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.

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Cited by in corpus (39)

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• Variable-order fractional calculus: a change of perspective
• A comment on some new definitions of fractional derivative
• General fractional calculus and Prabhakar's theory
• Generalised geometric Brownian motion: Theory and applications to option pricing
• A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators
• Generalized diffusion-wave equation with memory kernel
• The generalized Cattaneo (telegrapher's) equation and corresponding random walks
• Stochastic resetting on comb-like structures
• Stability of fractional-order systems with Prabhakar derivatives
• Storage and dissipation of energy in Prabhakar viscoelasticity
• Stochastic dynamics with multiplicative dichotomic noise: heterogeneous telegrapher's equation, anomalous crossovers and resetting
• Quenched and Annealed Disorder Mechanisms in Comb-Models with Fractional Operators
• Models for characterizing the transition among anomalous diffusions with different diffusion exponents
• Finite-velocity diffusion on a comb
• General approach to stochastic resetting
• Generalized fractional Poisson process and related stochastic dynamics
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• Some results on the complete monotonicity of the Mittag-Leffler functions of Le Roy type
• Bicomplex Mittag-Leffler Function and Properties
• The Volterra type equations related to the non-Debye relaxation
• The Havriliak-Negami and Jurlewicz-Weron-Stanislavsky relaxation models revisited: memory functions based study
• From continuous-time random walks to controlled-diffusion reaction
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• Generalised Diffusion and Wave Equations: Recent Advances
• Volterra-Prabhakar derivative of distributed order and some applications
• Studies on generalized Yule models
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• Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions
• Fractional diffusion equations interpolate between damping and waves
• Emergent time crystal from a fractional Langevin equation with white and colored noise
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