The Prabhakar or three parameter Mittag--Leffler function: theory and application

TL;DR

This paper explores the mathematical properties and applications of the Prabhakar function, a three-parameter Mittag-Leffler function, emphasizing its role in modeling anomalous dielectric behavior and solving nonlinear heat conduction equations with memory effects.

Contribution

It provides a comprehensive analysis of the Prabhakar function's properties, asymptotic behavior, and introduces fractional operators and equations involving Prabhakar derivatives.

Findings

Asymptotic expansion of the Prabhakar function in the complex plane.

Development of fractional integral and derivative operators of Prabhakar type.

Application to nonlinear heat conduction equations with memory effects.

Abstract

The Prabhakar function (namely, a three parameter Mittag-Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak-Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied.