Multistep variable methods for exact integration of perturbed stiff linear systems

A family of real and analytical functions with values within the ring of M(m, R) is introduced. The solution for linear systems of differential equations is expressed as a series of Phi-functions. This new multistep method is defined for variable-step and variable-order, maintains the good properties of the Phi-function series method. It incorporates to compute the coefficients of the algorithm a recurrent algebraic procedure, based in the existing relation between the divided differences and the elemental and complete symmetrical functions. In addition, under certain hypotheses, the new multistep method calculates the exact solution of the perturbed problem. The new method is implemented in a computational algorithm which enables us to resolve in a general manner some physics and engineering IVP's modeled by means systems of differential equations. The good behaviour and precision of the method is evidenced by contrasting the results with other-reputed algorithms and even with methods based on Scheifele's G-functions.