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

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Cited by in corpus (18)

• Spatiotemporal accessible solitons in fractional dimensions
• Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
• Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
• 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
• Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero
• 1/r potential in higher dimensions
• Large Lattice Fractional Fokker-Planck Equation
• Saturn rings: fractal structure and random field model
• Newtonian Fractional-Dimension Gravity and MOND
• Newtonian Fractional-Dimension Gravity and Disk Galaxies
• Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain
• Fractional Diffusion Equations for Lattice and Continuum: Grunwald-Letnikov Differences and Derivatives Approach
• Newtonian Fractional-Dimension Gravity and the External Field Effect
• General Fractional Vector Calculus
• Relativistic Fractional-Dimension Gravity
• Spatial integrals in non-standard dimensions via Gaussian measure and analytic continuation