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 alpha (0 < alpha less than or equal to 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 < alpha < 1 and in terms of the generalized Wright function in the case 1 < or < 2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo-type modification of the Erdelyi-Kober fractional differential operator. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 26A33; 33B20; 45J05; 45K05.