Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain

Abstract

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length.We obtain explicit expressions for this fractional Laplacianmatrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional derivative) on the finite periodic string.In this approach we introduce two material parameters, the particle mass $\mu$ anda frequency $\Omega \_\alpha$. The requirement of finiteness of the the total mass and total elastic energy in the continuum limit (lattice constant $h\to 0$) leads to scaling relations for the two parameters, namely$\mu \sim h$ and $\Omega \_{\alpha }^2\sim h^{-\alpha }$.The present approach can be generalized to define lattice fractional calculus on periodic lattices in full analogy to the usual `continuous' fractional calculus.

arXiv admin note: text overlap with arXiv:1412.5904

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• Random walks with long-range steps generated by functions of Laplacian matrices
• A fractional generalization of the classical lattice dynamics approach
• Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels