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.


35A22 Transform methods (e.g., integral transforms) applied to PDEs
26A33 Fractional derivatives and integrals
35S10 Initial value problems for PDEs with pseudodifferential operators
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