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

25 pages, LaTeX

References in corpus (3)

• Fractional Hydrodynamic Equations for Fractal Media
• Possible Experimental Test of Continuous Medium Model for Fractal Media
• Quantum mechanics in fractional and other anomalous spacetimes

Cited by in corpus (8)

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• Flow of Fractal Fluid in Pipes: Non-Integer Dimensional Space Approach
• Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space
• Lattice Model with Nearest-Neighbor and Next-Nearest-Neighbor Interactions for Gradient Elasticity
• Large Lattice Fractional Fokker-Planck Equation
• Fractional Diffusion Equations for Lattice and Continuum: Grunwald-Letnikov Differences and Derivatives Approach