Evolutionary Processes Solved with Lie Series and by Picard Iteration Approach

The solution of evolutionary Cauchy problems by means of Lie series expansion and its linkage to Picard iteration method, is presented. Thanks to a Taylor transform and to the introduction of a differential Lie-Groebner operator D, the initial generally non-linear and non-autonomous problem canbe reduced to a linear one, whose solution is given in terms of the Lie operator exp(tD). The Picard procedure applied to the Volterra integral equation that turns out from the initial problem, can rigorously introduce generalized Lie series since its steps are the partial sums of those series.