Fundamental solution of the tempered fractional diffusion equation

Abstract

In this paper, we consider the space-time fractional diffusion equation Dtβu(x,t)+K(−∞Dxα,λ)u(x,t)=0,x∈R,t>0, with the tempered Riemann-Liouville derivative of order 0 < α ≤ 1 in space and the Caputo derivative of order 0 < β ≤ 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 ∈ ℕ0. 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 Mα(x) of order 0 < α < 1 and arguments x∈R0+.