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The time-fractional diffusion-wave equation is revisited, where the timederivative is of order $2 \nu$ and $0 < \nu \le 1$. The behaviour of theequation is "diffusion-like" (respectively, "wave-like") when $0 < \nu \le\frac{1}{2}$ (respectively, $\frac{1}{2} < \nu \le 1$). Two types oftime-fractional derivatives are considered, namely the Caputo andRiemann-Liouville derivatives. Initial value problems and initial-boundaryvalue problems are investigated and handled in a unified way using an embeddingmethod. A two-parameter auxiliary function is introduced and its properties areinvestigated. The time-fractional diffusion equation is used to generate a newfamily of probability distributions, and that includes the normal distributionas a particular case.

A unified way to solve IVPs and IBVPs for the time-fractional diffusion-wave equation