Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

Abstract

This paper is devoted to an in-depth investigation of the first fundamental solution to the linear multi-dimensional space-time-fractional diffusion-wave equation. This equation is obtained from the diffusion equation by replacing the first order time-derivative by the Caputo fractional derivative of order $\beta$, $0 <\beta \leq 2$, and the Laplace operator by the fractional Laplacian $(-\Delta )^{\alpha /2}$ with $0<\alpha \leq 2$. First, a representation of the fundamental solution in form of a Mellin–Barnes integral is deduced by employing the technique of the Mellin integral transform. This representation is then used for establishing several subordination formulas that connect the fundamental solutions for different values of the fractional derivatives $\alpha$ and $\beta$. We also discuss some new cases of completely monotone functions and probability density functions that are expressed in terms of the Mittag-Leffler function, the Wright function, and the generalized Wright function.