Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity

Abstract

Models of three-dimensional lattices with long-range interactions of Grünwald-Letnikov type for fractional gradient elasticity of non-local continuum are suggested. The lattice long-range interactions are described by fractional-order difference operators. Continuous limit of suggested three-dimensional lattice equations gives continuum differential equations with the Grünwald-Letnikov derivatives of non-integer orders. The proposed lattice models give a new microstructural basis for elasticity of materials with power-law type of non-locality. Moreover these lattice models allow us to have a unified microscopic description for fractional and usual (non-fractional) gradient elasticity continuum.

Variational principle of stationary action for fractional nonlocal media and fields

“…( [31][32][33][34]40], since they directly connected with lattice fractional derivatives that are recently proposed [37,39].…”

“…As a result, proposed fractional variational principle allows us to get the Euler-Lagrange equations that are directly connected with microstructural lattice models of fractional nonlocal media [31][32][33][34]40], and the lattice field theories [35,38].…”

“…It should be emphasized that the main advantage of the suggested variational approach is that the proposed Riesz-type fractional derivatives naturally arise in the fractional continuum mechanics based on the lattice models [31][32][33][34]40], and are directly connected with lattice fractional derivatives suggested in [37,39].…”

Modified Kawahara equation within a fractional derivative with non-singular kernel

“…Fractional derivatives and integrals have been gaining more and more interest of scientists due to their extensive applications in different directions of science, social science, engineering and finance [1][2][3][4][5][6][7][8][9] when the relaxation process have to accounted for. In this context, Atangana [10] analyzed the fractional non-linear Fisher's reaction-diffusion equation associated with Caputo-Fabrizio (CF) fractional derivative.…”

Analysis of a New Fractional Model for Damped Bergers’ Equation

“…Tarasov [9] investigated the 3 D lattice equations pertaining to long-range interactions of Grünwald-Letnikov kind for fractional extension of gradient elasticity. Choudhary et al [10] studied the fractional order differential equations occurring in fluid dynamics.…”