Abstract

The timefractional diffusionwave equation is revisited, where the time derivative is of order��2 �� and 0 < ����� 1. The behaviour of the equation is ���diffusionlike��� (respectively, ���wavelike���) when 0<�����12 (respectively, 12<�����1). Two types of timefractional derivatives are considered, namely the Caputo and RiemannLiouville derivatives. Initial value problems and initialboundary value problems are studied and handled in a unified way using an embedding method. A twoparameter auxiliary function is introduced and its properties are investigated. The timefractional diffusion equation is used to generate a new family of probability distributions, and that includes the normal distribution as a particular case.