The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey
In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as 𝑀-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, t...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/104505 |
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| author | Francesco Mainardi Antonio Mura Gianni Pagnini |
| author_facet | Francesco Mainardi Antonio Mura Gianni Pagnini |
| author_sort | Francesco Mainardi |
| collection | DOAJ |
| description | In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as 𝑀-Wright function, entering as a probability density in a
relevant class of self-similar stochastic processes that we generally refer to as time-fractional
diffusion processes. Indeed, the master equations governing these processes
generalize the standard diffusion equation by means of time-integral operators interpreted
as derivatives of fractional order. When these generalized diffusion processes are properly
characterized with stationary increments, the 𝑀-Wright function is shown to play the
same key role as the Gaussian density in the standard and fractional Brownian motions.
Furthermore, these processes provide stochastic models suitable for describing phenomena
of anomalous diffusion of both slow and fast types. |
| format | Article |
| id | doaj-art-f0bf58c35f9a4881a94ea140e80f5253 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-f0bf58c35f9a4881a94ea140e80f52532025-02-03T05:48:23ZengWileyInternational Journal of Differential Equations1687-96431687-96512010-01-01201010.1155/2010/104505104505The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial SurveyFrancesco Mainardi0Antonio Mura1Gianni Pagnini2Department of Physics, University of Bologna and INFN, Via Irnerio 46, 40126 Bologna, ItalyCRESME Ricerche S.p.A, Viale Gorizia 25C, 00199 Roma, ItalyCRS4, Centro Ricerche Studi Superiori e Sviluppo in Sardegna, Polaris Building 1, 09010 Pula, ItalyIn the present review we survey the properties of a transcendental function of the Wright type, nowadays known as 𝑀-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the 𝑀-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.http://dx.doi.org/10.1155/2010/104505 |
| spellingShingle | Francesco Mainardi Antonio Mura Gianni Pagnini The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey International Journal of Differential Equations |
| title | The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey |
| title_full | The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey |
| title_fullStr | The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey |
| title_full_unstemmed | The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey |
| title_short | The 𝑀-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey |
| title_sort | 𝑀 wright function in time fractional diffusion processes a tutorial survey |
| url | http://dx.doi.org/10.1155/2010/104505 |
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