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The fundamental solution (Green function) for the Cauchy problem of thespace-time fractional diffusion equation is investigated with respect to itsscaling and similarity properties, starting from its Fourier-Laplacerepresentation. Then, by using the Mellin transform, a general representationof the Green function in terms of Mellin-Barnes integrals in the complex planeis derived. This allows us to obtain a suitable computational form of the Greenfunction in the space-time domain and to analyse its probabilityinterpretation.

Applications of integral transforms in fractional diffusion processes