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Fractional diffusion and wave equations. (English) Zbl 0692.45004
Summary: Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with ${t}^{\alpha -1}/{\Gamma }\left(\alpha \right)$, $\alpha =1,2$, respectively. Fractional diffusion and wave equations are obtained by letting $\alpha$ vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.

##### MSC:
 45K05 Integro-partial differential equations 35K30 Higher order parabolic equations, initial value problems