Yongyi Gu

Nonlinear differential equations are used for deciphering complex dynamical phenomena that cannot be captured by classical linear approximations, spanning physics, engineering, electronics, biomedicine, and computational mathematics. It is important to study nonlinear differential equations. This paper comprehensively analyzes the nonlinear dynamic characteristics of the [Formula: see text]-dimensional Kundu–Mukherjee–Naskar equation deploying the analytical method, Hamiltonian system framework, and numerical simulation. The wave solutions of the equation are derived by using the improved [Formula: see text]-expansion method, and its structural properties under different parameters are investigated. In order to better study the features of dynamics behavior of the equation, Hamiltonian system is established and nonlinear dynamics are systematically discussed with bifurcation, phase portraits, and Lyapunov exponents. Subsequently, detailed chaos analysis employing bifurcation diagrams and Poincaré maps demonstrates the complex response of the system under diverse initial conditions and parameters. In addition, we utilize the Runge–Kutta algorithm for numerical simulations to highlight the effectiveness and practical significance of these analytical solutions, and explore the attributes of the equation in depth through numerical solutions. We also compare this inclusive study with previous researches, highlighting the innovation and implication of this work. These exhaustive findings significantly enrich the application background of the [Formula: see text]-dimensional Kundu–Mukherjee–Naskar equation and provide solid theoretical assistance for the profound investigation of related nonlinear differential equations.

In this paper, the density-independent fractional diffusion-reaction (FDR) equation involving quadratic nonlinearity is investigated. The fractional derivative is illustrated in the beta derivative sense. We firstly propose Bernoulli [see formula in PDF]-expansion method to study nonlinear fractional differential equations (NFDEs). Subsequently, closed form solutions of the density-independent FDR equation are acquired successfully. In order to better understand the dynamic behaviors of these solutions, 3D, contour map and line plots are given by the computer simulation. The results show that the proposed method is a reliable and efficient approach.

In this paper, we investigate closed-form meromorphic solutions of the fifth-order Sawada-Kotera (fSK) equation and (3+1)-dimensional generalized shallow water (gSW) equation. The study of high-order and high-dimensional differential equations is pivotal for modeling complex nonlinear phenomena in physics and engineering, where higher-order dispersion, dissipation, and multidimensional dynamics govern system behavior. Constructing explicit solutions is of great significance for the study of these equations. The elliptic, hyperbolic, rational, and exponential function solutions for these high-order and high-dimensional differential equations are achieved by proposing the extended complex method. The planar dynamics behavior of the (3+1)-dimensional gSW equation and its phase portraits are analyzed. Using computational simulation, the chaos behaviors of the high-dimensional differential equation under noise perturbations are examined. The dynamic structures of some obtained solutions are revealed via some 2D and 3D graphs. The results show that the extended complex method is an efficient and straightforward approach to solving diverse differential equations in mathematical physics.

The fractional Klein–Gordon equation (fKGE) holds a crucial position in various fields of theoretical and applied physics, with wide applications covering different areas such as nonlinear optics, condensed matter physics, and quantum mechanics. In this paper, we carry out analytical investigation to the fKGE with beta fractional derivative by using the Bernoulli [ ]-expansion method and improved [ ]-expansion method. In order to better comprehend the physical structure of the obtained solutions, three-dimensional visualizations, contour diagrams, and line graphs of the exponent function solutions are depicted with the aid of Matlab. Moreover, the phase portraits and bifurcation behaviors of the fKGE under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These studies enhance our understanding of practical phenomena governed by the model of fKGE, and are crucial for examining the dynamic behaviors and phase portraits of the fKGE system. The strategies utilized here are more direct and effective, which can be effortlessly utilized to various fractional-order differential equations arising in nonlinear optics and quantum mechanics.

With the rapid expansion of the market and the increase in pig farming density, improving the automation and intelligence of pig farms has become key. Despite continuous progress in this field, there is still a lack of intelligent systems for cleaning and disinfecting pigs. In this paper, we conduct research from the perspective of product functional requirements. By conducting research to obtain raw data on user needs, the Analytic Hierarchy Process (AHP) is used to conduct a hierarchical analysis of user needs. The demand indicator system for the intelligent washing and disinfection integrated system of pig farms is summarized at three levels: goals, criteria, and indicators. Combined with competitor analysis and literature research methods, we obtained the operative words for the design requirements. Using Quality Function Deployment (QFD) to convert user requirements into performance indicators of product design, a quality house model for the product is constructed. We next analyzed the requirements of the intelligent washing and disinfection integrated system for pig farms in terms of functionality, usage, safety, and appearance, and this completed the conceptual design. Finally, we improved the mechanical structure of the mobile nozzle, supplemented the automation control system relied upon for nozzle movement, and enhanced the scientific and rational design. This study provides new ideas for the research and development of intelligent equipment in pig farms, promoting the development of intelligent and precision farming.

This paper explores the exact solutions of the fractional Hirota–Satsuma coupled KdV (fHScKdV) equation in the Beta fractional derivative. The logistic method is first proposed to construct analytical solutions for the fHScKdV equation. In order to better comprehend the physical structure of the solutions, three-dimensional visualizations and line graphs of the exponent function solutions are depicted with the aid of Matlab. Furthermore, the phase portraits and bifurcation behaviors of the fHScKdV model under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These investigations enhance our understanding of practical phenomena governed by the fHScKdV model, and are crucial for examining the dynamic behaviors and phase portraits of the fHScKdV system. The strategies utilized here are more direct and effective, and can be applied effortlessly to other fractional order differential equations.