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Zbl 1053.35008
Mainardi, Francesco; Pagnini, Gianni
The Wright functions as solutions of the time-fractional diffusion equation. (English)
[J] Appl. Math. Comput. 141, No. 1, 51-62 (2003). ISSN 0096-3003

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.
[Ismail Taqi Ali (Safat)]

MSC 2000:: *35A22 Transform methods (PDE)
26A33 Fractional derivatives and integrals (real functions)
35S10 Initial value problems for pseudodifferential operators

Keywords: Laplace transforms; Fourier transforms; Mellin-Barnes integrals; Mittag-Leffler functions; Wright functions; Fox $H$-functions

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Zentralblatt MATH Berlin [Germany]