System Of Nonlinear Equations Research Articles

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method

Determining the influence of wheelset arrangement in the model 18-100 bogies on the level of steering efforts in the wheel-rail flange contacts

The object of this study is the process of directing the bogies of model 18-100 freight cars along a rail track, in particular, along curved sections of the track. The task to be solved was to determine the influence of wheelset arrangement on the level of steering forces in the flange contacts. A calculation diagram and a mathematical model of fitting the bogie into the curved section of the track have been built. The bogie loading scheme by external forces, including lateral rocking forces acting on the car in a curve, has been refined. In this case, the method of pseudo-statics of the mechanical system, which is a system of nonlinear algebraic equations, was applied. The calculation module Given-Find in the Mathcad software package (USA) was used to solve the mathematical model. It was established that the misalignment of wheelsets in the frames of model 18-100 bogies was of an accumulative nature. At the maximum operating angles of misalignment of wheelsets, the lateral steering forces in the flange contacts increase by 40–60 % compared to the rated setting. These angles can be up to 0.015 rad (0.85 degrees). The field of practical application of the results is railroad transportation, in particular, the system of maintenance and repair of freight cars on bogies of the 18-100 model. At the same time, the condition for the practical application of the research results is the expediency of introducing into the maintenance system the technological operation of controlling the deviation of wheelset arrangement in the bogie relative to the rated one. The current study will contribute to the construction of a measuring system for monitoring the deviation of wheelset arrangement in the bogie relative to the rated one. This proves the expediency of introducing into the trolley maintenance system the technological operation of controlling the deviation of wheelsets and designing a device for monitoring this parameter

Research on acoustic localization method for multiple leakage sources of water-cooling wall in furnace

Abstract The water-cooling wall of the furnace often leaks under harsh conditions such as high temperature and pressure, which affects the safety and economic operation of the boiler. To achieve the localization of multiple leakage sources in water-cooling wall, an acoustic positioning method based on improved differential evolution and density-based spatial clustering of application with noise (IDE-DBSCAN) is proposed. The nonlinear equation system is constructed using time delay of arrival (TDOA) obtained by the generalized second cross-correlation algorithm, and the IDE algorithm is used to solve nonlinear equation system and obtain multiple spatial positions corresponding to multiple TDOA sequences. To eliminate false leakage sources, the paper presents IDE-DBSCAN to cluster the multiple estimated positions. The number of clusters obtained is the number of leakage sources, and the cluster centers are the location coordinates of the leakage sources. The numerical simulation results show that compared to traditional acoustic positioning methods, IDE-DBSCAN has higher positioning accuracy and better anti-noise ability. In addition, experiments are implemented on multiple leakage sound sources within a measurement area of 1m×1m×1.06m, and the positioning errors remain within 5cm. The proposed method provides a theoretical basis and experimental verification for further research on multiple leakages localization of water-cooling wall in furnace.

Flow analysis of temperature-dependent variable viscosity Phan Thien Tanner fluid thin film over a horizontally moving heated plate

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

Time Optimal and PID Controller for Armed Manipulator Robots

This paper is fourfold. First, three different time-optimal control problems for simulating the manipulator robots with one, two, and three arms are mathematically formulated. Corresponding to the related dynamical systems, the nonlinear system of ordinary differential equations is derived. It has been found that these problems are time-free and have distinct initial and boundary conditions, making them hard to solve. To find the minimum time with different payloads, a successful numerical method based on the finite difference and the three-stage Lobatto formula is proposed. Secondly, the related torquecontrolling problems are simulated, and then for one, two, and three armed manipulator robots, they are solved using the PID controller method. Thirdly, it is shown that, compared to the time-optimal controlling problem, the optimal PID torque controller solution takes more time to do the job than was expected. However, the solution in the PID controller method shows less oscillation than the time-optimal control problem. Fourthly, mathematical theories are used, and the numerical results for both methods and different payloads are compared.

Computational investigation of methanol-based hybrid nanofluid flow over a stretching cylinder with Cattaneo–Christov heat flux

Abstract This study investigates heat transfer rates in (AA7075-AA7072/Methanol) hybrid nanofluid flows, considering non-uniform heat sources and Cattaneo–Christov heat flux, with significant implications for aerospace engineering by enhancing thermal management in aircraft engines. The findings could revolutionize automotive cooling system efficiency, optimize heat dissipation in electronic devices, and advance the design of renewable energy systems such as concentrated solar power plants. The study aims to conduct a comparative analysis of (AA7075/Methanol) nanofluid and (AA7075-AA7072/Methanol) hybrid nanofluid flow, examining heat transfer rates, non-uniform heat sources, and Cattaneo–Christov heat flux theory around a stretching cylinder. Thermal radiation and the Biot number are also evaluated. Two different nanoparticles, AA7072 and AA7075, are used with methanol to create AA7075/Methanol nanofluid and AA7075-AA7072/Methanol hybrid nanofluid. The study compresses the resultant non-linear partial differential equation system and applies suitable similarity transformations to reduce the governing partial differential equations with boundary conditions to dimensionless form. The BVP4C shooting method in MATLAB is employed to numerically and graphically solve these dimensionless ordinary differential equations. The results indicate that higher curvature parameter values correlate with increased velocity and temperature distribution profiles. A rise in nanoparticle volume fraction reduces the radial velocity profile but increases the temperature profile. Temperature distribution profiles increase with higher thermal radiation parameter and Biot number values, while higher thermal relaxation parameter values decrease temperature. Additionally, thermal distribution profiles rise with increasing values of both the time-dependent heat source constant and space-dependent heat source parameter.

Talbot Effect for the Manakov System on the Torus

In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Logarithmic Chelyshkov functions for one- and two-dimensional nonlinear Caputo–Hadamard fractional Rosenau equation

In this work, the Caputo–Hadamard time fractional version of the Rosenau equation in one and two dimensions is introduced. The logarithmic Chelyshkov functions (LCFs), as a novel family of basis functions, are introduced and used together with the classical Chelyshkov polynomials (CCPs) to established a hybrid method to solve the expressed fractional equations. For this aim, the ordinary and fractional derivative matrices associated with these basis functions are determined. To establish the desired approach by considering a hybrid expansion of the fractional problem’s solution using the CCPs (for the spatial variables) and LCFs (for the temporal variable), and employing the derived derivative matrices, solving the fractional problem under consideration turns into solving an algebraic system of nonlinear equations. Some numerical examples are considered to investigate the capability and accuracy of this proposed scheme.

Superconvergent scheme for a system of Green nonlinear Fredholm integral equations

This study investigates a system of second-kind nonlinear Fredholm integral equations (SSNFIEs) featuring a Green's type kernel function. To solve this integral equations (IEs), we introduce Galerkin and iterated Galerkin (IG) methods based on piecewise polynomials. Convergence and error analysis are conducted for these methods, and we demonstrate superconvergence for the IG method. Numerical tests are employed to validate the theoretical findings.

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Mathematical Model of Basal Stem Rot (BSR) Disease Spread in Oil Palm Plants

This study discusses the stability analysis of mathematical models for the spread of Basar Stem Rot(BSR) in oil palm plants. In developing this mathematical model, several assumptions are taken to obtain a model that is suitable for the spread of BSR disease. The resulting model is a system of first-order nonlinear differential equations with three variables. This research includes both analytical and numerical analysis. Analytical analysis includes determination of equilibrium points and local stability analysis, while numerical analysis is conducted using Microsoft Excel application. From this study, two equilibrium points were found with stability conditions that depend on the fulfillment of certain conditions. One important result obtained is that the equilibrium point will be locally stable if and only if α > μ and b > √D, where D is the discriminant of a quadratic equation. After analyzing analytically, the study continued with numerical simulations to illustrate and test the analytical results. Numerical results in the form of graphs show that the solution of the system is stable, which indicates that the disease will be endemic under certain conditions and time. This research provides a deeper understanding of the dynamics of the spread of BSR and the conditions that affect the stability of the spread of the disease. With this analysis, it is expected to contribute to efforts to control BSR disease in oil palm plants. In addition, this research also opens opportunities for the development of similar mathematical models for other plant diseases.

Mathematical Assessment of the Role of Human Behavior Changes on SARS-CoV-2 Transmission Dynamics in the United States.

The COVID-19 pandemic has not only presented a major global public health and socio-economic crisis, but has also significantly impacted human behavior towards adherence (or lack thereof) to public health intervention and mitigation measures implemented in communities worldwide. This study is based on the use of mathematical modeling approaches to assess the extent to which SARS-CoV-2 transmission dynamics is impacted by population-level changes of human behavior due to factors such as (a) the severity of transmission (such as disease-induced mortality and level of symptomatic transmission), (b) fatigue due to the implementation of mitigation interventions measures (e.g., lockdowns) over a long (extended) period of time, (c) social peer-pressure, among others. A novel behavior-epidemiology model, which takes the form of a deterministic system of nonlinear differential equations, is developed and fitted using observed cumulative SARS-CoV-2 mortality data during the first wave in the United States. The model fits the observed data, as well as makes a more accurate prediction of the observed daily SARS-CoV-2 mortality during the first wave (March 2020-June 2020), in comparison to the equivalent model which does not explicitly account for changes in human behavior. This study suggests that, as more newly-infected individuals become asymptomatically-infectious, the overall level of positive behavior change can be expected to significantly decrease (while new cases may rise, particularly if asymptomatic individuals have higher contact rate, in comparison to symptomatic individuals).

