Integration algorithm for ordinary differential equations which produces analytical results

Abstract

Problems of integration of systems of ordinary linear differential equations on large intervals are attracting increasing attention. Such problems include, in particular, problems of celestial mechanics, calculation of asymptotic expansions, and evaluation of special functions for large argument values. The existing methods do not always guarantee high accuracy on large intervals. It is therefore relevant to develop methods whose errors are nonincreasing with the increase of the distance from the initial point and which allow integration of systems of arbitrary order. The developments in computer applications, and primarily the larger memories and faster speeds of modern computers, make it possible to perform symbolic transformations on the data, which produce a solution of the Cauchy problem in formula form. For a wide class of differential equations, this ensures asymptotically nonincreasing errors and also allows some analytical investigations of the solution. Moreover, this provides an opportunity for supplementing the existing libraries and software packages with qualitatively new programs that approximate the solutions of a large class of differential equations [1, 21. Consider a system of N ordinary differential equations ax u) = A (t) × X (t), (1) dt