Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s

The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented.