Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald–Letnikov–Riesz type

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 Grünwald–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 Grünwald–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 Grünwald–Letnikov derivatives, which is equal the Riesz’s derivative. This fact allows us to get particular analytical solutions of fractional elasticity equations.