Electrospinning is a complex process, and it can be modeled by a Bratu-type equation with fractal derivatives by taking into account the solvent evaporation. Though there are many analytical methods available for such a problem, e.g. the variational iteration method and the homotopy perturbation method, a straightforward method with a simple solution process and high accurate results is much needed. This paper applies the Taylor series technology to fractal calculus, and an analytical approximate solution is obtained. A fractal variational principle is also discussed. As the Taylor series is accessible to all non-mathematicians, this paper sheds a bright light on practical applications of fractal calculus.

A generalized KdV equation with fractal derivatives is suggested, and a special function is introduced to establish a fractal variational principle. A detailed derivation is elucidated step by step by the semi-inverse method, and some special cases are discussed.

Many nonlinear vibrations arising in the engineering of architecture include noise and uncertain properties, and this paper suggests a fractional model to elucidate the properties. The fractional sine-Gordon equation with the Riemann-Liouville fractional derivative is used as an example to solve its periodic solution by the homotopy perturbation method. The frequency-amplitude relationship is obtained, and the effect of the fractional derivative order on the vibration property is discussed. Additionally, the harmonic resonance is also discussed. This preliminary research can be further extended to real applications.

The dropping mechanism in the electrospinning process is elucidated. A moving jet becomes thinner at the initial stage due to the acceleration caused by the electrostatic force. When the jet diameter reaches a threshold, beyond which the jet breaks into drops and daughter jets, dropping occurs. The drops will finally form microspheres. Effects of applied voltage, flow rate, polymer?s concentration and receptor?s distance on the dropping process are theoretically analyzed and experimentally verified. This paper gives a general strategy for fabrication of smooth fiber, microspheres, and their mixture.

Site selection adaptability of traditional villages along rivers can be summarized as the local ecological wisdom accumulated and preserved by generations of villagers. The site selection and distribution pattern of traditional villages is closely related to the river. In this research, 97 traditional villages along the Yellow River in Shanxi and Shaanxi were selected as the research objects. Kernel density analysis, Area density analysis, River buffer zone and a self‐designed river location function (RLF) were used to explore, calculate, and evaluate the spatial distribution and site selection adaptation mechanism of traditional villages along the Yellow River. Most traditional villages are located within 15 km buffer area. Villages are densely distributed in one high‐density aggregation area and one middle‐density aggregation area. The aggregation areas also represent a highest area density of river. The results of RLF show that there are six villages with a value between 1.8 and 2.2, which is consistent with the results of kernel density and river buffer zone analyses. It is concluded that site selection adaptation factors are composed of area density of river, distance from river, river width and bank form. At the same time, it is found that traditional villages are clustered on convex banks, which reflects the scientific value of ancient Chinese feng shuitheory. Our findings improve the understanding of the overall environment and pattern of traditional villages, provide a quantitative basis for the study of site selection adaptability, and offer operational technical guidance for current and future efforts for village renewal, planning, and construction.