Implicit Methods for Enclosing Solutions of ODEs

The paper presents a new enclosure method for initial value problems in systems of ordinary di erential equations. Like the common enclosure methods (eg Lohner's algorithm AWA), it is based on Taylor expansion. In contrast to them, however, it is an implicit method. The solution sets of nonlinear inequalities have to be enclosed by a Newton-like algorithm. As the presented examples show, the new method sometimes yields much tighter bounds than any of the common explicit methods.