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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.

##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 26A33 Fractional derivatives and integrals 35A08 Fundamental solutions to PDEs
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##### References:
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