标题： 非整数维空间中的矢量微积分各向异性分形介质 显示英文标题

标题： Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space

摘要： 对描述各向异性分形介质的不同方法进行了综述。 在本文中，非整数维和多分数空间的微分和积分被考虑作为描述各向异性分形材料和介质的工具。 我们建议通过使用乘积测度方法，对非整数维空间进行矢量微积分的推广。 分数和非整数维空间的乘积使我们能够在连续模型框架内考虑分形介质的各向异性。 考虑了在非整数维空间上的积分。 在本文中，提出了分数空间和非整数维空间的一阶和二阶微分算子。 微分算子被定义为在非整数维空间中积分的逆运算。 作为不同维度空间乘积的非整数维空间，使我们能够为各向异性类型的介质提供连续模型。 作为应用所提出的各向异性分形材料和介质的矢量微积分推广的例子，考虑了分形介质的泊松方程、欧拉-伯努利分形梁以及分形材料的蒂莫申科梁方程。 显示英文摘要

摘要： 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.