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.
Applications of local fractional calculus to engineering in fractal time-space: Local fractional differential equations with local fractional derivative
Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media
ETH Zürich（苏黎世联邦理工学院）；Hong Kong Institute for Advanced Study, City University of Hong Kong（香港城市大学高等研究院）；University of California, Riverside（加州大学河滨分校）；City University of Hong Kong（香港城市大学）
