Wright functions as scale-invariant solutions of the diffusion-wave equation

The time-fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0<α⩽2). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional diffusion-wave equation are given in terms of the Wright function in the case 0<α<1 and in terms of the generalized Wright function in the case 1<α<2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo-type modification of the Erdélyi–Kober fractional differential operator.