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Mittag‐Leffler Functions and Their Applications

Author

Listed:
  • H. J. Haubold
  • A. M. Mathai
  • R. K. Saxena

Abstract

Motivated essentially by the success of the applications of the Mittag‐Leffler functions in many areas of science and engineering, the authors present, in a unified manner, a detailed account or rather a brief survey of the Mittag‐Leffler function, generalized Mittag‐Leffler functions, Mittag‐Leffler type functions, and their interesting and useful properties. Applications of G. M. Mittag‐Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag‐Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth. In this survey paper, nearly all types of Mittag‐Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag‐Leffler functions to make the reader familiar with the present trend of research in Mittag‐Leffler type functions and their applications.

Suggested Citation

  • H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag‐Leffler Functions and Their Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnljam:v:2011:y:2011:i:1:n:298628
    DOI: 10.1155/2011/298628
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    References listed on IDEAS

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    1. Saxena, R.K. & Mathai, A.M. & Haubold, H.J., 2004. "On generalized fractional kinetic equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 657-664.
    2. Saxena, R.K. & Mathai, A.M. & Haubold, H.J., 2004. "Astrophysical thermonuclear functions for Boltzmann–Gibbs statistics and Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 649-656.
    3. Pakes, Anthony G., 1998. "Mixture representations for symmetric generalized Linnik laws," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 213-221, March.
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    Citations

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    Cited by:

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    5. Kangqun Zhang, 2020. "Existence of Solution of Space–Time Fractional Diffusion‐Wave Equation in Weighted Sobolev Space," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).
    6. Roman G. Smirnov, 2026. "Caputo-Type Memory Invariants: A Fractional Generalization of the Cobb-Douglas Production Function," Papers 2605.20152, arXiv.org.
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    10. Om Ahuja & Asena Çetinkaya & Naveen Kumar Jain, 2022. "Mittag‐Leffler Operator Connected with Certain Subclasses of Bazilevic̆ Functions," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    11. Ahmad A Abubaker & Khaled Matarneh & Suha B. Al-Shaikh & Mohammad Faisal Khan, 2025. "Some new applications of the fractional integral and four-parameter Mittag-Leffler function," PLOS ONE, Public Library of Science, vol. 20(2), pages 1-18, February.
    12. Sara H. M. Hamed & Eltayeb A. Yousif & Arbab I. Arbab, 2014. "Analytic and Approximate Solutions of the Space‐Time Fractional Schrödinger Equations by Homotopy Perturbation Sumudu Transform Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    13. Marko Kostić, 2013. "Perturbation Theory for Abstract Volterra Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    14. Zhokh, O.O. & Strizhak, P.E., 2026. "A review of non-fickian reaction-diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 203(C).
    15. Johannes Muhle-Karbe & Youssef Ouazzani Chahdi & Mathieu Rosenbaum & Gr'egoire Szymanski, 2026. "A unified theory of order flow, market impact, and volatility," Papers 2601.23172, arXiv.org, revised Feb 2026.
    16. Edgardo Alvarez-Pardo & Carlos Lizama, 2012. "The Maximal Subspace for Generation of (a, k)‐Regularized Families," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    17. José Francisco Gómez Aguilar & Margarita Miranda Hernández, 2014. "Space‐Time Fractional Diffusion‐Advection Equation with Caputo Derivative," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    18. Neha Gupta & Arun Kumar, 2023. "Fractional Poisson Processes of Order k and Beyond," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2165-2191, December.
    19. Bakalis, Evangelos & Zerbetto, Francesco, 2025. "Barrier-crossing driven by fractional Gaussian noise in the context of reactive flux formalism: An exact result," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 661(C).
    20. B. B. Jaimini & Meenu & Suman Sharma & S. D. Purohit & D. L. Suthar, 2025. "A General Class of Multivariable Mittag–Leffler Function and Its Associated Applications," Abstract and Applied Analysis, John Wiley & Sons, vol. 2025(1).
    21. Bakalis, Evangelos & Zerbetto, Francesco, 2026. "Subdiffusion supports barrier crossing: Mittag-Leffler noise autocorrelation and reactive flux formalism as evidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 685(C).
    22. J. A. Tenreiro Machado, 2013. "Fractional Dynamics of Genetic Algorithms Using Hexagonal Space Tessellation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    23. Fen Qin & Fuyi Xu & Huizeng Qin, 2022. "Some Notes of the Fractional Integral and Derivatives and Its Applications," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).

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