Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel

Abstract

Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional calculus approach is a popular approach to analyze the anomalous relaxation processes and has been intensively studied in various dielectric materials. However, Capelas de Oliveira et al. proved that the memory kernels of Caputo type fractional derivatives must satisfy the initial value condition (IVC). But both kernels of the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative do not satisfy this condition. In this paper, we prove that the Caputo type derivative with a Prabhakar-like kernel satisfies the IVC and this derivative is in the framework of general Caputo fractional derivative (GC derivative). Corresponding anomalous relaxation model and its solution are discussed. Analysis result shows our model, as a direct extension of the Cole–Cole model, contains the Debye model and the fractional relaxation model as particular cases.