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# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

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Volume 18, Issue 1

# Some Analytical and Numerical Properties of the Mittag-Leffler Functions

Moreno Concezzi
/ Renato Spigler
Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0006

## Abstract

Some analytical properties of the Mittag-Leffler functions, ${e}_{\alpha }\left(t\right)\not\equiv {E}_{\alpha }\left(-{t}^{\alpha }\right)$, are established on some t-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Padé approximants of ${e}_{\alpha }\left(t\right)\right)$ for $t>0$ and $0<\alpha <1.$. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an adaptive predictor-corrector method, based on the K. Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB® and by Mathematica when $t$ is real and even possibly large

MSC 2010: Primary 33E12; Secondary 34A08; 65L99

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## About the article

Published Online: 2015-02-10

Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 64–94, ISSN (Online) 1314-2224,

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© 2015 Diogenes Co., Sofia.

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