The Wright functions as solutions of the time-fractional diffusion equation.
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.
scientific article; zbMATH DE number 1981433