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 Pi card iteration method, is presented. Thanks to a Taylor transform and to the introduction of a differential Lie-Groebner operato r D, the initial generally non-linear and non-autonomous problem can be reduced to a linear one, whose solution is given in terms of t he Lie operator exp(tD). The Picard procedure applied to the Volterra integr al equation that turns out from the initial problem, can rigorously introduce generalized Lie series since its steps are the part ial sums of those series. INDING solutions to partial differential equations (PDEs), in a general, fast and efficient way, is as importa nt as it is a difficult task. In previous papers (1)-(8) we faced the problem to find out a unique method capable to solve evolut ionary PDEs, both linear and nonlinear, both autonomous and non- autonomous, under the constraint of analyticity reg arding the evolutionary operator, i.e. the transformed functio n of the unknown one were analytic, developable in a multiple power series of its arguments. We reached this goal by im proving and extending a method, based on Lie series, proposed b y W. Groebner and others in the 70's (9)-(11). Inspired by those ideas, we started a systematic st udy both aimed at a better theoretical foundation and at a p ractical applicability of that method. So we fixed what can be considered a generalization of the Groebner's appro ach. The present paper is a further step in that direction. In it we study the integration of a linear or nonlinear PDE of the evolutionary type in the Cauchy's formulation of the problem. Fo r the sake of simplicity of exposition, here we treat a two di mensional problem, but the method is easily extendible to hig her dimensions, to systems of equations and to boundary value problems, as we partly already showed and partly ar e going to show in future papers. Anyway, in this work we util ize the classical Picard's iteration procedure in order to construct the unique solution, whose components will remain represented by Lie series. In fact, as a standard result from nonlinear functi onal analysis, we know Picard's iteration gives a genera l theorem on the existence (and uniqueness) of the solution. Therefore we are going to revisit that approach in order to s how that the Picard's procedure is constructive also because, by our point of view, it allows to write the solution in explici t form by means of Lie series. Finally we stress from a practical point of view, t hat the improved method gives an approaching polynomial, in the frame of an integration by series, with the require d precision, to the solution to the assigned Cauchy problem, eve n in situations when other methods fail. We want to rema rk that here, at the end, we shall find a link between the Picard's