The chapter proposes a mathematical model for a wide class of hereditary oscillators, which is a Cauchy problem in the local formulation. As an initial model equation, an integrodifferential equation of Voltaire type was introduced, which was reduced by means of a special choice of difference kernels to a differential equation with nonlocal derivatives of fractional-order variables. An explicit finite-difference scheme is proposed, and questions of its stability and convergence are investigated. A computer study of the proposed numerical algorithm on various test examples of the hereditary oscillators Airy, Duffing, and others was carried out. Oscillograms and phase trajectories are plotted and constructed
