Fractional diffusion and wave equations

Abstract

Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α 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.