  This paper deals with the investigation of the computational solutions of an
unified fractional reaction-diffusion equation, which is obtained from the
standard diffusion equation by replacing the time derivative of first order by
the generalized Riemann-Liouville fractional derivative defined in Hilfer et
al. , and the space derivative of second order by the Riesz-Feller fractional
derivative, and adding a function $\phi(x,t)$. The solution is derived by the
application of the Laplace and Fourier transforms in a compact and closed form
in terms of Mittag-Leffler functions. The main result obtained in this paper
provides an elegant extension of the fundamental solution for the space-time
fractional diffusion equation obtained earlier by Mainardi et al., and the
result very recently given by Tomovski et al.. At the end, extensions of the
derived results, associated with a finite number of Riesz-Feller space
fractional derivatives, are also investigated.
