A fast algorithm for computing lie series solutions of autonomous differential equations

Abstract

The solution of any real variable autonomous differential equation with the initial condition can be expressed as a Lie series under the assumption of convergence of the series, where and . Lie series solutions of multivariate systems of autonomous differential equations are defined analogously through partial, instead of ordinary derivatives. The main advantage of using truncated Lie series as approximate solutions of systems of autonomous differential equations is that it provides a systematic way of obtaining solutions that are in an explicit functional form. However, the actual computation of coefficients of Lie series solutions by directly using the mathematical definition of the operator is a computationally intractable process for most practically useful autonomous differential equations. In this paper we first present a fast algorithm that computes the coefficients of the first N terms of the Lie series solution of an autonomous differential equation. The algorithm is then extended, with an analysis of computational complexity and storage requirements, to Lie series solutions of multi-variable autonomous systems. An example illustrates the actual application of the algorithm to a simple yet useful type of autonomous nonlinear differential equation.