The Mittag-Leffler function and its application to the ultra-hyperbolic time-fractional diffusion-wave equation

In this paper we study an n-dimensional generalization of time-fractional diffusion-wave equation, where the Laplacian operator is replaced by the ultra-hyperbolic operator and the time-fractional derivative is taken in the Hilfer sense. The analytical solution is obtained in terms of the Fox's H-function, for which the inverse Fourier transform of a Mittag-Leffler-type function that contains in its argument a positive-definite quadratic form is calculated.