Historical notes on the M-Wright/Mainardi function

Definition and main properties of the M-Wright/Mainardi function, which is the Green function of the time-fractional diffusion equation in the Caputo sense, are briefly reported with its history since the first appearing in conferences and proceedings to the publication in peer-reviewed journals.

[1]  Francesco Mainardi,et al.  A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics , 2007, 0711.0665.

[2]  Radu Balescu,et al.  V-Langevin equations, continuous time random walks and fractional diffusion , 2007, 0704.2517.

[3]  E. Scalas,et al.  On the convergence of quadratic variation for compound fractional Poisson processes , 2012 .

[4]  E. Wright.  THE GENERALIZED BESSEL FUNCTION OF ORDER GREATER THAN ONE , 1940 .

[5]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[6]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[7]  R. Gorenflo,et al.  Analytical properties and applications of the Wright function , 2007, math-ph/0701069.

[8]  Gianni Pagnini,et al.  Short note on the emergence of fractional kinetics , 2014, 1404.0215.

[9]  Francesco Mainardi,et al.  The time fractional diffusion-wave equation , 1995 .

[10]  Gianni Pagnini,et al.  Characterizations and simulations of a class of stochastic processes to model anomalous diffusion , 2008, 0801.4879.

[11]  Francesco Mainardi,et al.  Fractional Diffusive Waves , 2001 .

[12]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[13]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[14]  Gianni Pagnini,et al.  The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes , 2013 .

[15]  Francesco Mainardi,et al.  The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey , 2010, 1004.2950.

[16]  Ralf Metzler,et al.  Anomalous Diffusion and Fractional Transport Equations , 2011 .

[17]  I. Podlubny.  Fractional differential equations , 1998 .

[18]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[19]  R. Gorenflo,et al.  Wright functions as scale-invariant solutions of the diffusion-wave equation , 2000 .

[20]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[21]  Francesco Mainardi,et al.  Parametric subordination in fractional diffusion processes , 2012, 1210.8414.

[22]  Francesco Mainardi,et al.  Renewal processes of Mittag-Leffler and Wright type , 2005 .