The Wright functions as solutions of the time-fractional diffusion equation

We revisit the Cauchy problem for the time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order @[email protected]?(0,2]. By using the Fourier-Laplace transforms the fundamentals solutions (Green functions) are shown to be high transcendental functions of the Wright-type that can be interpreted as spatial probability density functions evolving in time with similarity properties. We provide a general representation of these functions in terms of Mellin-Barnes integrals useful for numerical computation.

[1]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[2]  橋本 英典,et al.  A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi ; Higher Transcendental Functions, Vols. I, II, III. McGraw-Hill, New York-Toronto-London, 1953, 1953, 1955. xxvi+302, xvii+396, xvii+292頁. 16×23.5cm. $6.50, $7.50, $6.50. , 1955 .

[3]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

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

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

[6]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .

[7]  Arak M. Mathai,et al.  The H-Function with Applications in Statistics and Other Disciplines. , 1981 .

[8]  O. I. Marichev,et al.  Handbook of Integral Transforms of Higher Transcendental Functions , 1983 .

[9]  N. Leonenko,et al.  Spectral Analysis of Fractional Kinetic Equations with Random Data , 2001 .

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

[11]  H. Srivastava,et al.  Special Functions in Queuing Theory and Related Stochastic Processes , 1982 .

[12]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

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

[14]  Salvatore Pincherle: the pioneer of the Mellin-Barnes integrals , 2003, math/0702520.

[15]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[16]  R. P. Boas,et al.  Higher Transcendental Functions, vols. I and II.Based, in part, on notes left by Harry Bateman.Bateman Project Staff, A. Erdélyi, Ed. McGraw-Hill, New York-London, 1953. vol. I, xxvi + 302 pp.,$6.50; vol. II, xviii + 396 pp., $7.50 , 1954 .

[17]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[18]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[19]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[20]  R. Askey,et al.  HANDBOOK OF INTEGRAL TRANSFORMS OF HIGHER TRANSCENDENTAL FUNCTIONS: Theory and Algorithmic Tables (Ellis Horwood Series: Mathematics and Its Applications) , 1983 .

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

[22]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .