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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 $\alpha$ $\left(0<\alpha \le 2\right)$.

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 $0<\alpha <1$ and in terms of the generalized Wright function in the case $1<\alpha .

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.

##### MSC:
 35A25 Other special methods (PDE) 26A33 Fractional derivatives and integrals (real functions) 33E20 Functions defined by series and integrals 45J05 Integro-ordinary differential equations 45K05 Integro-partial differential equations