The M Wright function in time-fractional diffusion processes: a tutorial survey,

none3In 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.International Journal of Differential Equations, Vol. 2010, Article ID 104505, 28 pages. doi:10.1155/2010/104505 . http://www.hindawi.com/a104505.html Electronic Journal published by Hindawi Publishing Corporation, special issue devoted to Fractional Differential Equations, see http://www.hindawi.com/journals/ijde/contents.html under invitation of the Guest Editors: Fawang Liu, Mark M. Meerschaert, Shaher Momani, Nikolai N. Leonenko, Wen Chen, and Om P. AgrawalmixedF. MAINARDI; A. MURA; G. PAGNINIF. MAINARDI; A. MURA; G. PAGNIN