Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated (also called interval) methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.; To date, the only effective approach for computing guaranteed enclosures of the solution of an IVP for an ODE has been interval methods based on Taylor series. This thesis derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures.; Compared to interval Taylor series (ITS) methods, for the same order and stepsize, our IHO scheme has a smaller truncation error and better stability. As a result, the IHO method allows larger stepsizes than the corresponding ITS methods, thus saving computation time. In addition, since fewer Taylor coefficients are required by IHO than ITS methods, the IHO method performs better than the ITS methods when the function for computing the right side contains many terms.; The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.; This thesis also proposes a Taylor series method for validating existence and uniqueness of the solution, a simple stepsize control, and a program structure appropriate for a large class of validated ODE solvers.