A Theoretical Approach to Understand the Nonlinear Processes in Enzymatic Electrochemical Biosensors.

The enzymatic biosensors' response can be monitored based on the results of nonlinear differential equations. The nonlinear reaction-diffusion equations proposed for this enzyme-based electrochemical biosensor include a nonlinear term associated with Michaelis-Menten kinetics. Herein, the system of nonlinear reaction-diffusion equations is solved using a modified homotopy perturbation method. For all values of the rate constants, the approximate analytical expressions for the concentration profiles, current, sensitivity, and gradient of biosensor have been determined. Performance factors of an enzymatic electrochemical biosensor, such as response time, sensitivity, accuracy, and resistance, are discussed. The analytical results and numerically simulated outcomes using Matlab software have been compared.

Niche center identification differential evolution for multimodal optimization problems

Niching techniques are commonly incorporated into evolutionary computation (EC) algorithms to address multimodal optimization problems (MMOPs). Nevertheless, identifying proper individuals as niche centers remains the main challenge in niching techniques. Generally, niche centers should possess promising fitness (fitness aspect) and should be dispersedly distributed in different search regions (distance aspect). In this study, we propose a novel niching technique known as niche center identification (NCI) and integrate it with differential evolution (DE) for tackling MMOPs, termed NCIDE. In NCI, niche centers are first identified from both the fitness and distance aspects. Individuals that are not niche centers are added to their nearest niche centers to form niches. Moreover, we develop a niche-level archival-adaptive parameter scheme (NAAPS) to adaptively adjust the parameters at the niche level and reduce their sensitivity. Meanwhile, with the help of an archive, we can preserve the identified optima and reinitialize stagnant individuals for further exploration. The experimental results on the CEC2013 multimodal benchmark test suite demonstrate that NCIDE significantly outperforms several state-of-the-art multimodal algorithms, including multiple competition winners from CEC2015 and GECCO2017-GECCO2019. Finally, NCIDE is applied to solve multimodal nonlinear equation system (NES) problems to further illustrate its practical applicability.

Stability of Periodic solutions by Krasnoselskii fixed point theorem of neutral nonlinear system of dynamical equation with Variable Delays

The fixed point theorem is used in this study to provide stability results for the zero solution of a nonlinear neutral system of differential equations with functional delay.

Semi-analytical solution of MHD free convective flow of a nanofluid over an exponentially stretching porous sheet in the presence of heat source/sink

In this study, the magnetohydrodynamic flow and heat transfer of electrically conducting nanofluid over an exponentially stretching porous sheet is investigated. By introducing similarity variables, the non-linear set of partial differential governing equations, are transformed into a system of ordinary nonlinear differential equations. The equations are solved semi-analytically by the Optimal Collocation Method (OCM) which is a powerful method to solve equations containing infinite boundary conditions. The accuracy of the OCM results is examined by the Richardson method. The effects of the different physical parameters such as magnetic field strength, nanofluid volume fraction, suction strength, and the heat sink/source value are discussed on the flow and heat transfer characteristics of the problem. The obtained results show that the velocity and temperature of nanofluid increase with the increase in volume fraction. Also, when suction increases, the thickness of the boundary layers decreases. As the strength of the magnetic field (Hartmann) increases, the temperature slightly increases but the flow rate of the nanofluid decreases. The skin friction coefficient and Nusselt number increase when the values of suction parameters, volume fraction, magnetic field, and the field angle increase.

Two-step inertial derivative-free projection method for solving nonlinear equations with application

Large-scale systems of nonlinear equations play a fundamental role in various applications, spanning from differential equations to economics, engineering, management science, probability theory, and various other applied sciences. This paper proposes a derivative-free iterative method to find ϵ-approximate solutions of large-scale nonlinear equations. The proposed scheme combines the hyperplane projection technique with a two-step inertial extrapolation, presenting a new approach. The proposed method does not require Jacobian information and thus can be applied to solve nonsmooth equations. Under standard assumptions, we prove the proposed method’s global convergence and nonasymptotic O(1/k) convergence rate. Furthermore, we include numerical results and comparisons to demonstrate the efficiency of the proposed method. Finally, we demonstrate the applicability of the proposed method in solving regularized decentralized logistic regression, a popular problem in machine learning applications.

A Jacobian smoothing inexact Newton method for solving the nonlinear complementary problem

In this paper, we propose a Jacobian smoothing inexact Newton-type algorithm for solving the nonlinear complementarity problem by reformulating it as a system of nonlinear equations. We show that the algorithm converges up to q-quadratically and present numerical experiments that show its good local performance. In order to compare in terms of time our algorithm with other known algorithms, we introduce a new normalized measurement that we call time index.

A residual-based weighted nonlinear Kaczmarz method for solving nonlinear systems of equations

A residual-based weighted nonlinear Kaczmarz method for solving nonlinear systems of equations