Vector Calculus in Non-Integer Dimensional Space and its Applications to Fractal Media

TL;DR

This paper develops a generalized vector calculus for non-integer dimensional spaces to model fractal media, enabling analysis of elasticity, heat distribution, and electric fields in fractal structures.

Contribution

It introduces definitions of differential operators in non-integer dimensions and applies them to physical problems involving fractal materials.

Findings

Defined gradient, divergence, and Laplacian in non-integer dimensions

Solved elasticity problems for fractal hollow balls and pipes

Analyzed heat and electric field distributions in fractal media

Abstract

We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and vector fields that are independent of angles. We formulate a generalization of vector calculus for rotationally covariant scalar and vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.