Bounds for Initial Value Problems

Abstract

A new method has been developed for finding rigorous upper and lower bounds to the solution of a wide class of initial value problems. The method is applicable to initial value problems of the following type: ẍ(t) + f(t, x, ẋ) = 0, x(0) = X0, ẋ(0) = V0, where f is continuous with continuous first derivatives, Lipschitzian, and ∂f/∂x ≥ 0. An original bounding theorem has been formulated and proven and a numerical technique has been developed for finding the bounding functions in analytic form as linear combinations of Tchebyshev polynomials. The method has been applied to several problems of engineering interest.