System Of Nonlinear Equations Research Articles

Quasi-Classical Models of Nonlinear Relaxation Polarization and Conductivity in Electric, Optoelectric, and Fiber Optic Elements Based on Materials with Ionic–Molecular Chemical Bonds

A generalized scientific review with elements of additions and clarifications has been carried out on the methods of theoretical research on the electrophysical properties of crystals with ionic–molecular chemical bonds (CIMBs). The main theoretical tools adopted are the methods of quasi-classical kinetic theory as applied to ionic subsystems relaxing in layered dielectrics (natural silicates, crystal hydrates, various types of ceramics, and perovskites) in an electric field. A universal (applicable for any CIMBs class crystals) nonlinear quasi-classical kinetic equation of theoretical and practical importance has been constructed. This equation describes, in complex with the Poisson equation, the mechanism of ion-relaxation polarization and conductivity in a wide range of polarizing field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). The physical model is based on a system of non-interacting ions (due to the low concentration in the crystal) moving in a one-dimensional, spatially periodic crystalline potential field, perturbed by an external electric field. The energy spectrum of ions is assumed to be continuous. Elements of quantum mechanical theory in a quasi-classical model are used to mathematically describe the influence of tunnel transitions of hydrogen ions (protons) during the interaction of proton and anion subsystems in hydrogen-bonded crystals (HBC) on the polarization of the dielectric in the region of nitrogen (50–100 K) and helium (1–10 K) temperatures. The mathematical model is based on the solution of a system of nonlinear Fokker-Planck and Poisson equations, solved by perturbation theory methods (via expanding solutions into infinite power series in a small dimensionless parameter). Theoretical frequency and temperature spectra of the dielectric loss tangent were constructed and analyzed, the molecular parameters of relaxers were calculated, and the physical nature of the maxima of the experimental temperature spectra of dielectric losses for a number of HBC crystals was discovered. The low-temperature maximum, which is caused by the quantum tunneling of protons and is absent in the experimental spectra, was theoretically calculated and investigated. The most effective areas of scientific and technical application of the theoretical results obtained were identified. The application of the equations and recurrent formulas of the constructed model to the study of nonlinear optical effects in elements of laser technologies and nonlinear radio wave effects in elements of microwave signal control systems is of the greatest interest.

Functional Central Limits Theorem for epidemic model with varying infectivity and waning immunity

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Speci cally, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In [12], a functional law of large numbers (FLLN) is proved as the population size goes to in nity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic ™uctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.

Legendre wavelet based numerical method for the solution of marangoni convective flow of dusty ree-erying fluid over a riga plate with activation energy

AbstractLegendre wavelet based numerical method has been developed for the solution of the system of non-linear differential equations arising in the study of a Marangoni convective flow of dusty Ree-Erying fluid over a Riga plate in Darcy-Forchheimer medium with Soret, Dufour, non-linear radiation, activation energy effects and entropy generation analysis. The proposed method is validated by comparing the solutions of the test problem with the Bernoulli wavelet based numerical method and the exact solution and by comparing the results of the considered dusty fluid problem for some fixed parameters with the previously published results. The effects of various factors on the velocity, thermal, concentration attributes are studied by using the proposed method. It is observed that the Ree-Erying fluid parameter and Marangoni ratio parameter escalates the velocity and declines the temperature and concentration. The opposite behaviour is observed for Deborah number. The concentration declines for higher chemical reaction parameter and improves for activation energy. Both entropy generation and Bejan number escalates for radiation parameter and diffusion parameter. The proposed method is reliable, fast computable and powerful tool to solve the differential equations.

METHOD FOR CONSTRUCTING PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

A justification is given for the analytical method for constructing periodic solutions for nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series, in which the coefficients are polynomials depending on the parameter without the assumption of its smallness. Two examples are considered — the van der Pol equation and the Lorentz system.

ON PIECEWISE CUBIC ESTIMATES OF THE VALUE FUNCTION IN THE PROBLEM OF TARGET CONTROL FOR A NONLINEAR SYSTEM

A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set at a fixed time interval by selecting the appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem at a fixed time interval. The paper proposes formulas for calculating the values of a piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.

