On Translation Groups and Non-iterative Transformation Methods

The aim of this work is to point out that, within group invariance theory, some classes of boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of (free) boundary problems arises because these are (always) often nonlinear problems. The theoretical content of this paper is original: results already available in literature are related to the use of scaling or spiral groups of transformations; here we show how it is also possible to use the invariance with respect to two translation groups. As far as applications of the proposed approach are concerned, we solve two problems: a free boundary problem describing a rope configuration against an obstacle, where we compare the obtained numerical results with the exact solution, and a boundary problem modeling the fall of a parachutist, where we modify the classical formulation of the problem in order to prescribe the total falling time.