Michelitsch, T.M., Collet, B., Nowakowski, A.F. orcid.org/0000-0002-5018-2661 et al. (1 more author) (2016) Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain. Chaos, Solitons and Fractals, 82. pp. 38-47. ISSN 0960-0779
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 Laplacian matrix 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 μ and a frequency Ωα. The requirement of finiteness of the the total mass and total elastic energy in the continuum limit (lattice constant h → 0) leads to scaling relations for the two parameters, namely μ ∼ h and View the MathML source. The present approach can be generalized to define lattice fractional calculus on periodic lattices in full analogy to the usual ‘continuous’ fractional calculus.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 Elsevier B.V. This is an author produced version of a paper subsequently published in Chaos, Solitons and Fractals. Uploaded in accordance with the publisher's self-archiving policy. Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) |
Keywords: | Lattice fractional Laplacian; Fractional Laplacian matrix; Riesz fractional derivative; Discrete fractional calculus; Periodic fractional Laplacian; Power-law matrix functions |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Mechanical Engineering (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 20 Apr 2016 10:00 |
Last Modified: | 28 Nov 2017 01:38 |
Published Version: | http://dx.doi.org/10.1016/j.chaos.2015.10.035 |
Status: | Published |
Publisher: | Elsevier |
Refereed: | Yes |
Identification Number: | 10.1016/j.chaos.2015.10.035 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:97732 |