AN ALGORITHM FOR THE DETECTION AND INTEGRATION OF HIGHLY OSCILLATORY ORDINARY DIFFERENTIAL EQUATIONS USING A GENERALIZED UNIFIED MODIFIED DIVIDED DIFFERENCE REPRESENTATION

This thesis describes an algorithm which automatically integrates systems of ordinary differential equations which have highly oscillatory solutions. Natural variable step derivations of the Generalized Adams and Generalized BDF methods are presented. An efficient numerical algorithm for the evaluation of the local period of an oscillation is presented along with a corresponding algorithm which detects behavior that indicates the system may be amenable to solution by the generalized methods. A code, which implements the algorithm and exploits the overwhelming similarity between the generalized methods and conventional integration methods, is discussed along with some numerical results.