Journal of Applied Mathematics

Volume 2011, Article ID 298628, 51 pages

http://dx.doi.org/10.1155/2011/298628

Review Article

## Mittag-Leffler Functions and Their Applications

^{1}Office for Outer Space Affairs, United Nations, Vienna International Centre, P.O. Box 500, 1400 Vienna, Austria^{2}Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6^{3}Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala, Kerala 686574, India^{4}Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342005, India

Received 15 December 2010; Accepted 28 February 2011

Academic Editor: Ch Tsitouras

Copyright © 2011 H. J. Haubold et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Higher Transcendental Functions*, vol. 3, McGraw-Hill, New York, NY, USA, 1955. - G. M. Mittag-Leffler, “Une generalisation de l'integrale de Laplace-Abel,”
*Comptes Rendus de l'Académie des Sciences Série II*, vol. 137, pp. 537–539, 1903. View at Google Scholar - G. M. Mittag-Leffler, “Sur la nouvelle fonction ${E}_{\alpha}(x)$,”
*Comptes Rendus de l'Académie des Sciences*, vol. 137, pp. 554–558, 1903. View at Google Scholar - G. Mittag-Leffler, “Mittag-Leffler, Sur la representation analytiqie d’une fonction monogene (cinquieme note),”
*Acta Mathematica*, vol. 29, no. 1, pp. 101–181, 1905. View at Publisher · View at Google Scholar · View at Scopus - A. Wiman, “Über den fundamental satz in der theorie der funcktionen, ${E}_{\alpha}(x)$,”
*Acta Mathematica*, vol. 29, pp. 191–201, 1905. View at Google Scholar - A. Wiman, “Über die Nullstellun der Funktionen ${E}_{\alpha}(x)$,”
*Acta Mathematica*, vol. 29, pp. 217–234, 1905. View at Google Scholar - R. P. Agarwai, “A propos d’une note de M. Pierre Humbert,”
*Comptes Rendus de l'Académie des Sciences*, vol. 236, pp. 203–2032, 1953. View at Google Scholar - P. Humbert, “Quelques resultants retifs a la fonction de Mittag-Leffler,”
*Comptes Rendus de l'Académie des Sciences*, vol. 236, pp. 1467–1468, 1953. View at Google Scholar - P. Humbert and R. P. Agarwal, “Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations,”
*Bulletin of Science and Mathematics Series II*, vol. 77, pp. 180–185, 1953. View at Google Scholar - M. M. Dzherbashyan,
*Integral Transforms and Representations of Functions in the Complex Plane*, Nauka, Moscow, Russia, 1966. - K. R. Lang, “Astrophysical Formulae,” in
*Gas Processes and High-Energy Astrophysics*, vol. 1, Springer, New York, NY, USA, 3rd edition, 1999. View at Google Scholar · View at Zentralblatt MATH - K. R. Lang, “Astrophysical formulae,” in
*Space, Time, Matter and Cosmology*, vol. 2, Springer, New York, NY, USA, 1999. View at Google Scholar · View at Zentralblatt MATH - R. Hilfer, “Fractional diffusion based on Riemann-Liouville fractional derivatives,”
*Journal of Physical Chemistry B*, vol. 104, no. 3, pp. 914–924, 2000. View at Google Scholar · View at Scopus - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, Singapore, 2000. - R. K. Saxena, “Certain properties of generalized Mittag-Leffler function,” in
*Proceedings of the 3rd Annual Conference of the Society for Special Functions and Their Applications*, pp. 77–81, Chennai, India, 2002. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach, New York, NY, USA, 1993. - E. Hille and J. D. Tamarkin, “On the theory of linear integral equations,”
*Annals of Mathematics*, vol. 31, pp. 479–528, 1930. View at Google Scholar - G. W. S. Blair, “Psychorheology: links between the past and the present,”
*Journal of Texture Studies*, vol. 5, pp. 3–12, 1974. View at Google Scholar - P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behaviour of real materials,”
*Journal of Applied Mechanics, Transactions ASME*, vol. 51, no. 2, pp. 294–298, 1984. View at Google Scholar · View at Scopus - M. Caputo and F. Mainardi, “Linear models of dissipation in anelastic solids,”
