An algorithm for exact integration of some forced and damped oscillatory problems, based in the τ-functions

This article creates a new method for the numerical integration of forced and damped oscillators, and their computational implementation. It also provides a generalisation of methods based on G-function and φ-function series.The algorithm produced in this paper integrates the non-perturbed problem with no truncation error, in which the perturbation parameter is a factor in the local truncation error. Under certain hypotheses, the new method calculates the exact solution of the perturbed problem as a series of τ-functions, the coefficients of which are obtained using simple algebraic recurrences involving the perturbation function.The new τ-function series method makes it possible to provide general solutions for certain problems in physics and engineering that are modelled using forced and damped oscillators. The method is more accurate than the well-known LSODE, MGEAR and GEAR methods in the way it resolves stiff and highly oscillatory problems, as the applications in this paper demonstrate.