Fractal calculus and its geometrical explanation

• Published in: Results in physics Vol. 10; pp. 272 - 276
• Main Author: He, Ji-Huan
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-09-2018; Elsevier
• Subjects: Fractal derivative; Fractal temperature gradient; Fractional derivative; Fractional differential equation; Hausdorff derivative; Hierarchical structure; Nanofiber membrane; Porous medium; Thermal resistance; Fractional differential equation; Nanofiber membrane; Fractal derivative; Thermal resistance; Fractal temperature gradient; Porous medium; Hausdorff derivative; Hierarchical structure; Fractional derivative
• ISSN: 2211-3797; 2211-3797
• DOI: 10.1016/j.rinp.2018.06.011

•Definition of fractional derivative is such a mess that a new replacement is needed.•Fractal calculus is tutorially introduced from very beginning, and it is accessible to all audience.•Its geometrical explanation and applications are elucidated.
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.