*La Rivista del Nuovo Cimento*, vol. 1, no. 2, pp. 161–198, 1971. View at Publisher · View at Google Scholar · View at Scopus - R. Gorenflo and S. Vessella,
*Abel Integral Equations: Analysis and Applications*, vol. 1461, Springer, Berlin, Germany, 1991. - R. Gorenflo and R. Rutman, “On ultraslow and intermediate processes,” in
*Transform Methods and Special Functions, Sofia*, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 171–183, Science Culture Technology, Singapore, 1995. View at Google Scholar - A. A. Kilbas and M. Saigo, “On solutions of integral equations of Abel-Volterra type,”
*Differential and Integral Equations*, vol. 8, pp. 933–1011, 1995. View at Google Scholar - R. Gorenflo and Y. Luchko, “Operational methods for solving generalized Abel equations of second kind,”
*Integral Transforms and Specisl Functiond*, vol. 5, pp. 47–58, 1997. View at Google Scholar - R. Gorenflo and F. Mainardi, “Fractional oscillations and Mittag-Leffler functions,” Tech. Rep. 1-14/96, Free University of Berlin, Berlin, Germany, 1996. View at Google Scholar
- R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpinteri and F. Mainardi, Eds., pp. 223–276, Springer, Berlin, Germany, 1997. View at Google Scholar - F. Mainardi and R. Gorenflo, “The Mittag-Leffler function in the Riemann-Liouville fractional calculus,” in
*Boundary Value Problems, Special Functions and Fractional Calculus*, A. A. Kilbas, Ed., pp. 215–225, Byelorussian State University, Minsk, Belarus, 1996. View at Google Scholar - F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,”
*Journal of Computational and Applied Mathematics*, vol. 118, no. 1-2, pp. 283–299, 2000. View at Google Scholar · View at Scopus - R. Gorenflo, Y. Luchko, and S. V. Rogosin, “Mittag-Leffler type functions, notes on growth properties and distribution of zeros,” Tech. Rep. A04-97, Freie Universität Berlin, Berlin, Germany, 1997. View at Google Scholar
- R. Gorenflo, A. A. Kilbas, and S. V. Rogosin, “On the generalized Mittag-Leffler type function,”
*Integral Transform Special Functions*, vol. 7, no. 3-4, pp. 215–224, 1998. View at Google Scholar - Y. U. Luchko, “Operational method in fractional calculus,”
*Fractional Calculus & Applied Analisys*, vol. 2, pp. 463–488, 1999. View at Google Scholar - Y. U. F. Luchko and H. M. Srivastava, “The exact solution of certain differential equations of fractional order by using operational calculus,”
*Computers and Mathematics with Applications*, vol. 29, no. 8, pp. 73–85, 1995. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. A. Kilbas, M. Saigo, and R. K. Saxena, “Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels,”
*Journal of Integral Equations and Applications*, vol. 14, no. 4, pp. 377–386, 2002. View at Google Scholar - A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,”
*Integral Transforms and Special Functions*, vol. 15, no. 1, pp. 31–49, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. K. Saxena and M. Saigo, “Certain properties of fractional calculus operators associated with generalized Wright function,”
*Fractional Calculus & Applied Analisys*, vol. 6, pp. 141–154, 2005. View at Google Scholar - V. Kiryakova, “Some special functions related to fractional calculus and fractional non-integer order control systems and equations,”
*Facta Universitatis. Series: Mechanics, Automatic Control and Robotics*, vol. 7, no. 1, pp. 79–98, 2008. View at Google Scholar - V. S. Kiryakova, “Special functions of fractional calculus: recent list, results, applications,” in
*Proceedings of the 3rd IFC Workshop: Fractional Differentiation and Its Applications (FDA '08)*, pp. 1–23, Cankaya University, Ankara, Turkey, November 2008. - R. K. Saxena, S. L. Kalla, and V. S. Kiryakova, “Relations connecting multiindex Mittag-Leffler functions and Riemann-Liouville fractional calculus,”
*Algebras, Groups and Geometries*, vol. 20, pp. 363–385, 2003. View at Google Scholar - R. K. Saxena, A. M. Mathai, and H. J. Haubold, “On fractional kinetic equations,”
