Anisotropic fractal media by vector calculus in noninteger dimensional space
Abstract
A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration noninteger dimensional and multifractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for noninteger dimensional space by using a product measure method. The product of fractional and noninteger dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over nonintegerdimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and noninteger dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with noninteger dimensions. Noninteger dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the EulerBernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.
 Authors:
 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991 (Russian Federation)
 Publication Date:
 OSTI Identifier:
 22306099
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANISOTROPY; BEAMS; MATHEMATICAL SPACE; POISSON EQUATION; VECTOR FIELDS; VECTORS
Citation Formats
Tarasov, Vasily E., Email: tarasov@theory.sinp.msu.ru. Anisotropic fractal media by vector calculus in noninteger dimensional space. United States: N. p., 2014.
Web. doi:10.1063/1.4892155.
Tarasov, Vasily E., Email: tarasov@theory.sinp.msu.ru. Anisotropic fractal media by vector calculus in noninteger dimensional space. United States. doi:10.1063/1.4892155.
Tarasov, Vasily E., Email: tarasov@theory.sinp.msu.ru. Fri .
"Anisotropic fractal media by vector calculus in noninteger dimensional space". United States.
doi:10.1063/1.4892155.
@article{osti_22306099,
title = {Anisotropic fractal media by vector calculus in noninteger dimensional space},
author = {Tarasov, Vasily E., Email: tarasov@theory.sinp.msu.ru},
abstractNote = {A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration noninteger dimensional and multifractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for noninteger dimensional space by using a product measure method. The product of fractional and noninteger dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over nonintegerdimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and noninteger dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with noninteger dimensions. Noninteger dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the EulerBernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.},
doi = {10.1063/1.4892155},
journal = {Journal of Mathematical Physics},
number = 8,
volume = 55,
place = {United States},
year = {Fri Aug 15 00:00:00 EDT 2014},
month = {Fri Aug 15 00:00:00 EDT 2014}
}

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