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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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