Application of Lie Series to the Solution of Ordinary Differential Equations and to Algebraic Functions

摘要

Research has been conducted on the numerical solution of ordinary differential equations with given initial conditions using the method of Lie series. This is similar to Fehlberg's modification of the Runge-Kutta method, that is, it is a perturbation method which allows one to improve upon arbitrary approximate solutions. In most cases, the first terms of the power series expansions of the solutions are taken as approximate solutions. For this reason, the technique of recursive generation of the Taylor coefficients of these solutions is considered first. A new derivation of the formulas of W. Groebner with the error term of H. Knapp is given, which allows improvement of arbitrary approximate solutions and error estimates. Finally, Gaussian quadrature formulas for the numerical calculation of the occuring integrals are tabulated and some generalizations sketched. (Author)