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

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.

References in corpus (3)

• On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics
• On the coefficients of the asymptotic expansion of n!
• Fractional calculus and generalized Mittag-Leffler type functions

Cited by in corpus (8)

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• From continuous-time random walks to controlled-diffusion reaction
• Generalised Diffusion and Wave Equations: Recent Advances
• On the mistake in defining fractional derivative using a non-singular kernel