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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 18, Issue 1

Issues

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, eα(t)Eα(-tα), 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 eα(t)) for t>0 and 0<α<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

Key Words and Phrases: Mittag-Leffler functions; fractional ordinary differential equations; predictor-corrector methods for fractional differential equations; adaptive methods for fractional differential equations

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About the article

Received: 2014-03-21

Published Online: 2015-02-10


Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 64–94, ISSN (Online) 1314-2224, DOI: https://doi.org/10.1515/fca-2015-0006.

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