Mellin integral transform approach to analyze the multidimensional diffusion-wave equations

In this paper, a family of the multidimensional time- and space-fractional diffusion-wave equations with the Caputo time-fractional derivative of the order β, 0 < β ⩽ 2 and the fractional Laplacian (−Δ)α2 with 1 < α ⩽ 2 is considered. A representation of the first fundamental solution to this equation is deduced in form of a Mellin-Barnes integral by employing the technique of the Mellin integral transform. The Mellin-Barnes representation is used to derive some new identities for the fundamental solutions in different dimensions and to identify already known and some new particular cases of the fundamental solution that have especially simple closed form.