*Astrophysics and Space Science*, vol. 282, no. 1, pp. 281–287, 2002. View at Publisher · View at Google Scholar · View at Scopus - R. K. Saxena, A. M. Mathai, and H. J. Haubold, “On generalized fractional kinetic equations,”
*Physica A*, vol. 344, no. 3-4, pp. 657–664, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. K. Saxena, A. M. Mathai , and H. J. Haubold, “Unified fractional kinetic equations and a fractional diffusion equation,”
*Astrophysics & Space Science*, vol. 290, pp. 241–245, 2004. View at Google Scholar - R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Astrophysical thermonuclear functions for Boltzmann-Gibbs statistics and Tsallis statistics,”
*Physica A*, vol. 344, no. 3-4, pp. 649–656, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Fractional reaction-diffusion equations,”
*Astrophysics and Space Science*, vol. 305, no. 3, pp. 289–296, 2006. View at Publisher · View at Google Scholar · View at Scopus - R. K. Saxena and S. L. Kalla, “On the solutions of certain fractional kinetic equations,”
*Applied Mathematics and Computation*, vol. 199, no. 2, pp. 504–511, 2008. View at Publisher · View at Google Scholar · View at Scopus - A. M. Mathai, R. K. Saxena, and H. J. Haubold, “A certain class of Laplace transforms with application in reaction and reaction-diffusion equations,”
*Astrophysics & Space Science*, vol. 305, pp. 283–288, 2006. View at Google Scholar - H. J. Haubold and A. M. Mathai, “The fractional kinetic equation and thermonuclear functions,”
*Astrophysics and Space Science*, vol. 273, no. 1–4, pp. 53–63, 2000. View at Google Scholar · View at Scopus - H. J. Haubold, A. M. Mathai, and R. K Saxena, “Solution of fractional reaction-diffusion equations in terms of the H-function,”
*Bulletin of the Astronomical Society of India*, vol. 35, pp. 381–689, 2007. View at Google Scholar - H. M. Srivastava and R. K. Saxena, “Operators of fractional integration and their applications,”
*Applied Mathematics and Computation*, vol. 118, no. 1, pp. 1–52, 2001. View at Publisher · View at Google Scholar · View at Scopus - M. N. Berberan-Santos, “Relation between the inverse Laplace transforms of I(${t}^{\beta}$) and I(t): application to the Mittag-Leffler and asymptotic inverse power law relaxation functions,”
*Journal of Mathematical Chemistry*, vol. 38, no. 2, pp. 265–270, 2005. View at Publisher · View at Google Scholar · View at Scopus - H. Pollard, “The completely monotonic character of the Mittag-Leffler function ${E}_{\alpha}(-x)$,”
*Bulletin of the American Mathematical Society*, vol. 54, pp. 1115–116, 1948. View at Google Scholar - R. Gorenflo, Y. Luchko, and H. M. Srivastava, “Operational method for solving Gauss' hypergeometric function as a kernel,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 6, pp. 179–200, 1997. View at Google Scholar - R. Gorenflo, J. Loutschko, and Y. Luchko, “Computation of the Mittag-Leffler function and its derivatives,”
*Fractional Calculus & Applied Analisys*, vol. 5, no. 4, pp. 491–518, 2002. View at Google Scholar - I. S. Gupta and L. Debnath, “Some properties of the Mittag-Leffler functions,”
*Integral Transforms and Special Functions*, vol. 18, no. 5, pp. 329–336, 2007. View at Publisher · View at Google Scholar · View at Scopus - A. M. Mathai and R. K. Saxena,
*The H-Function with Applications in Statistics and Other Disciplines*, John Wiley & Sons, New York, NY, USA, 1978. - A. P. Prudnikov, Y. U. Brychkov, and O. I. Mariche,
*Integrals and Series*, vol. 3 of*More Special Functions*, Gordon and Breach, New York, NY, USA, 1990. - A. A. Kilbas and M. Saigo,
*H-Transforms: Theory and Applications, Analytic Methods and Special Functions*, Chapman & Hall, CRC Press, Boca Raton, Fla, USA, 2004. - E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,”
