In this paper, we consider the space-time fractional diffusion equation D(t)(beta)u(x, t) + K(D--infinity(x)alpha, lambda)u(x, t) = 0, x is an element of R, t > 0, with the tempered Riemann-Liouville derivative of order 0 < alpha <= 1 in space and the Caputo derivative of order 0 < beta <= 1 in time. The fundamental solution, which turns out to be a spatial probability density function, is given in computable series form as well as in integral representation. The spatial moments of the probability density function are determined explicitly for an arbitrary order n is an element of N-0. Moreover, Green's function of the untempered neutral-fractional diffusion equation is analyzed in view of absolute and relative extreme points. At the end of this article, we point out a remarkably and important integral representation for accurate evaluation of the M-Wright/Mainardi function M-alpha(x) of order 0 < alpha < 1 and arguments x is an element of R-0(+). (C) 2015 AIP Publishing LLC.
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