Nonlinear Mathematical Model Research Articles

Constructing mathematical models of thermal conductivity in individual elements and units of electronic devices at local heating considering thermosensitivity

This paper considers a heat conduction process for an isotropic medium with local external and internal thermal heating. It was necessary to construct linear and non-linear mathematical models for determining the temperature field, and consequently, for the analysis of temperature regimes in these heat-active environments. To solve the linear boundary value problems and the resulting linearized boundary value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which the analytical solutions to these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this environment were established. To determine the numerical values of the temperature and analyze the heat exchange processes in the given structure, caused by the external heat load, a geometric image of the temperature distribution was constructed depending on spatial coordinates. The resulting linear and non-linear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but the entire structure as well

Role of informal sector to combat unemployment in developing economy: A modeling study

In developing countries, informal sector is the primary job provider for a significant portion of the workforce. This study aims to analyze how jobs in the informal sector affect the unemployment dynamics of developing nations. To achieve this goal, we formulate a nonlinear mathematical model by categorizing the considered workforce into three distinct classes: unemployed, employed, and self-employed, and include a separate dynamic variable to represent vacancies within the informal sector. The proposed model is analyzed using the qualitative theory of dynamical systems. A threshold quantity known as the reproduction number is derived and using this, one can compute the job generation rate necessary to stabilize the system. It is observed that variations in the reproduction number lead to qualitative changes, such as transcritical (forward or backward) and saddle-node bifurcations in the formulated system. Moreover, we propose an optimal control problem to determine an optimal strategy for government policy implementation in enhancing the employment rate of unemployed individuals and promoting the self-employment of informal employees inside the informal sector. Further, the analytical findings are validated numerically. The obtained results suggest that promoting the self-employment of informal employees for job generation effectively reduces unemployment.

Hybrid Nonlinear Model Predictive Motion Control of a Heavy-duty Bionic Caterpillar-like Robot

This paper investigates the motion control of the heavy-duty Bionic Caterpillar-like Robot (BCR) for the maintenance of the China Fusion Engineering Test Reactor (CFETR). Initially, a comprehensive nonlinear mathematical model for the BCR system is formulated using a physics-based approach. The nonlinear components of the model are compensated through nonlinear feedback linearization. Subsequently, a fuzzy-based regulator is employed to enhance the receding horizon optimization process for achieving optimal results. A Deep Neural Network (DNN) is trained to address disturbances. Consequently, a novel hybrid controller incorporating Nonlinear Model Predictive Control (NMPC), the Fuzzy Regulator (FR), and Deep Neural Network Feedforward (DNNF), named NMPC-FRDNNF is developed. Finally, the efficacy of the control system is validated through simulations and experiments. The results indicate that the Root Mean Square Error (RMSE) of the controller with FR and DNNF decreases by 33.2 and 48.9%, respectively, compared to the controller without these enhancements. This research provides a theoretical foundation and practical insights for ensuring the future highly stable, safe, and efficient maintenance of blankets.

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis.

Height Control Strategy Design and Simulation of Electronic Control Air Suspension for Trucks

To address the large height error and attitude destabilization phenomenon in regulating the frame height of trucks with electronic control air suspension (ECAS), a height control strategy was designed. Firstly, the fundamental principles of height control were elucidated based on the single degree-of-freedom (DOF) vehicle model. The limitations of the classic non-linear mathematical model for the air spring were also highlighted. Thus, a dynamic model was constructed, consisting of an AEMSim model for the ECAS and a Simulink model for the truck. A frame height fuzzy controller was designed based on the fuzzy control theory to improve the height control accuracy and to solve the control conflict problem of the solenoid valves. Additionally, three typical control modes of the height and corresponding control strategies were proposed based on the practical requirements of usage scenarios for trucks. Finally, dynamic simulations were conducted under different modes. The results show that, compared to the existing switching control method, the proposed control approach can reduce height control errors by an order of magnitude and decrease the pitch angle by over 30%. The steady-state error remains nearly unchanged under the 30% variation of the sprung mass. The proposed control approach exhibits the superior control performance and robustness. It effectively reduces height errors and avoids the posture instability during the adjustment of the ECAS.

Research on Surveying and Mapping Data Processing Based on Nonlinear Mathematical Models and Deep Learning Optimization

In order to deeply understand the mapping data processing of nonlinear mathematical model optimization, this paper uses nonlinear model optimization theory to process mapping data. When the precision of parameter approximation is high, the calculation results of each algorithm are the same, and the iteration times of the fastest descent method and simplex method are significantly increased compared with the other three algorithms. Therefore x01 = 5.42 in the parameter is kept unchanged near the truth value, and the convergence range of Newton method, the fastest descent method, conjugate gradient method and simplex method away from the truth value of parameter x02 = −0.25 is investigated. When the approximate value of undetermined parameters is high, the results of Newton algorithm and simplex algorithm are consistent, but the Newton iterative algorithm has faster convergence speed and higher computational efficiency than the simplex algorithm. Lower value when the undetermined parameter approximation precision, namely the undetermined parameter approximation and its true value is far off, may complete failure type Newton iteration algorithm, the simplex algorithm can be a supplement of the Newton method for the most part, the simplex method significantly reduces the requirements for initial value of parameters, data calculation efficiency has improved significantly.

Investigation of friction induced vibrations on a nonlinear two degree of freedom mathematical model and experimental validation

This article investigates the dynamic behavior of a low order nonlinear mathematical model, which is developed based on an existing mass-sliding belt experiment. First, the proposed model is developed with kinematic and friction nonlinearities, and the corresponding governing equations are obtained. Governing equations are then solved numerically for certain operating parameters, and the characteristics of different dynamic responses are investigated. It is observed from the numerical solutions that the dynamic response of the system can be deterministic or chaotic based on the set of operating parameters. Furthermore, the period doubling mechanism is found to be the path from deterministic to chaotic behavior. Second, experiments are performed at a wide range of operating parameters in order to calculate the dynamic friction coefficient, which are used in artificial neural network models for the generation of friction models as functions of operating parameters. It is observed that these operating parameters have significant influence on the variation of the dynamic friction coefficient. Third, the friction models developed with artificial neural network approach are implemented in the nonlinear mathematical model, and numerical solutions are obtained at the same operating parameters. Finally, the numerical results of the low order nonlinear mathematical model are compared to the experimental data, and it is concluded that there is a good correlation between the model predictions and measurements from the perspectives of fundamental frequencies and the characteristics of the dynamic responses. Hence, a better insight about the dynamic response behavior of the mass-sliding belt experiment is achieved through the bifurcation analyses.

EVALUATION OF THE SCATTER OF LIQUID LAUNCH VEHICLE POGO OSCILLATION AMPLITUDES DUE TO THE INFLUENCE OF THE SCATTER OF INTERNAL FACTORS

Almost all liquid launch vehicle developers faced the problem of ensuring stability in relation to POGO oscillations. The level of POGO amplitudes oscillations of the launch vehicle can be significantly affected by the scatter of internal factors. The study aims to create a mathematical model that can determine the range of POGO amplitudes in liquid launch vehicles. This will be demonstrated through the example of the Dnipro launch vehicle, which is affected by a variety of internal factors that cause its POGO amplitudes to vary. We developed the non-linear non-stationary mathematical model of POGO oscillations of the prototype of the Dnipro space launch vehicle. The model is built by taking into account the two lower vibration modes of the LV structure, two lower oscillation modes of the oxidizer feedline, and the first oscillation mode of the fuel feedline of the propulsion system. Modeling of dynamic processes was conducted in a combination of four liquid rocket engines based on the schematic of the staged rocket engine. The simulation takes into account cavitation phenomena in the engine pumps and delay times in the gas generators’ chambers. We have developed a method for determining the scatter of the POGO oscillations caused by the action of internal factors, which is based on the use of the LP uniformly distributed sequences. As internal factors, the frequencies, decrements, and shapes of LV structural oscillation modes, the values of pressurization of the propellant tanks, and the engines’ specific thrust impulses were considered. Based on the results of the calculations, the dependence of the POGO amplitudes in two regions of LV instability was determined, and the lower and upper enveloping curves for the POGO amplitudes were constructed. It is shown that the maximum POGO amplitudes oscillations in the first region of instability lie in the range from 0.23 g to 0.72 g and in the second region of instability — from 0 g to 0.60 g. Variants of combinations of internal factors, which provided the largest and smallest values of POGO amplitudes, were analyzed. This made it possible to determine the internal factors, the scatter of which has the greatest effect on the POGO amplitudes scatter: frequency, decrement, shape coefficients of oscillations of the oxidizer feedlines and the LV 1st mode structural longitudinal oscillations in the payload cross-section.

