Advanced Search
Article Contents
Article Contents

On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$

Abstract Related Papers Cited by
  • We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_\alpha(t) := E_\alpha(-t^\alpha)$ for $0<\alpha<1$ and $t>0$, which is known to be completely monotone (CM) with a non-negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that (surprisingly) these two spectra coincide so providing a universal scaling property of this function, not well pointed out in the literature. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_\alpha(t)$ as $t\to 0$ and $t\to +\infty$. The first set is given by the stretched exponential for small times and the power law for large times, following a standard approach. For the second set we chose two rational CM functions in $t^\alpha$, obtained as the Pad\`e Approximants (PA) $[0/1]$ to the convergent series in positive powers (as $t\to 0$) and to the asymptotic series in negative powers (as $t\to \infty$), respectively. From numerical computations we are allowed to the conjecture that the second set provides upper and lower bounds to the Mittag-Leffler function.
    Mathematics Subject Classification: Primary: 26A33, 33E12; Secondary: 35S10, 45K05.


    \begin{equation} \\ \end{equation}
  • [1]

    G. A. Baker, Essentials of Padè Approximants, Academic Press, New York, 1975.


    D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific, Singapore, 2012.doi: 10.1142/9789814355216.


    L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709.doi: 10.1214/EJP.v15-762.


    E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J., Special Topics, 193 (2011) 161-171. [E-print arxiv.org/abs/1106.1761]


    M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134-147. [Reprinted in Fract. Calc. Appl. Anal.,10 (2007), 309-324.]


    M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1 (1971), 161-198.doi: 10.1007/BF02820620.


    K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics, II. Direct current characteristics, J. Chemical Physics, 10 (1942), 98-105.doi: 10.1063/1.1723677.


    H. T. Davis, The Theory of Linear Operators, The Principia Press, Bloomington, Indiana, 1936.


    K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Lecture Notes in Mathematics No 2004, Heidelberg, 2010.doi: 10.1007/978-3-642-14574-2.


    M. M. Dzherbashyan, , Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow., 1966 [in Russian].


    A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.


    W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second Edition, Wiley, New York, 1971.


    A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus, {First Annual Report, NASA/TM-2002-211914}, Gleen Research Center, 2002, pp. XIV - 121.


    R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), 491-518.


    R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, (eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, 1997, pp. 223-276. [E-print arxiv.org/abs/0805.3823]


    B. Gross, On creep and relaxation, J. Appl. Phys., 18 (1947), 212-221.doi: 10.1063/1.1697606.


    R. Hilfer (editor), Fractional Calculus, Applications in Physics, World Scientific, Singapore, 2000.


    E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Ann. Math., 31 (1930), 479-528.doi: 10.2307/1968241.


    A. A. Kilbas, A . A. Koroleva and S. V. Rogosin, Multi-parametric Mittag-Leffler functions and their extensions, Fract. Calc. Appl. Anal., 16 (2013), 378-404.doi: 10.2478/s13540-013-0024-9.


    A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type, Differential and Integral Equations, 8 (1995), 993-1011.


    A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2004.doi: 10.1201/9780203487372.


    A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.


    V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - New York, 1994.


    V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus, Comp. Math. Appl., 59 (2010), 1885-1895.doi: 10.1016/j.camwa.2009.08.025.


    V. Kiryakova and Y. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis, In American Institute of Physics - Conf. Proc., 1301 (2010), 597-613.doi: 10.1063/1.3526661.


    J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.


    R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006.


    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore, 2010.doi: 10.1142/9781848163300.


    F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.


    O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.


    A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008.doi: 10.1007/978-0-387-75894-7.


    A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi, 1978.


    A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer Verlag, New York, 2010.doi: 10.1007/978-1-4419-0916-9.


    K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions, 12 (2001), 389-402.doi: 10.1080/10652460108819360.


    I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.


    I. Podlubny, Mittag-Leffler function, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, Matlab File Exchange www.mathworks.com/matlabcentral/fileexchange, 2006.


    H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_\alpha (-x)$, Bull. Amer. Math. Soc., 54 (1948), 1115-1116.doi: 10.1090/S0002-9904-1948-09132-7.


    S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993. [English translation and revised version from the Russian edition, Integrals and Derivatives of Fractional Order and Some of Their Applications Nauka i Tekhnika, Minsk, 1987]


    T. Sandev, R. Metzler and Z. Tomovski, Velocity and displacement correlation functions for fractional generalized Langevin equations, Fract. Calc. Appl. Anal., 15 (2012), 426-450.doi: 10.2478/s13540-012-0031-2.


    G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. I. Holomorphic Functions, Nordhoff, Groningen, 1960.


    R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2-nd ed., De Gruyter, Berlin, 2012.doi: 10.1515/9783110269338.


    T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25. [E-print arXiv:1310.1888]doi: 10.1214/EJP.v19-3058.


    H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.


    A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions, Sbornik Mathematics, 198 (2007), 1011-1023.doi: 10.1070/SM2007v198n07ABEH003871.


    V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2010.doi: 10.1007/978-3-642-14003-7.


    Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814.doi: 10.1080/10652461003675737.


    V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013.doi: 10.1007/978-3-642-33911-0.


    R. Wong and Y.-Q Zhao, Exponential asymptotics of the Mittag-Leffler function, Constructive Approximation, 18 (2002), 355-385.doi: 10.1007/s00365-001-0019-3.


    C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse, E-print arXiv:1310.5592 [math.CA] (2013), pp. 17.

  • 加载中

Article Metrics

HTML views() PDF downloads(2398) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint