System Of Nonlinear Equations Research Articles

Rate Lifting for Stochastic Process Algebra by Transition Context Augmentation

This article presents an algorithm for determining the unknown rates in the sequential processes of a Stochastic Process Algebra (SPA) model, provided that the rates in the combined flat model are given. Such a rate lifting is useful for model reverse engineering and model repair. Technically, the algorithm works by solving systems of nonlinear equations and—if necessary—adjusting the model’s synchronisation structure, without changing its transition system. The adjustments cause an augmentation of a transition’s context and thus enable additional control over the transition rate. The complete pseudo-code of the rate lifting algorithm is included and discussed in the article, and its practical usefulness is demonstrated by two case studies. The approach taken by the algorithm exploits some structural and behavioural properties of SPA systems, which are formulated here for the first time and could be very beneficial also in other contexts, such as compositional system verification.

Mathematical Analysis of Nonlinear Reaction Diffusion Process at Carbon Dioxide Absorption in Concentrated Mixtures of 2-Amino-2-Methyl-1-Proponal and 1,8-Diamino-p-Methane

A mathematical model of carbon dioxide (CO2) absorption in an aqueous solution consisting of two reactants, 2-amino-2-methyl-1-proponal and 1,8-diamino-p-methane is considered. Akbari Ganji Method and Differential Transform Method are implemented to resolve the system of nonlinear equations, yielding an analytical formulation for the concentration of carbon dioxide, 2-amino-2-methyl-1-proponal, 1,8-diamino-p-methane and the molar flux in-terms of reaction rate constants. The obtained analytical findings are used to evaluate the different diffusion parameters and compared with numerical results. By assessing Matlab results with analytical findings, a successful outcome is discovered. Moreover, the influence of parameters on molar flux is examined. