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Zbl 0973.35012
Gorenflo, Rudolf; Luchko, Yuri; Mainardi, Francesco
Wright functions as scale-invariant solutions of the diffusion-wave equation. (English)
[J] J. Comput. Appl. Math. 118, No.1-2, 175-191 (2000). ISSN 0377-0427

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 $\alpha$ $(0<\alpha\le 2)$.\par 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 $1<\alpha< z$.\par 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.
[Ismail Taqi Ali (Safat)]

MSC 2000:: *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

Keywords: time-fractional diffusion-wave equation; similarity method; Laplace transform; mixed problem; reduced equation for the scale-invariant solutions; Erdélyi-Kober fractional differential operator

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