Large scale singularly perturbed boundary value problems

In this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear non-singular transformation that transforms an arbitrary n-time scale system into a diagonal form. Furthermore, a dichotomy transformation is employed to decompose the faster subsystems into stable and unstable modes. Fast, slow, stable and unstable modes decomposition processes provide a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system. This method yields a constructive and computationally attractive way to investigate the system.