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Fractional Calculus and Applied Analysis

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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


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


  • [1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, 55, U. S. Government Printing Office, Washington, D. C. (1964).Google Scholar

  • [2] C. Alsina and M. S. Tomlaa, A geometrical proof of a new inequality for the gamma function. J. Ineq. Pure Appl. Math. 6, No 2 (2005), Article # 48.Google Scholar

  • [3] K. Diethelm and A. D. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order. In: S. Heinzel, T. Plesser (Eds.), Forschung und Wissenschaftliches Rechnen 1998, Gessellschaft fur Wissenschaftliche Datenverarbeitung, Gottingen (1999), 57.71.Google Scholar

  • [4] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods. Computing 71 (2003), 305.319; at http://www.scielo.br/pdf/cam/v23n1/a02v23n1.pdf.Google Scholar

  • [5] K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, No 1 (2004), 31.52.CrossrefGoogle Scholar

  • [6] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions [Bateman Manuscript Project], Vols. 1, 2, and 3. McGraw-Hill, New York (1953).Google Scholar

  • [7] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations. Internat. J. Comput. Math. 87, No 10 (2010), 2281.2290.Google Scholar

  • [8] R. Gorenflo, J. Loutchko, and Yu. Luchko, Computation of the Mittag- Leffler function E-z) and its derivative. Fract. Calc. Appl. Anal. 5, No 4 (2002), 491.518.Google Scholar

  • [9] R. Gorenflo, J. Loutchko, and Yu. Luchko, Correction to Computation of the Mittag-Leffler function E-z) and its derivative [Fract. Calc. Appl. Anal. 5, No 4 (2002), 491.518]. Fract. Calc. Appl. Anal. 6, No 1 (2003), 111.112.Google Scholar

  • [10] E. Hairer, C. Lubich, and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Statist. Comput. 6, No 3 (1985), 532.541.CrossrefGoogle Scholar

  • [11] F. Mainardi, On some properties of the Mittag-Leffler function E completely monotone for t 0 with 0 Discrete Continuous Dynamical Systems - Series B (DCDS-B), To appear; see also FRACALMO Preprint at http://www.fracalmo.org; and http://arxiv.org/pdf/1305.0161.pdf.Google Scholar

  • [12] http://reference.wolfram.com/mathematica/ref/MittagLefflerE.html.Google Scholar

  • [13] http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.Google Scholar

  • [14] F. W.J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Eds.), NIST Digital Library of Mathematical Functions. Cambridge University Press, Cambridge (2010).Google Scholar

  • [15] T. Simon, Comparing Frechet and positive stable laws. Electron. J. Probab. 19, No 16 (2014), 1-25; see also at http://arxiv.org/pdf/1310.1888v1.pdf.Web of ScienceGoogle Scholar

  • [16] D. Valerio and J. T. Machado, On the numerical computation of the Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 3419-3424; doi:10.1016/j.cnsns.2014.03.014.CrossrefGoogle Scholar

  • [17] Caibin Zeng and YangQuan Chen, Global Pade approximations of the generalized Mittag-Leffler function and its inverse; at http://arxiv.org/abs/1310.5592.Google Scholar

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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