American Institute of Mathematical Sciences

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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

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 & 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, (1975). Google Scholar

[2]

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

[3]

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

[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., 193 (2011), 161. Google Scholar

[5]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism,, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134. Google Scholar

[6]

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

[7]

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

[8]

H. T. Davis, The Theory of Linear Operators,, The Principia Press, (1936). Google Scholar

[9]

K. Diethelm, The Analysis of Fractional Differential Equations,, Springer, (2004). doi: 10.1007/978-3-642-14574-2. Google Scholar

[10]

M. M. Dzherbashyan,, Integral Transforms and Representations of Functions in the Complex Plane,, Nauka, (1966). Google Scholar

[11]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. III. Based, (1955). Google Scholar

[12]

W. Feller, An Introduction to Probability Theory and its Applications,, Vol. II, (1971). Google Scholar

[13]

A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus,, {First Annual Report, (2002), 2002. Google Scholar

[14]

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

[15]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, in Fractals and Fractional Calculus in Continuum Mechanics, (1997), 223. Google Scholar

[16]

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

[17]

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

[18]

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

[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. doi: 10.2478/s13540-013-0024-9. Google Scholar

[20]

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

[21]

A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications,, Chapman and Hall/CRC, (2004). doi: 10.1201/9780203487372. Google Scholar

[22]

A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006). Google Scholar

[23]

V. Kiryakova, Generalized Fractional Calculus and Applications,, Longman & J. Wiley, (1994). Google Scholar

[24]

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

[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. doi: 10.1063/1.3526661. Google Scholar

[26]

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

[27]

R. L. Magin, Fractional Calculus in Bioengineering,, Begell House Publishers, (2006). Google Scholar

[28]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300. Google Scholar

[29]

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

[30]

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions,, Theory and Algorithmic Tables, (1983). Google Scholar

[31]

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

[32]

A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines,, Wiley Eastern Ltd, (1978). Google Scholar

[33]

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

[34]

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

[35]

I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). Google Scholar

[36]

I. Podlubny, Mittag-Leffler function,, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, (2006). Google Scholar

[37]

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

[38]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications,, Gordon and Breach, (1993). Google Scholar

[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. doi: 10.2478/s13540-012-0031-2. Google Scholar

[40]

G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable,, Vol. I. Holomorphic Functions, (1960). Google Scholar

[41]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications,, 2-nd ed., (2012). doi: 10.1515/9783110269338. Google Scholar

[42]

T. Simon, Comparing Fréchet and positive stable laws,, Electron. J. Probab., 19 (2014), 1. doi: 10.1214/EJP.v19-3058. Google Scholar

[43]

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications,, South Asian Publishers, (1982). Google Scholar

[44]

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

[45]

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

[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. doi: 10.1080/10652461003675737. Google Scholar

[47]

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

[48]

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

[49]

C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse,, E-print , (2013). Google Scholar

show all references

References:
[1]

G. A. Baker, Essentials of Padè Approximants,, Academic Press, (1975). Google Scholar

[2]

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

[3]

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

[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., 193 (2011), 161. Google Scholar

[5]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism,, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134. Google Scholar

[6]

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

[7]

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

[8]

H. T. Davis, The Theory of Linear Operators,, The Principia Press, (1936). Google Scholar

[9]

K. Diethelm, The Analysis of Fractional Differential Equations,, Springer, (2004). doi: 10.1007/978-3-642-14574-2. Google Scholar

[10]

M. M. Dzherbashyan,, Integral Transforms and Representations of Functions in the Complex Plane,, Nauka, (1966). Google Scholar

[11]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. III. Based, (1955). Google Scholar

[12]

W. Feller, An Introduction to Probability Theory and its Applications,, Vol. II, (1971). Google Scholar

[13]

A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus,, {First Annual Report, (2002), 2002. Google Scholar

[14]

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

[15]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, in Fractals and Fractional Calculus in Continuum Mechanics, (1997), 223. Google Scholar

[16]

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

[17]

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

[18]

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

[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. doi: 10.2478/s13540-013-0024-9. Google Scholar

[20]

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

[21]

A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications,, Chapman and Hall/CRC, (2004). doi: 10.1201/9780203487372. Google Scholar

[22]

A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006). Google Scholar

[23]

V. Kiryakova, Generalized Fractional Calculus and Applications,, Longman & J. Wiley, (1994). Google Scholar

[24]

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

[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. doi: 10.1063/1.3526661. Google Scholar

[26]

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

[27]

R. L. Magin, Fractional Calculus in Bioengineering,, Begell House Publishers, (2006). Google Scholar

[28]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300. Google Scholar

[29]

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

[30]

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions,, Theory and Algorithmic Tables, (1983). Google Scholar

[31]

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

[32]

A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines,, Wiley Eastern Ltd, (1978). Google Scholar

[33]

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

[34]

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

[35]

I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). Google Scholar

[36]

I. Podlubny, Mittag-Leffler function,, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, (2006). Google Scholar

[37]

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

[38]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications,, Gordon and Breach, (1993). Google Scholar

[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. doi: 10.2478/s13540-012-0031-2. Google Scholar

[40]

G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable,, Vol. I. Holomorphic Functions, (1960). Google Scholar

[41]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications,, 2-nd ed., (2012). doi: 10.1515/9783110269338. Google Scholar

[42]

T. Simon, Comparing Fréchet and positive stable laws,, Electron. J. Probab., 19 (2014), 1. doi: 10.1214/EJP.v19-3058. Google Scholar

[43]

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications,, South Asian Publishers, (1982). Google Scholar

[44]

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

[45]

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

[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. doi: 10.1080/10652461003675737. Google Scholar

[47]

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

[48]

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

[49]

C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse,, E-print , (2013). Google Scholar

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