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Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order  α  ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

Subordination in a class of generalized time-fractional diffusion-wave equations