2004], to unbounded vector fields. Based on a sequence of approximations of the vector field, we construct two sequences of piecewise linear functions that converge exponentially fast from above and below to the classical solution of the initial value problem. We then show how to construct approximations of the vector field. First, we show that fast convergence is preserved under composition of approximations, if the approximated functions satisfy an additional property, which we call “Hausdorff Lipschitz from below”. In particular, this frees us from the need to work with maximal extensions of classical functions. In a second step, we show how to construct approximations that satisfy this condition from a given computable vector field.