The possibility of using non-local models to simulate the behavior of materials at the micro-scale and nano-scale has been the subject of numerous studies in recent years. The peridynamic-based approach, proposed by Silling, introduces the concept of a horizon for modeling composites with non-local behavior. The horizon concept introduced by Silling is used in the model proposed by the Authors in a previous work, which employs the regional fractional Laplacian operator to model a non-local behavior. In particular, the model accounts for both local and non-local behaviors in a two-phase approach, allowing the relative weight of the two to vary. In the present work, the results obtained from the study of the eigenvalues and eigenfunctions of the problem are presented, through a numerical estimate based on the integral representation of the fractional Laplacian, which can be solved using a finite difference approach. Additionally, an application of this model is shown to solve the case of free vibrations with a non-linear damping term.