A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grünwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

We study the reaction front for the process $A+\overrightarrow{B}C$ in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

We consider the coagulation dynamics $A+A\to A$ and $A+A\rightleftharpoons A$ and the annihilation dynamics $A+A\to 0$ for particles moving subdiffusively in one dimension. This scenario combines the “anomalous kinetics” and “anomalous diffusion” problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. Our analysis is based on the fractional diffusion equation.

The radial distribution function g(r) provided by the solution of the Percus-Yevick (PY) equation for hard spheres is rederived in terms of the simplest Padé approximant of a function defined in the Laplace space that is consistent with the following physical requirements: g(r) is continuous for r>1, the isothermal compressibility is finite, and the zeroth- and first-order coefficients in the density expansion of g(r) must be exact. An explicit expression for the solution of the generalized mean-spherical approximation (GMSA) is obtained as a simple extension involving two new parameters, which are determined by imposing two conditions: (i) the virial and the compressibility routes to the equation of state agree consistently, and (ii) this equation of state coincides with that of Carnahan and Starling [J. Chem. Phys. 51, 635 (1969)]. The second- and third-order coefficients in the density expansion of g(r) given by the GMSA are compared with the exact ones and with those given by the PY equation.