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.

Fractal calculus and its geometrical explanation

The yang-fourier transforms to heat-conduction in a semi-infinite fractal bar

Construction of Si nanowires/ZnFe2O4/Ag photocatalysts with enhanced photocatalytic activity under visible light and magnetic field

Eco-friendly and economical preparation of permanent ferrite magnet with ultrapure magnetite concentrate: Focus on the action mechanism of silicon and aluminum impurities

Thin ferroelectric layers are capable of generating large dielectric and electromechanical responses at relatively low voltages, thus can be utilized in various applications including energy storage, energy harvesting, sensors, and actuators. Among ferroelectrics, relaxor ferroelectrics (1 − x)Pb(Mg1/3Nb2/3)O3 − xPbTiO3 are particularly interesting due to their superior dielectric and piezoelectric properties. However, dielectric and piezoelectric responses of the ferroelectric thin films are significantly suppressed relative to their bulk counterparts. The physical mechanism of the properties degradation still remains elusive, which greatly hinders their technological applications. Herein, this work systematically investigates the dielectric nonlinearity and the relationship between composition and intrinsic/extrinsic contributions of high-quality (1 − x)Pb(Mg1/3Nb2/3)O3 − xPbTiO3 epitaxial thin films (x = 0, 0.1, 0.32) fabricated by pulsed laser deposition. Through the Rayleigh analysis, it is found that the suppression of domain wall motion by substrate clamping is the main cause of thin film performance degradation and is further confirmed by continuous application of background dc bias to extract the dielectric response's intrinsic and extrinsic contributions. This work will pave the way for practical applications of (1 − x)Pb(Mg1/3Nb2/3)O3 − xPbTiO3 thin films in energy storage and electromechanical devices. 

Nonlinear Dielectric Response of Relaxor Ferroelectric (1 −
 x
 )Pb(Mg
 1/3
 Nb
 2/3
 )O
 3
 −
 x
 PbTiO
 3
 Epitaxial Thin Films

Enhanced magnetoresistance and electroresistance at high temperature in a nano-matrix manganite

Hairs of a polar bear (Ursus maritimus) possess special membrane-pore
 structure. The structure enables the polar bear to survive in the harsh
 Arctic regions. In this paper, the membrane-pore structure be approximately
 considered as fractal space, 1-D heat conduction equation of the polar bear
 hair is established and the solution of the equation is obtained.

/pdf/one-dimensional-heat-conduction-equation-of-the-polar-bear-1uplaigwpe.pdf

Yong-Yan Pan

Papers

Enhanced magnetoresistance and electroresistance at high temperature in a nano-matrix manganite

One-dimensional heat conduction equation of the polar bear hair