*Journal London Mathematical Society*, vol. 10, pp. 286–293, 1935. View at Google Scholar - E. M. Wright, “The asymptotic expansion of the integral functions defined by Taylor series,”
*Philosophical Transactions of the Royal Society A*, vol. 238, pp. 423–451, 1940. View at Google Scholar - A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Higher Transcendental Functions*, vol. 1, McGraw-Hill, New York, NY, USA, 1953. - A. A. Kilbas, M. Saigo, and J. J. Trujillo, “On the generalized Wright function,”
*Fractional Calculus & Applied Analisys*, vol. 5, no. 4, pp. 437–460, 2002. View at Google Scholar - E. M. Wright, “On the coefficients of power series having exponential singularities,”
*Journal London Mathematical Society*, vol. 8, pp. 71–79, 1933. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V .S. Kiryakova,
*Generalized Fractional Calculus and Applications*, vol. 301 of*Pitman Research Notes in Mathematics*, Longman, Harlow, UK; John Wiley & Sons, New York, NY, USA, 1994. - F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Wien, Germany, 1997. View at Google Scholar · View at Zentralblatt MATH - E. Buckwar and Y. Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations,”
*Journal of Mathematical Analysis and Applications*, vol. 227, no. 1, pp. 81–97, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. U. Luchko and R. Gorenflo, “Scale-invariant solutins of a partial differential equation of fractional order,”
*Fractional Calculus & Applied Analisys*, vol. 1, pp. 63–78, 1998. View at Google Scholar - A. A. Kilbas, “Fractional calculus of the generalized Wright function,”
*Fractional Calculus & Applied Analisys*, vol. 8, no. 2, pp. 113–126, 2005. View at Google Scholar - T. R. Prabhakar, “A singular integral equation with a generalized Mittag-Leffler function in the kernel,”
*Yokohama Mathematical Journal*, vol. 19, pp. 7–15, 1971. View at Google Scholar - E. M. Wright, “The asymptotic expansion of the generalized Bessel function,”
*Proceedings London Mathematical Society*, vol. 38, pp. 257–270, 1934. View at Google Scholar - Y. U. Luchko and S. B. Yaskubovich, “Operational calculus for the generalized fractional differential operator and applications,”
*Mathematica Balkanica—New Series*, vol. 4, no. 2, pp. 119–130, 1990. View at Google Scholar - Y. U. Luchko and S. B. Yakubovich, “An operational method for solving some classes of integro-differential equations,”
*Differentsial'nye Uravneniya*, vol. 30, pp. 269–280, 1994 (Russian). View at Google Scholar - M. A. Al-Bassam and Y. F. Luchko, “On generalized fractional calculus and its application to the solution of integro-differential equations,”
*Journal of Fractional Calculus*, vol. 7, pp. 69–88, 1995. View at Google Scholar · View at Zentralblatt MATH - S. B. Hadid and Y. Luchko, “An operational method for solving fractional differential equations of anarbitrary real order,”
*Pan-American Mathematical Journal*, vol. 6, pp. 57–73, 1996. View at Google Scholar - R. Gorenflo, A. Iskenderov, and Y. Luchko, “Mapping between solutions of fractional diffusion wave equations,”
*Fractional Calculus & Applied Analisys*, vol. 3, pp. 75–86, 2000. View at Google Scholar - R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation,”
*Journal of Computational and Applied Mathematics*, vol. 118, no. 1-2, pp. 175–191, 2000. View at Google Scholar · View at Scopus - Y. U. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with a Caputo derivative,”
*Acta Mathematica Vietnamica*, vol. 24, pp. 207–234, 1999. View at Google Scholar - A. A. Kilbas and M. Saigo, “Fractional integrals and derivatives of Mittag-Leffler type function,”
*Doklady Akademii Nauk Belarusi*, vol. 39, no. 4, pp. 22–26, 1995 (Russian). View at Google Scholar - A. A. Kilbas and M. Saigo, “On mittag-leffler type function, fractional calculus operators and solutions of integral equations,”
