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

Abstract

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