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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 \(\alpha\) \((0<\alpha\leq 2)\).
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 \(1<\alpha< z\).
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.

35A25 Other special methods applied to PDEs
26A33 Fractional derivatives and integrals
33E20 Other functions defined by series and integrals
45J05 Integro-ordinary differential equations
45K05 Integro-partial differential equations
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