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Zbl 0879.35036
Mainardi, F.
The fundamental solutions for the fractional diffusion-wave equation.
(English)
[J] Appl. Math. Lett. 9, No.6, 23-28 (1996). ISSN 0893-9659

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.
[R.Vaillancourt (Ottawa)]
MSC 2000:
*35G10 Initial value problems for linear higher-order PDE
26A33 Fractional derivatives and integrals (real functions)
35A08 Fundamental solutions of PDE

Keywords: fractional derivatives; diffusion equation; wave equation; Green's function; Wright function; signalling problems; Laplace transform

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