A Tutorial on the Basic Special Functions of Fractional Calculus

Abstract

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright function that generalizes the Gaussian and is related with the time-fractional diffusion equation.

26 pages, 16 figures

References in corpus (5)

• The fundamental solution of the space-time fractional diffusion equation
• Analytical properties and applications of the Wright function
• The Wright functions of the second kind in Mathematical Physics
• Alternative numerical computation of one-sided Levy and Mittag-Leffler distributions
• On the Evolution of Fractional Diffusive Waves

Cited by in corpus (7)

• Why the Mittag-Leffler function can be considered the Queen function of the Fractional Calculus?
• Subdiffusion equation with Caputo fractional derivative with respect to another function
• The Wright functions of the second kind in Mathematical Physics
• Efficient computation of the Wright function and its applications to fractional diffusion-wave equations
• Stochastic foundations of $g$-subdiffusion process
• Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion
• Subdiffusion with particle immobilization process described by differential equation with Riemann--Liouville type fractional time derivative