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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 $\beta \in \left(0,2\right)$. 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