Generalized exponential time differencing for fractional oscillation models

Oscillations occur in various processes and are of great significance for understanding, analyzing and simulating real-world phenomena. Fractional evolution equations of oscillatory type provide an effective tool to model some anomalous oscillatory behaviors. Generally, solutions of these equations exhibit oscillatory behavior which can sometimes be erratic. Therefore, developing efficient numerical methods that adequately capture the oscillatory behavior of these solutions can be challenging. In this paper, an efficient novel second-order numerical scheme is developed for a class of fractional oscillation models. The scheme is based on the exponential time differencing technique, special approximations of Mittag-Leffler function, and the non-uniform mesh. Convergence and the stability analysis are conducted and verified through numerical experiments. In particular, we illustrate the potential of the numerical scheme as a time integrator for fractional diffusion-wave equations.