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Zbl 0692.45004
Schneider, W.R.; Wyss, W.
Fractional diffusion and wave equations. (English)
[J] J. Math. Phys. 30, No.1, 134-144 (1989). ISSN 0022-2488; ISSN 1089-7658/e

Summary: Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with $t\sp{\alpha -1}/\Gamma (\alpha)$, $\alpha =1,2$, 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 2000:: *45K05 Integro-partial differential equations
35K30 Higher order parabolic equations, initial value problems

Keywords: wave equations; convolution; Fractional diffusion; Green's functions; Fox functions

Cited in: Zbl 1137.37322 Zbl 1086.60049 Zbl 0979.65121 Zbl 0970.35174 Zbl 0799.35232

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