An Approximate Solution for Heat Transfer in the Entrance Region of Laminar Newtonian Pipe Flow

The study of the simultaneously developing pipe flow requires facing nonlinear systems of partial differential equations. In this framework, the aim of this paper is to demonstrate that the integral method can be an effective procedure to obtain analytic-approximate solutions that are easy to handle while allowing the recovery of a satisfactory accordance with the exact solution. To prove the above statement this paper will present a comparison between the approximate solution and the corresponding numerical solution in the entrance region of Newtonian pipe flow. Third-kind thermal boundary conditions are included, while velocity and temperature profiles at the inlet are assumed uniform. Numerical results demonstrate that the proposed approximate solution is quite accurate and readily implemented, both in terms of developing velocity and temperature profiles. Moreover, the expected functional dependence on the main parameters of the problem at hand is retained. As a consequence, the developing Fanning friction coefficient and Nusselt curves are satisfactory and accurate for different thermal boundary conditions at the wall.

On Non-Linear Differential Systems with Mixed Boundary Conditions

For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions.

On the efficacy of sparse representation approaches for determining nonlinear structural system equations of motion

Abstract A sparsity-based optimization approach is presented for determining the equations of motion of stochastically excited nonlinear structural systems. This is done by utilizing measured excitation-response realizations in the formulation of the related optimization problem, and by considering a library of candidate functions for representing the system governing dynamics. Note that a novel aspect of the approach relates to treating, also, systems endowed with fractional derivative elements. Clearly, this is of significant importance to a multitude of diverse applications in engineering mechanics taking into account the enhanced modeling capabilities of fractional calculus. Further, the fundamental theoretical and computational aspects of various representative, state-of-the-art, numerical schemes for solving the derived sparsity-based optimization problem are reviewed and discussed. A Bayesian compressive sampling approach that exhibits the additional advantage of quantifying the uncertainty of the estimates is considered as well. Furthermore, comparisons and a critical assessment of the employed numerical schemes are provided with respect to their efficacy in determining the nonlinear structural system equations of motion. In this regard, two illustrative numerical examples are considered pertaining to a nonlinear tuned-mass-damper-inerter vibration control system, and to a nonlinear electromechanical energy harvester, both endowed with fractional derivative elements.

An inertial-double-projection-based algorithm for large-scale nonlinear systems of monotone equations

An inertial-double-projection-based algorithm for large-scale nonlinear systems of monotone equations

On Large Amplitude Vibrations of the Softening Duffing Oscillator at Low Excitation Frequencies—Some Fundamental Considerations

The Duffing equation containing a cubic nonlinearity is probably the most popular example of a nonlinear oscillator. For its harmonically excited, slightly damped, and softening version, stationary large amplitude solutions at subcritical excitation frequencies are obtained when standard semi-analytical methods like Harmonic Balance or Perturbation Analysis are applied. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary when applying the Harmonic Balance or other semi-analytical methods to make them converge. Some of these solutions are observed to be asymptotically stable, while in most cases, they are unstable. The current paper aims to give a descriptive explanation for this behavior of the nose solutions, which is mainly related to the exact solution of the free undamped vibrations. Based on this, approximations of the nose solutions are calculated with a procedure combining properties of Perturbation Analysis and Harmonic Balance. Therein, the exact solution of the free undamped vibrations is taken as the zeroth approximation, while higher-order solution parts are calculated by balancing the harmonics, and the phase shift of the zeroth approximation is calculated by a residuum minimization. This method just requires the solution of a system of linear algebraic equations, while systems of nonlinear algebraic equations have to be solved in the case of directly applying Harmonic Balance.

