Effect of Silicon Carbide Nanoparticles on the Friction-Wear Properties of Copper-Based Friction Discs

To study the influence of nano-additives on the friction-wear characteristics of friction materials, the nano-sized silicon carbide particles which have excellent chemical and physical properties are considered to add in composite to form the modified friction material. The influence of the silicon carbide nanoparticles (SCN) on the friction-wear characteristics of copper-based friction materials (CBFM) is investigated via the SAE#2 (made in Hangzhou, China) clutch bench test with the applied pressure, rotating speed, and automatic transmission fluid (ATF) temperature taken into account. Moreover, the variations of friction torque and temperature are considered to evaluate the friction performance, and the variable coefficient is employed to describe the friction stability. The wear characteristics of friction materials are investigated by the disc changes in thickness and micro-morphology. The results show that the CBFM with SCN can provide a higher friction torque, which increased by 30% to 50% compared with CBFM. The variable coefficient of CBFM with SCN changes from 674 to 52 with the rotating speed raised from 600 rpm to 3000 rpm, which shows that the friction stability is relatively worse. Furthermore, the micromorphology shows that the CBFM with SCN has lower porosity and surface roughness, which increases the microscopic contact area and the coefficient of friction (COF). Simultaneously, the reduction in porosity also leads to a decrease in the cooling quality, bringing about a rapid temperature rise. Thus, the wear amount of CBFM with SCN increases significantly, especially for the friction disc in the axial middle position.

Bifurcation Analysis and Single Traveling Wave Solutions of the Variable-Coefficient Davey–Stewartson System

This paper mainly studies the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system. By employing the traveling wave transformation, the variable-coefficient Davey–Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations. On the one hand, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. On the other hand, we use the polynomial complete discriminant method to obtain the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.

Parallel Solving Method for the Variable Coefficient Nonlinear Equation

In view of the inaccuracy of traditional methods for solving nonlinear equations with variable coefficients in parallel, a new method for solving nonlinear equations with variable coefficients is proposed. Using the generalized symmetry group, the variable coefficient of the equation is taken as a new variable which is the same as the state of the original actual physical field. Some relations between variable coefficient equations and their solutions are found. This paper analyzes the meaning of linear differential equation and nonlinear differential equation, the difference between linear differential equation and nonlinear differential equation and their role in physics, and the necessity of solving nonlinear differential equation. By solving the nonlinear equation with variable coefficients, it can be seen that the general methods to solve the nonlinear equation include scattering inversion, Backlund transform and traveling wave solution. Based on the existing methods for solving nonlinear equations with variable coefficients, the homogeneous balance method is combined with the improved truncated expansion method, truncated expansion method and function reduction method, and the Hopf Cole transform and trial function are combined respectively. The above three methods are used to solve nonlinear equations with variable coefficients. Based on KdV Painleve principle, a parallel method for solving nonlinear equations with variable coefficients is proposed. Finally, it is proved that the method is accurate and effective for the parallel solution of nonlinear equations with variable coefficients.

Lie Group Classification of Generalized Variable Coefficient Korteweg-de Vries Equation with Dual Power-Law Nonlinearities with Linear Damping and Dispersion in Quantum Field Theory

Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, this article presents a complete Lie group analysis of a generalized variable coefficient damped wave equation in quantum field theory with time-dependent coefficients having dual power-law nonlinearities. Lie group classification of two distinct cases of the equation was performed to obtain its kernel algebra. Thereafter, symmetry reductions and invariant solutions of the equation were obtained. We also investigate various soliton solutions and their dynamical wave behaviours. Further, each class of general solutions found is invoked to construct conserved quantities for the equation with damping term via direct technique and homotopy formula. In addition, Noether’s theorem is engaged to furnish more conserved currents of the equation under some classifications.

An Empirical Study of National Education and Economic Development Based on a Variable Coefficient Model

This paper studies the relationship between the level of national education and economic development in 31 provinces and cities in China from 2012 to 2019. Principal component analysis is used to comprehensively evaluate the level of national education, on the basis of which a variable coefficient model is tested and a constant coefficient model is established for comparison. The results show that: (1) there is a co-integration relationship between the level of national education and economic development, i.e. the two are in equilibrium in the long run. (2) In some provinces and cities in China, economic development is positively correlated with the level of national education, and some provinces and cities are negatively correlated. (3) Economic development has different effects on the level of national education in the east, middle and west. (4) Economic development and national education level are positively correlated in the average sense in China.

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

The aim of this paper is to establish the existence and uniqueness results for differential equations of Hilfer-type fractional order with variable coefficient. Firstly, we establish the equivalent Volterra integral equation to an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. Secondly, we obtain the existence and uniqueness results for a class of Hilfer fractional differential equations with variable coefficient. We verify our results by providing two examples.