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Wright functions as scale-invariant solutions of the diffusion-wave equation. (English) Zbl 0973.35012

The authors obtain the time-fractional diffusion-wave equation from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0<α2).

They show by using the similarity method and the method of the Laplace transform that the scale-invariant solutions of the mixed problem of signaling type for 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<z.

The authors give the reduced equation for the scale-invariant solutions in terms of the Caputo-type modification of the Erdélyi-Kober fractional differential operator.

35A25Other special methods (PDE)
26A33Fractional derivatives and integrals (real functions)
33E20Functions defined by series and integrals
45J05Integro-ordinary differential equations
45K05Integro-partial differential equations