Interaction of induced magnetic field, double diffusion convection and multiple slips for thermal radiative biological flow of six-constant Jeffreys nanofluid: Advancements in mechanics

ABSTRACT The mathematical modeling of biological fluids holds paramount importance, given its diverse applications in the medical field. Understanding the peristaltic mechanism is crucial for gaining insights into various biological flows. This study focuses on numerically estimating the double-diffusive peristaltic flow of a non-Newtonian six constant Jefferys nanofluid within an irregular medium. The investigation considers the influences of nonlinear thermal emission, viscous dissolution, and induced magnetization, incorporating multiple slip conditions. The Buongiorno model is employed to underscore key features related to thermophoresis and Brownian diffusion coefficients. The convection of double diffusivity is elucidated through the Soret and Dufour variables. Lubrication technique is employed for mathematical simulation, and the resulting nonlinear coupled system of differential equations is solved numerically through the Mathematica program’s built-in command (ND Solve function). The graphical representation of the impact of crucial flow variables, such as nanoparticle volume proportion, pressure gradient, solute density, temperature, pressure fluctuation per wavelength, and velocity distribution, provides valuable insight. The findings of the current study show that an increase in the Prandtl number and thermophoresis parameter leads to the expansion of thermal curves. Moreover, it is also noted that rising heat frequencies of radiation cause the concentration of nanoparticles to increase.

Line-search based optimization using function approximations with tunable accuracy

This paper develops a line-search algorithm that uses objective function models with tunable accuracy to solve smooth optimization problems with convex constraints. The evaluation of objective function and its gradient is potentially computationally expensive, but it is assumed that one can construct effective, computationally inexpensive models. This paper specifies how these models can be used to generate new iterates. At each iteration, the model has to satisfy function error and relative gradient error tolerances determined by the algorithm based on its progress. Moreover, a bound for the model error is used to explore regions where the model is sufficiently accurate. The algorithm has the same first-order global convergence properties as standard line-search methods, but only uses the models and the model error bounds. The algorithm is applied to problems where the evaluation of the objective requires the solution of a large-scale system of nonlinear equations. The models are constructed from reduced order models of this system. Numerical results for partial differential equation constrained optimization problems show the benefits of the proposed algorithm.

An Effective Hybrid Metaheuristic Approach Based on the Genetic Algorithm

This paper presents an effective hybrid metaheuristic algorithm combining the genetic algorithm (GA) and a simple algorithm based on evolutionary computation. The evolutionary approach (EA) is applied to form the initial population of the GA, thus improving the algorithm’s performance, especially its convergence speed. To assess its effectiveness, the proposed hybrid algorithm, the EAGA, is evaluated on selected benchmark functions, as well as on a real optimisation process. The EAGA is used to identify parameters in a nonlinear system of differential equations modelling an E. coli fed-batch fermentation process. The obtained results are compared against published results from hybrid metaheuristic algorithms applied to the selected optimisation problems. The EAGA hybrid outperforms the competing algorithms due to its effective initial population generation strategy. The risk of premature convergence is reduced. Better numerical outcomes are achieved. The investigations validate the potential of the proposed hybrid metaheuristic EAGA for solving real complex nonlinear optimisation tasks.

Optimized five-by-five block preconditioning for efficient GMRES convergence in curvature-based image deblurring

We introduce an enhanced preconditioning technique designed to expedite the convergence of Krylov subspace methods when dealing with non-linear systems of equations featuring a block five-by-five format. This scenario often arises in the context of cell centered finite difference discretizations applied to the mean curvature based image deblurring problem. A thorough spectral analysis of the preconditioned matrices reveals a favorable eigenvalue distribution, leading to accelerated convergence of preconditioned Generalized Minimal Residual (GMRES) methods. Furthermore, we present numerical experiments to showcase the effectiveness of this preconditioner when combined with the flexible GMRES solver for addressing non-linear systems of equations originating from a image deblurring problems. Codes can be obtained at https://github.com/shahbaz1982/Preconditioning.

