Long time integration for initial value problems of ordinary differential equations using power series arithmetic

In this paper, we present a numerical method with guaranteed accuracy to so lve initial value problems (IVPs) of normal form simultaneous first order o rdinary differential equations (ODES) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi [1], [2], by which we can obt ain inclusions of exact values at several discrete points of the solution c urve of ODES. The method can be regarded as an extension of the Lohner's me thod [3]. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We re alize it by combining Kashiwagi's algorithm with the mean value form. We al so consider the wrapping effects [4] to obtain tighter bounds.