Some Analytical and Numerical Properties of the Mittag-Leffler Functions

Abstract

Some analytical properties of the Mittag-Leffler functions, e α (t) ≡ E α (−t α ), 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 α (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 R 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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