Random-order fractional differential equation models

This paper proposes a new concept of random-order fractional differential equation model, in which a noise term is included in the fractional order. We investigate both a random-order anomalous relaxation model and a random-order time fractional anomalous diffusion model to demonstrate the advantages and the distinguishing features of the proposed models. From numerical simulation results, it is observed that the scale parameter and the frequency of the noise play a crucial role in the evolution behaviors of these systems. In addition, some potential applications of the new models are presented.