## International Journal of Differential Equations

### The $M$-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

#### Abstract

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.

#### Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 104505, 29 pages.

Dates
Received: 13 September 2009
Accepted: 8 November 2009
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399865

Digital Object Identifier
doi:10.1155/2010/104505

Zentralblatt MATH identifier
1222.60060

#### Citation

Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni. The $M$ -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 104505, 29 pages. doi:10.1155/2010/104505. https://projecteuclid.org/euclid.ijde/1485399865

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