Unraveling dengue dynamics with data calibration from Palu and Jakarta: Optimizing active surveillance and fogging interventions

Dengue fever is a complex infectious disease driven by multiple factors, including viral dynamics, mosquito behavior, environmental conditions, and human behaviours. The intricate nature of its transmission and outbreaks necessitates an interdisciplinary approach, integrating expertise from fields such as mathematics and public health. In this research, we examine the role of active case finding and mosquito population reduction through fogging in dengue control using a mathematical model approach. Active case finding aims to identify undetected dengue cases, both asymptomatic and symptomatic, which can help prevent further transmission and reduce the likelihood of severe symptoms by enabling earlier treatment. The model was developed using a system of nine-dimensional nonlinear ordinary differential equations. We conducted a mathematical analysis of the equilibria and their stability based on the basic reproduction number (R0). Our analysis shows that the disease-free equilibrium is locally asymptotically stable when R0<1. Furthermore, when R0=1, the model may exhibit backward bifurcation , depending on the death rate induced by dengue. The higher the dengue-induced death rate, the greater the likelihood of backward bifurcation at R0=1. We used dengue incidence data from two Indonesian provinces, Jakarta and Palu, to calibrate the model parameter values. Our global sensitivity analysis on the basic reproduction number indicates that active case findings are more crucial in Palu compared to Jakarta. Conversely, Jakarta is more sensitive to the infection parameter than Palu. Our numerical continuation simulation shows that implementing fogging to control the mosquito population should carefully consider the intensity, timing, and duration of the intervention to achieve a more optimal results.

Newton-type methods for solving vertical linear complementarity problems

In this paper, Newton-type methods are proposed to solve the vertical linear complementarity problems. This kind of methods is different from the existing iteration methods such as the modulus-based matrix splitting methods. By analyzing the sign pattern for the solution and reformulating the vertical linear complementarity problem to an equivalent system of nonlinear equations, we propose the Newton’s method and a practical-Newton’s method that combine the Newton’s method and modulus-based matrix splitting method. Numerical experiments are given to show that these Newton-type methods have much higher efficiency than the existing modulus-based matrix splitting methods.

Learning dynamical systems from noisy data with inverse-explicit integrators

We introduce the mean inverse integrator (MII), a novel approach that improves accuracy when training neural networks to approximate vector fields of dynamical systems using noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge–Kutta methods. We show that the class of mono-implicit Runge–Kutta methods (MIRK) offers significant advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained by inserting the training data in the MIRK formulae. This allows the use of symmetric and high-order integrators without requiring the solution of non-linear systems of equations. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with extensive numerical experiments using data from chaotic and high-dimensional Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, providing theoretical support for the favorable performance of MII and symmetric MIRK methods.

Modulation optimization method for seven-level SHEPWM inverter based on EPSO algorithm

Selective Harmonic Elimination Pulse Width Modulation (SHEPWM) has excellent harmonic characteristics, but its nonlinear transcendental system of equations is difficult to be solved, and the practical application encounters a bottleneck. In this paper, a modulation optimization method for seven-level SHEPWM inverter based on the Evolutionary Particle Swarm Optimization (EPSO) algorithm is proposed to address this problem, so that the algorithm quickly converges to the global optimum solution. The EPSO algorithm incorporates a population optimization strategy in two phases to improve the population diversity in real time. In the initialization phase, the initialized population is optimized using Opposition-Based Learning (OBL) to improve the quality of the initial population. In the iterative stage, we combine the adaptive Particle Swarm Optimization (PSO) algorithm, Tunicate Swarm Algorithm (TSA), Adaptive Gaussian Variation, Quasi-Opposition-Based Learning (QOBL) and other optimization methods to solve the problem of insufficient population diversity in the process of searching for the optimal solution, to break through the local optimum, and to improve the convergence speed and accuracy of the algorithm. Experiments of the algorithm in 19 benchmark functions and seven-level SHEPWM inverter optimization modulation show that the optimization ability of the EPSO algorithm is ahead of TSA, INFO, MA (Mayfly Algorithm), EO (Equilibrium Optimizer) and other optimization algorithms. The solution speed is about three times that of PSO, which achieves fast and highly accurate convergence, with a small error in the output of inverter, and better harmonic distortion rate than the standard requirement.

