Publications

In this paper we investigate the transport equations in fractal porous media by using the fractional complex transform method. The local fractional linear and nonlinear transport equations with local fractional time and space fractional derivatives are obtained. The proposed models adequately describe the fractal transport processes.

By the semi-inverse method proposed by He, a variational principle is established for three-dimensional MHD equations with high Hartmann number. In order to incorporate the no-slip condition and far distance boundary condition as natural boundary conditions, a special technique is proposed in this paper. Lagrange crisis are also illustrated.

This paper speculates on the possibility that a proton or an electron could have a hierarchical structure, consisting of quarks and infinite sub-quarks, which have fractional charge values. The charge of a proton or an electron could be seen as a collection fractional charge of quarks and infinite sub-quarks. On extremely small scales or at very high observational resolution equivalent to a very high energy, fractal charge might become foam-like, which can, however, be described using a continued fraction.

According to an ancient Chinese algorithm, the Ying Buzu Shu, in about second century BC, known as the rule of double false position in West after 1202 AD, two trial roots are assumed to solve algebraic equations. The solution procedure can be extended to solve nonlinear differential equations by constructing an approximate solution with an unknown parameter, and the unknown parameter can be easily determined using the Ying Buzu Shu. An example in heat transfer is given to elucidate the solution procedure.

A very simple and effective approach to nonlinear oscillators is suggested. Anyone with basic knowledge of advanced calculus can apply the method to finding approximately the amplitude-frequency relationship of a nonlinear oscillator. Some examples are given to illustrate its extremely simple solution procedure and an acceptable accuracy of the obtained solutions.

No abstract is provided for this article.

In this paper, a new iteration technique is proposed to solve autonomous ordinary differential systems. In this method, general Lagrange multipliers are introduced to construct correction functionals for the systems. The multipliers in the functionals can be identified by the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. Some examples are given. The results reveal that the method is very effective and convenient.

A series of CrO x –Y2O3 catalysts were prepared by a deposition–precipitation method and tested for the fluorination of 2-chloro-1,1,1-trifluoroethane (CF3CH2Cl) to synthesize 1,1,1,2-tetrafluoroethane (CF3CH2F). The highest activity was obtained on a pre-fluorinated catalyst calcined at 400 °C, with 19% of CF3CH2Cl conversion at 320 °C. The effect of the calcination temperature on the CrO x species was investigated. X-ray diffraction and Raman results indicated that the CrO x species (Cr(VI)) were well dispersed on the catalyst surface when the catalyst was calcined at 400 °C. With increasing calcination temperature, most of the CrO x species changed from high oxidation state Cr(VI) to low oxidation state Cr(V) or Cr(III) species, which resulted in difficulty in pre-fluorination of the catalyst. It was also found that the CrF x , CrO x F y or Cr(OH) x F y phases originated from high oxidation state Cr(VI) species were the active sites for the fluorination reaction.

The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary initial conditions. The tangent oscillator, the hyperbolic tangent oscillator, a singular oscillator, and a MEMS oscillator are chosen to elucidate the simple solving process. The results, when compared with those obtained by the homotopy perturbation method, exhibit a good agreement. This paper introduces a very convenient procedure for attaining quick and accurate insight into the vibration property of a nonlinear vibration system.

No abstract is provided for this article.

The paper is a condensed but accurate account of El Naschie’s theory of Cantorian space-time which was used by him to clarify some major problems in theoretical physics and cosmology. In particular El Naschie’s revision and completion of relativity theory and demystification of dark energy are destined to be two milestones in the history of theoretical physics.

Physics is mathematical, new physical phenomena require new mathematical tools. This Research Topic is an attractive introduction of some new mathematical concepts, e.g., the two-scale fractal geometry, the fractal-fractional models, the homotopy perturbation method and the frequency formulation appliable to physics. The new findings of physical phenomena using the new mathematical tools have excited physicists with their potential to reveal secrets in physics and have triggered new research frontiers in physics. Here is an overview of the Research Topic.

Spider-spun fiber is of extraordinary strength and toughness comparable to those of electrospun fiber, the later needs a very high voltage (from several thousands voltage to several ten thousands voltages) applied to water-soluble protein “soup” that was produced by a spider, furthermore, its mechanical strength dramatically decreases comparable to spider silk. A possible mechanism in spider-spinning process is given, the distinct character in spider-spinning is that its spinneret consists of millions of nano scale tubes, and a bubble can be produced at the apex of each nano-tube. The surface tension of each bubble is extremely small such that it can be spun into nanofibers with an awfully small force, either by the spider’s body weight or tension created by the rear legs. We mimic the spider-spinning in electrospinning using an aerated solution, which leads to various small bubbles on surface with very small surface tension, as a result the bubble can be easily electrospun into nanofibers with low applied voltage. This fabrication process possesses features of high productivity, versatility, in addition, the minimum diameter of nanofibers produced by this process can reach as small as 50nm.

This study investigates the frequency-amplitude relationship in nonlinear oscillators with irrational nonlinearities, which pose significant challenges to traditional analytical methods such as the homotopy perturbation method and variational iteration method. By employing a simplified frequency formulation, we analyze the impact of irrational nonlinearities on oscillation dynamics and derive a frequency-amplitude relationship in a single-step calculation. For small amplitudes, our results are consistent with those obtained by the homotopy perturbation method, validating the proposed approach. This research provides new insights into the complex dynamics of nonlinear oscillators and offers practical implications for fields such as physics, engineering, and applied mathematics. Additionally, it contributes to the design and optimization of MEMS systems and suggests future research directions, including the incorporation of damping and external forces for a more comprehensive understanding of system behavior.

Polymer bubbles are ubiquitously used for the fabrication of nanofibers by the bubble electrospinning. When a bubble is broken, the fragments tend to be wrinkled. The wrinkle angle plays an important in controlling the fiber morphology during the bubble electrospinning. This paper shows the maximal angle is about 49°, which is close to the experimental value of 50°. This maximal angle can be used for the optimal design of the nozzle in the bubble electrospinning for the fabrication of non-smooth nanofibers.

In this paper, a new perturbation method is proposed. In contrast to the traditional perturbation methods, this technique does not require a small parameter in an equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter p∈[0,1] is constructed, and the imbedding parameter is considered as a “small parameter”, so the method is called the homotopy perturbation method, which can take the full advantages of the traditional perturbation methods and homotopy techniques. To illustrate its effectiveness and its convenience, a Duffing equation with high order of nonlinearity is used; the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter, and is more accurate than the perturbation solutions.