A new method for exact integration of some perturbed stiff linear systems of oscillatory type

This article presents a new family of real functions with values within the ring of M(m,R) matrices, Φ-functions for perturbed linear systems and a numerical method adapted for integration of this type of problem. This method permits the system solution to be expressed as a series of Φ-functions. The coefficients of this series are obtained through recurrences in which the perturbation intervenes.The Φ-functions series method has the advantage of being exactly integrated in the perturbed problem. For this purpose an appropriate B matrix is selected and used to construct the operator described in this article, thus annihilating the disturbance terms, transforming the system into a homogenous second-order system, which is exactly integrated with the two first Φ-functions.The article ends with a detailed study of four perturbed systems which illustrate how the method is used in stiff problems or in highly oscillatory problems, contrasting its behaviour by studying its accuracy in comparison with other well-known codes.