  Lattice model with long-range interaction of power-law type that is connected
with difference of non-integer order is suggested. The continuous limit maps
the equations of motion of lattice particles into continuum equations with
fractional Grunwald-Letnikov-Riesz derivatives. The suggested continuum
equations describe fractional generalizations of the gradient and integral
elasticity. The proposed type of long-range interaction allows us to have
united approach to describe of lattice models for the fractional gradient and
fractional integral elasticity. Additional important advantages of this
approach are the following: (1) It is possible to use this model of long-range
interaction in numerical simulations since this type of interactions and the
Grunwald-Letnikov derivatives are defined by generalized finite difference; (2)
The suggested model of long-range interaction leads to an equation containing
the sum of the Grunwald-Letnikov derivatives, which is equal the Riesz's
derivative. This fact allows us to get particular analytical solutions of
fractional elasticity equations.
