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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 2β u/t 2β =D 2 u/x 2 , where 0<β1, D>0, x and t are the space-time variables and u(x,t;β) 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;β) of Wright type, where z=|x|/t β is the similarity variable.
MSC:
35G10Initial value problems for linear higher-order PDE
26A33Fractional derivatives and integrals (real functions)
35A08Fundamental solutions of PDE