%0 Journal Article
%T Fractional Diffusion Equations for Lattice and Continuum: Gr¨¹nwald-Letnikov Differences and Derivatives Approach
%A Vasily E. Tarasov
%J International Journal of Statistical Mechanics
%D 2014
%R 10.1155/2014/873529
%X Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Gr¨¹nwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Gr¨¹nwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Gr¨¹nwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media. 1. Introduction Fractional calculus and differential equation of noninteger orders [1¨C5] have a long history that is connected with the names of famous scientists such as Liouville, Riemann, Gr¨¹nwald, Letnikov, Marchaud, and Riesz. Derivatives and integrals of noninteger orders have a lot of applications in different areas of physics [6¨C10]. Fractional calculus is a powerful tool to describe processes in continuously distributed media with nonlocal properties. As it was shown in [11, 12], the continuum equations with fractional derivatives are directly connected [7] to lattice models with long-range interactions. The lattice equations for fractional nonlocal media and the correspondent continuum equations have been considered recently in [13¨C15]. Fractional-order differences and the correspondent derivatives have been first proposed by Gr¨¹nwald [16] and by Letnikov [17]. At the present time these generalized differences and derivatives are called the Gr¨¹nwald-Letnikov fractional differences and derivatives [1¨C3, 18]. One-dimensional lattice models with long-range interactions of the Gr¨¹nwald-Letnikov type and the correspondent fractional differential and integral continuum equations have been suggested in [19]. The suggested form of long-range interaction is based on the form of the left-sided and right-sided Gr¨¹nwald-Letnikov fractional differences. A possible form of lattice vectors calculus based on the fractional-order differences of the Gr¨¹nwald-Letnikov type has been suggested in [20]. In this paper, we apply this approach to describe diffusion on lattices with long-range jumps and to derive fractional diffusion equations for nonlocal continuum with power-law nonlocality. The diffusion equations describe the change of probability of a random function in space and time in transport processes, and they usually
%U http://www.hindawi.com/journals/ijsm/2014/873529/