Fractional Liouville equation on lattice phase-space

Abstract

In this paper we propose a lattice analog of phase-space fractional Liouville equation. The Liouville equation for phase-space lattice with long-range jumps of power-law types is suggested. We prove that the continuum limit transforms this lattice equation into Liouville equation with conjugate Riesz fractional derivatives of non-integer orders with respect to coordinates of continuum phase-space. An application of the fractional Liouville equation with these Riesz fractional derivatives to describe properties of plasma-like nonlocal media is considered.

Well posedness for semidiscrete fractional Cauchy problems with finite delay

“…Finally, we remark that the study done in this paper was motivated by recent studies on semidiscrete mathematical models that prove that they can serve as a new microstructural basis for fractional nonlocal continuum mechanics and physics [33], [34]. Fractional order semidiscrete equations can be also used to formulate adequate models in nanomechanics [34], [36] and therefore further studies in this class of semidiscrete equations deserve to be investigated.…”

“…mathematical models that represent evolutions at a nano-level of interest, and which become nowadays more important because of the rapid development of nano-technologies. First studies are due to Tarasov [33,34] in the field of nano-mechanics and physics, and to Wu, Baleanu et.al. [36], [37] in the study of the chaotic behavior of discrete fractional models.…”

“…Fractional order semidiscrete equations can be also used to formulate adequate models in nanomechanics [34], [36] and therefore further studies in this class of semidiscrete equations deserve to be investigated. Our contribution in this paper provides a new qualitative advance in this line of research, that incorporates tools from operator theory and allows the analysis of well posedness for a very general but still simple model.…”

Maximal regularity in l spaces for discrete time fractional shifted equations

“…Recently, Tarasov [31], [32] began to study fractional models with the Grünwald-Letnikov fractional difference. As suggested by Tarasov, these models can serve as a new microstructural basis for the fractional nonlocal continuum mechanics and physics.…”

“…As suggested by Tarasov, these models can serve as a new microstructural basis for the fractional nonlocal continuum mechanics and physics. Fractional-order semidiscrete equations can also be used to formulate adequate models in nanomechanics [32], [34].…”