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The Wright Functions of the Second Kind in Mathematical Physics

Author

Listed:
  • Francesco Mainardi

    (Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, I-40126 Bologna, Italy)

  • Armando Consiglio

    (Institut für Theoretische Physik und Astrophysik, Universität Würzburg, D-97074 Würzburg, Germany)

Abstract

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.

Suggested Citation

  • Francesco Mainardi & Armando Consiglio, 2020. "The Wright Functions of the Second Kind in Mathematical Physics," Mathematics, MDPI, vol. 8(6), pages 1-26, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:884-:d:366089
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    References listed on IDEAS

    as
    1. Boyadjiev, Lyubomir & Luchko, Yuri, 2017. "Mellin integral transform approach to analyze the multidimensional diffusion-wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 127-134.
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    Cited by:

    1. Stephen H. Lihn, 2024. "Generalization of the Alpha-Stable Distribution with the Degree of Freedom," Papers 2405.04693, arXiv.org.
    2. Muhey U. Din & Mohsan Raza & Qin Xin & Sibel Yalçin & Sarfraz Nawaz Malik, 2022. "Close-to-Convexity of q -Bessel–Wright Functions," Mathematics, MDPI, vol. 10(18), pages 1-12, September.
    3. Kreer, Markus, 2022. "An elementary proof for dynamical scaling for certain fractional non-homogeneous Poisson processes," Statistics & Probability Letters, Elsevier, vol. 182(C).
    4. Oraby, T. & Suazo, E. & Arrubla, H., 2023. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    5. Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    6. M. A. Pathan & Maged G. Bin-Saad, 2023. "Mittag-leffler-type function of arbitrary order and their application in the fractional kinetic equation," Partial Differential Equations and Applications, Springer, vol. 4(2), pages 1-25, April.

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