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

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September 2014, 19(7): 2267-2278. doi: 10.3934/dcdsb.2014.19.2267

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

Francesco Mainardi ^1,

1.	Department of Physics and Astronomy, University of Bologna, and INFN, Via Irnerio 46, Bologna, I-40126, Italy

Received April 2013 Revised July 2013 Published August 2014

Abstract
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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.

Keywords: rational approximations., fractional relaxation, Mittag-Leffler function, complete monotonicity, asymptotic analysis.

Mathematics Subject Classification: Primary: 26A33, 33E12; Secondary: 35S10, 45K0.

Citation: Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267

References:

[1]	G. A. Baker, Essentials of Padè Approximants, Academic Press, New York, 1975.
[2]	D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific, Singapore, 2012. doi: 10.1142/9789814355216.
[3]	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.
[4]	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]
[5]	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.]
[6]	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.
[7]	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.
[8]	H. T. Davis, The Theory of Linear Operators, The Principia Press, Bloomington, Indiana, 1936.
[9]	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.
[10]	M. M. Dzherbashyan, , Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow., 1966 [in Russian].
[11]	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.
[12]	W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second Edition, Wiley, New York, 1971.
[13]	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.
[14]	R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), 491-518.
[15]	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]
[16]	B. Gross, On creep and relaxation, J. Appl. Phys., 18 (1947), 212-221. doi: 10.1063/1.1697606.
[17]	R. Hilfer (editor), Fractional Calculus, Applications in Physics, World Scientific, Singapore, 2000.
[18]	E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Ann. Math., 31 (1930), 479-528. doi: 10.2307/1968241.
[19]	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.
[20]	A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type, Differential and Integral Equations, 8 (1995), 993-1011.
[21]	A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203487372.
[22]	A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[23]	V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - New York, 1994.
[24]	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.
[25]	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.
[26]	J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.
[27]	R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006.
[28]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore, 2010. doi: 10.1142/9781848163300.
[29]	F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.
[30]	O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.
[31]	A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. doi: 10.1007/978-0-387-75894-7.
[32]	A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi, 1978.
[33]	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.
[34]	K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions, 12 (2001), 389-402. doi: 10.1080/10652460108819360.
[35]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[36]	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.
[37]	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.
[38]	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]
[39]	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.
[40]	G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. I. Holomorphic Functions, Nordhoff, Groningen, 1960.
[41]	R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2-nd ed., De Gruyter, Berlin, 2012. doi: 10.1515/9783110269338.
[42]	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.
[43]	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.
[44]	A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions, Sbornik Mathematics, 198 (2007), 1011-1023. doi: 10.1070/SM2007v198n07ABEH003871.
[45]	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.
[46]	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.
[47]	V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013. doi: 10.1007/978-3-642-33911-0.
[48]	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.
[49]	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.

show all references

References:

[1]	G. A. Baker, Essentials of Padè Approximants, Academic Press, New York, 1975.
[2]	D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific, Singapore, 2012. doi: 10.1142/9789814355216.
[3]	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.
[4]	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]
[5]	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.]
[6]	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.
[7]	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.
[8]	H. T. Davis, The Theory of Linear Operators, The Principia Press, Bloomington, Indiana, 1936.
[9]	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.
[10]	M. M. Dzherbashyan, , Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow., 1966 [in Russian].
[11]	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.
[12]	W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second Edition, Wiley, New York, 1971.
[13]	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.
[14]	R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), 491-518.
[15]	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]
[16]	B. Gross, On creep and relaxation, J. Appl. Phys., 18 (1947), 212-221. doi: 10.1063/1.1697606.
[17]	R. Hilfer (editor), Fractional Calculus, Applications in Physics, World Scientific, Singapore, 2000.
[18]	E. Hille and J. D. Tamarkin, On the theory of linear integral equations, Ann. Math., 31 (1930), 479-528. doi: 10.2307/1968241.
[19]	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.
[20]	A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type, Differential and Integral Equations, 8 (1995), 993-1011.
[21]	A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203487372.
[22]	A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[23]	V. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow - New York, 1994.
[24]	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.
[25]	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.
[26]	J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.
[27]	R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006.
[28]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore, 2010. doi: 10.1142/9781848163300.
[29]	F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.
[30]	O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983.
[31]	A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. doi: 10.1007/978-0-387-75894-7.
[32]	A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd, New Delhi, 1978.
[33]	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.
[34]	K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions, 12 (2001), 389-402. doi: 10.1080/10652460108819360.
[35]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[36]	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.
[37]	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.
[38]	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]
[39]	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.
[40]	G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. I. Holomorphic Functions, Nordhoff, Groningen, 1960.
[41]	R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2-nd ed., De Gruyter, Berlin, 2012. doi: 10.1515/9783110269338.
[42]	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.
[43]	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.
[44]	A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions, Sbornik Mathematics, 198 (2007), 1011-1023. doi: 10.1070/SM2007v198n07ABEH003871.
[45]	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.
[46]	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.
[47]	V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013. doi: 10.1007/978-3-642-33911-0.
[48]	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.
[49]	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.

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