A FAST ALGORITHM FOR COMPUTING LIE SERIES SOLUTIONS OF AUTONOMOUS DIFFERENTIAL-EQUATIONS

The solution of any real variable autonomous differential equation dx/ dt = F(x) with the initial condition x(0) = alpha can be expressed as a Lie series x(t) = Sigma(n=0)(infinity)(t(n)/n!)M((n))(alpha) under t he assumption of convergence of the series, where M((0))(alpha) = alph a, M((1))(alpha) = F(alpha), and M((n))(alpha) = F(alpha)dM((n-1))(alp ha)/d alpha for n greater than or equal to 2. 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 s ystems of autonomous differential equations is that it provides a syst ematic way of obtaining solutions that are in an explicit functional f orm. However, the actual computation of coeff cients of Lie series sol utions by directly using the mathematical definition of the operator M ((n))(alpha) is a computationally intractable process for most practic ally 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 equati on. 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 applica tion of the algorithm to a simple yet useful type of autonomous nonlin ear differential equation.