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 studied 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.