Lattice fractional diffusion equation in terms of a Riesz–Caputo difference

• A Riesz–Caputo fractional difference is proposed within the discrete fractional calculus.
• A lattice diffusion equation is defined on discrete finite domains.
• The fractional order is varied to numerically depict the diffusion behaviors of long interactions.

A fractional difference is defined by the use of the right and the left Caputo fractional differences. The definition is a two-sided operator of Riesz type and introduces back and forward memory effects in space difference. Then, a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains. A lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders. The result shows that the Riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media.