Development of New Methods for the Solution of Nonlinear Differential Equations by the Method of Lie Series and Extension to New Fields

摘要

Chapter 1 of the report gives an application of the well experienced method of Lie series to the theory of Lie groups. First, find a representation of these functions by Lie series. The operators are commutative and contain a matrix of functions w sub ik(x;y) the infinitesimal transformations and the connected Lie algebra, or Lie ring more generally, of a given Lie group are derived. Linear infinitesimal operators are developed in detail the construction of the invariants belonging to these groups with the help of Lie series is demonstrated. A new method for finding subgroups is shown. A new derivation of the Campbell-Baker-Hausdorff-Formula and improvement to the Cayley Theorem is given. Chapter 2 clears the connection between the perturbation formulas of Groebner (1960) and Alexseev (1961) for the solution of ordinary differential equations. These formulas are generalized and iteration methods are given, which include the Methods of Picard, Groebner-Knapp, Poincare, Chen, as special cases. Chapter 3 generalizes an iterated integral equation of Chen and indicates an iteration method based on this generalization. A compound form combining the generalization with Groebner's perturbation formula is furnished. (Author)