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The Wright functions as solutions of the time-fractional diffusion equation. (English) Zbl 1053.35008
The authors consider the Cauchy problem for the time-fractional diffusion equation obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta\in (0,2)$. They use the Fourier-Laplace transforms to show that the fundamental solutions (Green functions) are higher transcendental functions of the Wright-type and can be interpreted as spatial probability density functions evolving in time with similarity properties. They also provide a general presentation of these functions in terms of Mellin-Barnes integrals useful for numerical computation.

##### MSC:
 35A22 Transform methods (PDE) 26A33 Fractional derivatives and integrals (real functions) 35S10 Initial value problems for pseudodifferential operators
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##### References:
 [1] Anh, V. V.; Leonenko, N. N.: Spectral analysis of fractional kinetic equations with random data. Journal of statistical physics 104, No. 5/6, 1349-1387 (2001) · Zbl 1034.82044 [2] Braaksma, B. L. J.: Asymptotic expansions and analytical continuations for a class of Barnes-integrals. Compositio Mathematica 15, No. 3, 239-341 (1962) · Zbl 0129.28604 [3] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: Higher transcendental functions. Bateman project 1--3 (1953--1955) · Zbl 0051.30303 [4] Gorenflo, R.; Iskenderov, A.; Luchko, Yu.: Mapping between solutions of fractional diffusion-wave equations. Fractional calculus and applied analysis 3, No. 1, 75-86 (2000) · Zbl 1033.35161 [5] Gorenflo, R.; Luchko, Yu.; Mainardi, F.: Analytical properties and applications of the wright function. Fractional calculus and applied analysis 2, No. 4, 383-414 (1999) · Zbl 1027.33006 [6] Gorenflo, R.; Luchko, Yu.; Mainardi, F.: Wright functions as scale-invariant solutions of the diffusion-wave equation. Journal of computational and applied mathematics 118, 175-191 (2000) · Zbl 0973.35012 [7] R. Gorenflo, Yu. Luchko, S. Rogozin, Mittag--Leffler type functions: notes on growth properties and distribution of zeros, Pre-print A-04/97, Fachbereich Mathematik und Informatik, Freie Universität, Berlin, 1997. Available from [8] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. Fractals and fractional calculus in continuum mechanics, 223-276 (1997) [9] Gorenflo, R.; Mainardi, F.; Srivastava, H. M.: Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena. Proceedings VIII international colloquium on differential equations, plovdiv 1997, 195-202 (1998) · Zbl 0921.33009 [10] Kiryakova, V.: Generalized fractional calculus and applications. Pitman research notes in mathematics 301 (1994) [11] Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Applied mathematics letters 9, No. 6, 23-28 (1996) · Zbl 0879.35036 [12] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, solitons and fractals 7, 1461-1477 (1996) · Zbl 1080.26505 [13] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004 [14] Mainardi, F.; Gorenflo, R.: On Mittag--Leffler-type functions in fractional evolution processes. Journal of computational and applied mathematics 118, 283-299 (2000) · Zbl 0970.45005 [15] Mainardi, F.; Luchko, Yu.; Pagnini, G.: The fundamental solution of the space--time fractional diffusion equation. Fractional calculus and applied analysis 4, No. 2, 153-192 (2001) · Zbl 1054.35156 [16] F. Mainardi, G. Pagnini, Salvatore Pincherle: the pioneer of the Mellin--Barnes integrals, J. Comput. Appl. Math., in press. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications, Roma-Ostia, Italy, June 18--22, 2001 [17] Marichev, O. I.: Handbook of integral transforms of higher transcendental functions, theory and algorithmic tables. (1983) · Zbl 0494.33001 [18] Mathai, A. M.; Saxena, R. K.: The H-function with applications in statistics and other disciplines. (1978) · Zbl 0382.33001 [19] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports 339, 1-77 (2000) · Zbl 0984.82032 [20] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008 [21] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003 [22] Srivastava, H. M.; Gupta, K. C.; Goyal, S. P.: The H-functions of one and two variables with applications. (1982) · Zbl 0506.33007 [23] Srivastava, H. M.; Kashyap, B. R. K.: Special functions in queuing theory and related stochastic processes. (1982) · Zbl 0492.60089