Design, Modeling and Simulation of Mini Rocket Launching Systems Applied to the Reforestation of Areas Impacted by Deforestation

Objective: This project aims to develop reforestation operations in areas impacted by deforestation, with the objective of designing mini-rocket launch systems, computational modeling and simulation for flight prediction, promoting biodiversity and environmental recovery Theoretical framework: The use of rockets for ecological restoration purposes presents unique challenges. Precision in seed deposition, selection of the most suitable species for each environment, and assessment of environmental impacts are issues that require in-depth research. This technology, traditionally associated with space exploration, has the potential to optimize agricultural practices, monitor the environment, and address challenges such as deforestation and desertification. Method and material: The materials used in the project were a dynamometer, specifically the AMF 5 model, capable of measuring both tensile and compressive forces up to 5 N with a precision of 0.01 N, a RAVEN brand smoke generator with a smoke flow rate of 13 liters per minute, a custom-designed wind tunnel constructed in the laboratory, a CB-8909 anemometer, and custom-designed rocket launch bases. The methodology began with the design and construction of a wind tunnel to determine the drag coefficient of the miniature rockets. A launch pad for miniature rockets propelled by water and air was developed and used. The thrust of various miniature rockets was experimentally measured on a test bench while keeping the rocket stationary. Flight prediction modeling and simulation were based on two sets of differential equations derived from the conservation of mass and momentum, solved numerically using the Runge-Kutta method. Two sets of nonlinear differential equations were formulated and solved: one for the Propulsion Phase and another for oblique launch. The computer program allows user-defined input and generates various outputs in matrix or graphical format. Launch tests were conducted to validate the developed computer program. Results and Discussion: The computational simulation of the mini-rocket flight prediction resulted in figures containing the most important flight parameters. The results of the computer program were validated through field launches of the mini-rockets. The differential of this work in relation to those found in the literature was that it addressed dynamic modeling for flight prediction, using two systems of nonlinear equations with twenty-two dynamic variables, with the obtaining of results for fifteen flight parameters. Research Implications: This study found that the use of rockets in agriculture and environmental fields opens up a range of innovative possibilities for natural resource management and environmental preservation. The projects in this study revealed the importance of studying mini-rockets, wind tunnels, mini-rocket launch systems, and the development of simulation software, opening up new frontiers in the aerospace sector for reforestation, restoration of degraded areas, and applications in agriculture. Originality/Value: The project, modeling and simulation of mini-rocket launch systems applied to reforestation revealed a new application area for rockets. The research was conducted at the Integrated Engineering Laboratory of the Department of Chemical Engineering at the Federal University of São João del Rey, Ouro Branco Campus, Minas Gerais, Brazil. This project, which was supported by FAPEMIG, involved the sharing of activities with post-doctoral, master's, undergraduate research, and final course work students, generating knowledge in the areas of research and teaching

Numerical computation of tangent hyperbolic magnetohydrodynamic Darcy–Forchheimer Williamson hybrid nanofluid flow configuring variable thermal conductivity

The current investigation delves into the tangent hyperbolic Williamson hybrid nanofluid flow featuring varying thermal conductivity through the Darcy–Forchheimer medium across an exponentially stretching cylinder. Incorporating the activation energy and chemical reaction effects into consideration also strengthens the mathematical model's vitality. The considered hybrid nanofluid comprises silver and molybdenum disulfide nanoparticles submerged in the water. The highly non-linear system of equations is solved utilizing the MATLAB bvp4c approach. The influences of the leading variables versus involved fields are demonstrated through graphical delineations and tables. The core findings demonstrated that a strong Weissenberg number and Darcy-Forchheimer factor decay the velocity curve and strengthen the thermal curve. Also, the fluid concentration is enhanced for escalating activation energy and tangent hyperbolic factor. Additionally, the hybrid nanofluid betokens substantially enhance the thermal transportation rate of up to 8.8 % in contrast to nanofluid. Also, in contrast to the nanofluid and Williamson hybrid nanofluid, the hybrid nanofluid exhibits notably higher mass and thermal transport rate. This study is a noteworthy advancement in the disciplines of fluid dynamics and nanofluid research, as it provides promising potential for optimizing the transfer of mass and heat in a wide range of engineering and industrial contexts. The findings exhibit good agreement when contrasted with previously published work.