Fractal calculus and its geometrical explanation
Fractal calculus and its geometrical explanation
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DOI：
10.1016/j.rinp.2018.06.011
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发表时间：
2018-09-01
影响因子：
5.3
通讯作者：
He, Ji-Huan
中科院分区：
文献类型：
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作者：
He, Ji-Huan
分形微积分是一种简单而有效的方法，它可以用来处理分层或多孔介质中的现象。它的运算与高等微积分几乎相同，使非数学家也能很容易地理解它。本文从温度分形梯度的基本概念出发，分形介质中两点之间的温度变化，揭示分形微积分的基本性质。引入分形速度和分形材料导数，推导出分形空间中的流体力学和热传导规律。从几何学上解释了分形空间中的质量守恒，并以纳米纤维膜为例说明了较小尺度上的分形空间到较大尺度上的连续空间的近似变换，纳米纤维膜在任何可观察到的尺度上都是光滑的，但其透气性必须在纳米尺度上进行研究，在这样的小尺度下，纳米纤维膜变成了多孔膜。最后举例说明了用高等微积分无法揭示的蚕茧的耐热性。
Fractal calculus is very simple but extremely effective to deal with phenomena in hierarchical or porous media. Its operation is almost same with that by the advanced calculus, making it much accessible to all non-mathematicians. This paper begins with the basic concept of fractal gradient of temperature, i.e., the temperature change between two points in a fractal medium, to reveal the basic properties of fractal calculus. The fractal velocity and fractal material derivative are then introduced to deduce laws for fluid mechanics and heat conduction in fractal space. Conservation of mass in a fractal space is geometrically explained, and an approximate transform of a fractal space on a smaller scale into its continuous partner on a larger scale is illustrated by a nanofiber membrane, which is smooth on any observable scales, but its air permeability has to studied in a nano scale, under such a small scale, the nanofiber membrane becomes a porous one. Finally an example is given to explain cocoon's heat-proof property, which cannot be unveiled by advanced calculus.