NONLINEAR DYNAMICS METHODS APPLICATIONS TO PRODUCT LIFECYCLE MANAGEMENT

The research being carried out is aimed at applying nonlinear dynamics methods in product life cycle management to improve communication between processes and achieve environmental sustainability. The purpose of the article is to create a nonlinear mathematical model for analysing product lifecycle management. Chaotic nonlinear dynamical processes of product lifecycle are deterministic but have the property that even small uncertainties in initial conditions lead to exponentially growing errors in the management that limit long-term forecasts. The emergence of chaotic behaviour in processes of product lifecycle was explored, focusing on fluctuations. The study employed methods such as systems analysis, correlation analysis, nonlinear dynamics, and the Surrogate model of chaotic Lorenz waterwheel. It was identified that sharp technological innovations as the primary drivers of short-term fluctuations impacting fixed efficiency and effectiveness of product life cycle process development. The resulting nonlinear dynamic model allowed for flexible operation, transitioning between equilibrium, periodic, and chaotic states based on coefficient values for asset growth rates and time constants reflecting product life cycle process dynamics. The research results are valuable for companies seeking to reduce their environmental impact and implement circular economy principles.

Simultaneous consideration of time and cost impacts of machine failures on cellular manufacturing systems

Cellular manufacturing systems (CMS) are essential for achieving efficient production processes, and their success relies on effective cell formation and layout design. This research paper presents a novel nonlinear mathematical programming model that employs the rectilinear distance notion to define the layout within a continuous space. The proposed model incorporates the stochastic nature of machine failures, considering the unreliability with the inclusion of a stochastic time between failures. With a bi-objective approach, the model aims to minimize inter and intra-cell movements of parts, as well as the associated costs related to exceptional elements (EEs), cell reconfigurations, and machine failures. To optimize the proposed model, several multi-objective meta-heuristic algorithms, including the multi-objective particle swarm optimization (MOPSO), multi-objective taboo search (MOTS), and non-dominated sorting genetic algorithm (NSGA-II), are introduced. The effectiveness of these algorithms is validated through numerical instances and a real case study. The results indicate that the MOPSO algorithm exhibits superior performance in optimizing various multi-objective criteria. The presented mathematical model and algorithms provide valuable tools for manufacturers to optimize their CMS, resulting in reduced costs, enhanced production efficiency, and increased competitiveness. By considering the simultaneous effects of time and cost associated with machine failures, this research offers practical insights and solutions for improving the performance of cellular manufacturing systems.

Analysis of output characteristics of positive feedback piezoelectric energy harvester based on nonlinear magnetic coupling.

In light of the limitations of the current piezoelectric energy harvesters and the demand for self-power supply in wireless sensor nodes, a novel positive feedback piezoelectric energy harvester based on nonlinear magnetic coupling is proposed. The operational characteristics of this energy harvester are investigated from three perspectives: theory, simulation, and experiment. First, a nonlinear electromechanical coupling mathematical model that describes the dynamic response of the energy harvester system is established by combining the Hamilton variational principle with the piezoelectric theory. This provides a theoretical foundation for subsequent research. Second, finite element method simulations are employed to optimize the structural parameters of the energy harvester and study the impact of nonlinear magnetic force on its output performance. Finally, an experimental prototype is fabricated and an experimental test system is constructed to validate the designed positive feedback piezoelectric energy harvester. The results demonstrate that changes in the longitudinal beam angle have minimal effect on energy capture efficiency. By appropriately increasing the bending surface length, reducing initial magnetic moment, and augmenting mass block weight, wider working frequency bands and higher power generation capacity can be achieved when vibrating in low-energy orbits. The experimental findings align closely with theoretical design values and contribute to advancing broadband multi-directional piezoelectric energy harvesting technology in order to provide high-performance vibration-based power solutions for wireless applications.

