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

Authors:
Francesco Mainardi;Gianni Pagnini
Affiliations:
Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy;Istituto per le Scienze dell'Atmosfera e del Clima del CNR, Via Gobetti 101, I-40129 Bologna, Italy
Venue:
Applied Mathematics and Computation - Special issue: Advanced special functions and related topics in differential equations, third Melfi workshop, proceedings of the Melfi school on advanced topics in mathematics and physics
Year:
2003

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Abstract

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 β ∈ (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.