An Interval Method for Initial Value Problems in Linear Ordinary Differential Equations

Interval numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs) compute bounds that contain the solution of an IVP for an ODE. This paper develops and studies an interval method for computing bounds on the solution of a linear ODE. Our method is based on enclosing the terms in the formula for the closed form solution using Taylor series and various interval techniques. We also present an alternative approach, for bounding solutions, based on following vertices that specify a parallelepiped enclosing the solution of a linear ODE. We propose a simple combination of two existing methods—the parallelepiped and QR-factorization methods—for reducing the wrapping effect, to obtain a better method for reducing it. The resulting method computes bounds than are as tight as the bounds produced by the QR-factorization method and often much tighter.