# Anisotropic fractal media by vector calculus in non-integer dimensional space

## Abstract

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer 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 Euler-Bernoulli 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., E-mail: tarasov@theory.sinp.msu.ru.
```*Anisotropic fractal media by vector calculus in non-integer dimensional space*. United States: N. p., 2014.
Web. doi:10.1063/1.4892155.

```
Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru.
```*Anisotropic fractal media by vector calculus in non-integer dimensional space*. United States. doi:10.1063/1.4892155.

```
Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru. Fri .
"Anisotropic fractal media by vector calculus in non-integer dimensional space". United States.
doi:10.1063/1.4892155.
```

```
@article{osti_22306099,
```

title = {Anisotropic fractal media by vector calculus in non-integer dimensional space},

author = {Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru},

abstractNote = {A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer 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 Euler-Bernoulli 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}

}