WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 19, 2020
A Tutorial on the Basic Special Functions of Fractional Calculus
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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 functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.
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Keywords: Mittag-Leffler functions, Wright functions, Fractional Calculus, Laplace, Fourier and Mellin transforms, Probability theory, Stable distributions.
Pages: 74-98
DOI: 10.37394/23206.2020.19.8