*Integral Transforms and Special Functions*, vol. 4, no. 4, pp. 355–370, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. - A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Tables of Integral Transforms*, vol. 2, McGraw-Hill, New York, NY, USA, 1954. - K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. - M. Caputo,
*Elasticitá e Dissipazione*, Zanichelli, Bologna, Italy, 1969. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,”
*Physics Report*, vol. 339, no. 1, pp. 1–77, 2000. View at Google Scholar · View at Scopus - A. Compte, “Stochastic foundations of fractional dynamics,”
*Physical Review E*, vol. 53, no. 4, pp. 4191–4193, 1996. View at Google Scholar · View at Scopus - B. N. Al-Saqabi and VU. K. Tuan, “Solution of a fractional differintegral equation,”
*Integral Transforms and Special Functions*, vol. 4, no. 4, pp. 321–326, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - V. S. Kiryakova, “Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus,”
*Journal of Computational and Applied Mathematics*, vol. 118, no. 1-2, pp. 241–259, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. M. Dzherbashyan, “On the integral transformations generated by the generalzied Mittag-Leffler fuction,”
*Izdatel'stvo Akademii Nauk Armyanskoi SSR*, vol. 13, no. 3, pp. 21–63, 1960 (Russian). View at Google Scholar - V. S. Kiryakova, “Multiindex Mittag-Leffler functions related to Gelfond-Leontiev operators and Laplace type integral transforms,”
*Fractional Calculus & Applied Analisys*, vol. 2, pp. 4445–462, 1999. View at Google Scholar - S. Sharma, “Fractional differentiation and fractional integration of the M-series,”
*Fractional Calculus & Applied Analisys*, vol. 11, pp. 187–191, 2008. View at Google Scholar - R. K. Saxena, “A remark on a paper on M-series,”
*Fractional Calculus & Applied Analisys*, vol. 12, no. 1, pp. 109–110, 2009. View at Google Scholar - R. N. Pillai, “On Mittag-Leffler functions and related distributions,”
*Annals of the Institute of Statistical Mathematics*, vol. 42, no. 1, pp. 157–161, 1990. View at Publisher · View at Google Scholar · View at Scopus - A. M. Mathai and H. J. Haubold,
*Special Functions for Applied Scientists*, Springer, New York, NY, USA, 2008. - K. Jayakumar, “Mittag-Leffler process,”
*Mathematical and Computer Modelling*, vol. 37, no. 12-13, pp. 1427–1434, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. Jayakumar and R. P. Suresh, “Mittag-Leffler distribution,”
*Journal of the Indian Society of Probability and Statistics*, vol. 7, pp. 51–71, 2003. View at Google Scholar - A. M. Mathai,
*A Handbook of Generalized Special Functions for Statistical and Physical Sciences*, Clarendon Press, Oxford, UK, 1993. - G. D. Lin, “On the Mittag-Leffler distribution,”
*Journal of Statistical Planning and Inference*, vol. 74, pp. 1–9, 1998. View at Google Scholar - A. M. Mathai, “A pathway to matrix-variate gamma and normal densities,”
*Linear Algebra and Its Applications*, vol. 396, no. 1–3, pp. 317–328, 2005. View at Publisher · View at Google Scholar · View at Scopus - R. N. Pillai and K. Jayakumar, “Discrete Mittag-Leffler distributions,”
*Statistics and Probability Letters*, vol. 23, no. 3, pp. 271–274, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. G. Pakes, “Mixture representations for symmetric generalized Linnik laws,”
*Statistics and Probability Letters*, vol. 37, no. 3, pp. 213–221, 1998. View at Google Scholar · View at Scopus - F. Mainardi and G. Pagnini, “Mellin-Barnes integrals for stable distributions and their convolutions,”
*Fractional Calculus & Applied Analysis*, vol. 11, no. 4, pp. 443–456, 2008. View at Google Scholar - S. C. Lim and L. P. Teo, “Analytic and asymptotic properties of multivariate generalized Linnik's probability densities,”
*Journal of Fourier Analysis and Applications*, pp. 1–33, 2009. View at Publisher · View at Google Scholar · View at Scopus