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https://doi.org/10.1515/fca-2020-0002(free access)

The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

Ultrafast diffusion process characterized by unusually large diffusivities is often occurs on porous media and the mean square displacement grows exponentially in time. This paper clarifies the characteristics of ultrafast diffusion and tackles this perplexing problem using the fractional Brownian motion run with a nonlinear clock model. We employ the Mittag-Leffler function as the nonlinear clock, the increments are dependent and obey Gaussian distribution, and the derived corresponding mean square displacement is more widely than the exponential function. A comparison between the power law model and the proposed model with respect to available experimental data verifies that the proposed model is more effective and accurate. Ultraslow diffusion is also studied with the inverse Mittag-Leffler function as the nonlinear clock. The results show that it can capture the ultraslow diffusion process better than the case of the logarithmic model. As the generalization of fractional Brownian motion, fractional Brownian motion run with a nonlinear clock is an alternative model method for extreme anomalous diffusion in complex systems.