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The fundamental solutions for the fractional diffusion-wave equation. (English) Zbl 0879.35036
The author provides the fundamental solutions of the Cauchy and signalling problems for the fractional evolution equation $\partial^{2\beta}u/\partial t^{2\beta}=D\partial^2u/\partial x^2$, where $0<\beta\leq1$, $D>0$, $x$ and $t$ are the space-time variables and $u(x,t;\beta)$ is the field variable, which is assumed to be causal, that is, vanishing for $t<0$. These solutions are expressed by the corresponding Green’s functions and are analyzed by the Laplace transform and are expressed in terms of an entire auxiliary function $M(z;\beta)$ of Wright type, where $z=|x|/t^\beta$ is the similarity variable.

35G10Initial value problems for linear higher-order PDE
26A33Fractional derivatives and integrals (real functions)
35A08Fundamental solutions of PDE
Full Text: DOI
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