A novel mathematical model for simultaneous optimization of desalination plant location and water distribution network; A case study

In recent decades, water scarcity has turned into a serious problem spanning many countries, now even capable of causing or inflaming ethnic and national conflicts. While our planet has very limited freshwater resources, it has huge amounts of saltwater in seas and oceans. There is a very limited number of ways that can make saltwater drinkable, the most important of them is desalination. This study aimed to provide a method for the simultaneous optimization of desalination plant location and its water distribution network based on mathematical modeling. For this purpose, the authors formulated a non-linear mathematical model with the objective of minimizing the costs of water production and transmission. A genetic algorithm was also developed for solving the proposed nonlinear model. The method was used in a case study of Sistan and Baluchestan, which is one of Iran's most water stressed provinces. The proposed genetic algorithm managed to provide an acceptable solution for this problem in 3.74 s. The best solution was found to be constructing a desalination facility with a capacity of 394,052 cubic meters per day in a single location, that is, the city of Chabahar. The water transmission lines needed for transporting water to other parts of the province and their capacities were also determined.

Dynamical Analysis of Peptic Ulcer Disease Model in Nigeria with the Effect of Vaccination and Treatment Program

The study developed a non-linear deterministic mathematical model to investigate the transmission dynamics of peptic ulcer disease in human population by considering both direct and indirect contact transmission with vaccine and treatment as control. In developing the model, the population was compartmentalized into susceptible-vaccinated-exposed-infected-treated-helicobacter pylori concentration in the environment recovered. The model developed is a system of differential equations. The rate of change of the system, existence and uniqueness of solution, region of absolute stability and positivity of the solution was established. Existence of disease-free equilibrium state and basic reproduction number was also established. Mathematical analysis was determined by the basic reproduction number if , the disease-free equilibrium is locally asymptotically stable whereas if then the equilibrium is unstable whereas the numerical analysis for the estimated basic reproduction number of the model is greater than unity in perspective of Nigeria, for which the Herd Immunity Threshold indicated that vaccinating 99.47% of Nigeria population can control spreading of Peptic ulcer disease in the country. The paper recommended amongst other things that if significant changes concerning the issue of threshold target are observed to improve the rate at which peptic ulcer can be minimized from the population where Government and non-governmental organization should encourage the use of peptic ulcer drugs and vaccines among individuals.

Cross-border political competition.

Individuals are increasingly exposed to news and opinion from beyond national borders. This news and opinion are often concentrated in clusters of ideological homophily, such as political parties, factions, or interest groups. But how does exposure to cross-border information affect the diffusion of ideas across national and ideological borders? Here, we develop a non-linear mathematical model for the cross-border spread of two ideologies. First, we describe the standard deterministic model where the populations of each country are assumed to be constant and homogeneously mixed. We solve the system of differential equations numerically by the Runge-Kutta method and show how small changes in the influence of a minority ideology can trigger shifts in the global political equilibrium. Second, we simulate recruitment as a stochastic differential process for each political affiliation and fit model solutions to population growth rates and voting populations in US presidential elections from 1932 to 2020. We also project the dynamics of several possible scenarios from 2020 to the end of the century. We show that cross-border influence plays a fundamental role in determining election outcomes. An increase in foreign support for a national party's ideas could change the election outcome, independent of domestic recruitment capacity. One key finding of our study suggests that voter turnout in the US will grow at a faster rate than non-voters in the coming decades. This trend is attributed to the enhanced recruitment capabilities of both major parties among non-partisans over time, making political disaffection less prominent. This phenomenon holds true across all simulated scenarios.

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V–L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

An adaptive integral sliding mode control strategy for medical linear accelerator vacuum system

Medical linear accelerators (MLAs) are critical components for radiation therapy, providing a state-of-the-art treatment platform for cancer therapy. The vacuum system is one of the most important MLA subsystems and its stable operation is necessary to generate high-quality beams. For vacuum system pressure control, traditional proportional-integral-derivative (PID) strategies have disadvantages such as inaccurate and imprecise control response due to its simple calculation. This paper presents an innovative adaptive integral sliding mode control (AISMC) strategy aimed at enhancing the response time, precision of control, and capability to reduce disturbances within the MLA vacuum system. In addition, a nonlinear MLA vacuum system mathematical model is established based on mechanism method. Stability of the developed vacuum control system is validated using Lyapunov stability theory. Simulation results illustrate that the proposed AISMC strategy has better response speed and accuracy than traditional PID-based systems, achieving better pressure tracking performance than traditional sliding mode control strategy with PID control. Most important for the proposed controller, system chattering is effectively mitigated.

Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics.

Waste consumption of bio-degradable products through a secondary supply chain

Food and agriculture organization estimates that 33% of the food is wasted in the world. It is essential to optimize the food supply chains to mitigate the economic and environmental impacts of this large amount of food waste. The targets of food waste reduction can be achieved by increasing lifetime and consuming the inevitably produced food waste for producing secondary products. This research proposes a primary and secondary supply chain model, where effective preservation policy increases lifetime of food items, and the yet produced food waste is converted into secondary products. This way, the waste from the system is removed and it is converted into a sustainable zero-waste system. A nonlinear mathematical model is proposed to set the objective function of the research, which best determines the production plan, inventory schedule and investment in preservation technology while maximizing profit. The model is solved using the Kuhn-Tucker method, and the model optimality is obtained by proposing an algorithm. Numerical analysis is carried out to validate the presented model, and sensitivity analysis is performed to demonstrate the effects of variation in cost parameters on the profit. The outcome of the analysis shows that the proposed secondary supply chain increases profit by 35% and the preservation policy increases the profit by 17.9%. Besides, the proposed policies minimize and remove the food waste from system. Managerial implications are provided for the practitioners to apply the proposed model in food supply chains.

From primary HPV infection to carcinoma in situ: A mathematical approach to cervical intraepithelial neoplasia

AbstractCervical intraepithelial neoplasia (CIN) is the development of abnormal cells on the surface of the cervix, caused by a human papillomavirus (HPV) infection. Although in most of the cases it is resolved by the immune system, a small percentage of people might develop a more serious CIN which, if left untreated, can develop into cervical cancer. Cervical cancer is the fourth most common cancer in women globally, for which the World Health Organization (WHO) recently adopted the Global Strategy for cervical cancer elimination by 2030. With this research topic being more imperative than ever, in this paper, we develop a nonlinear mathematical model describing the CIN progression. The model consists of partial differential equations describing the dynamics of epithelial, dysplastic, and immune cells, as well as the dynamics of viral particles. We use our model to explore numerically three important factors of dysplasia progression, namely, the geometry of the cervix, the strength of the immune response, and the frequency of viral exposure.

DESIGN OF A 3/2-WAY SELF-RETAINING (SELF-LOCKING) INDIRECT-ACTING ELECTRO-HYDRAULIC VALVE AND ITS MATHEMATICAL MODEL

A current topic of scientific research and development for the subsea production system, minemill industry, for the nuclear industry and other industries that require the use of hydraulic distributors operating in conditions of limited electrical power and/or explosive environment is currently the problem of designing self-retaining (self-locking) indirect-action electro-hydraulic valves (distributors). The article describes the created design of a 3/2-way self-retaining (self-locking) indirect-acting electro-hydraulic valve (distributor). The principle of operation of the distributor is described in detail. Its basic hydraulic diagram is given. The developed object-oriented nonlinear mathematical model of a 3/2-way self-retaining (self-locking) indirect-acting electro-hydraulic valve is presented. Dynamic equations are presented that describe the movements of the shut-off and control element (SCE) of the main distributor and two pilot valves; equations describing the flow of liquid through the main distributor's SCE and the law of changes in the overlap area of the main distributor's SCE flow sections; equations that determine the dynamic characteristics of an electromechanical converter (EMC). Transient processes were obtained and presented, with the help of which the dynamic quality indicators of the created a 3/2-way self-retaining (self-locking) indirect-acting electro-hydraulic valve were found. Based on the analysis, appropriate conclusions were drawn. The scientific novelty of the work includes the created object-oriented nonlinear mathematical model and the developed design of a 3/2-way self-retaining (self-locking) indirect-acting electro-hydraulic valve.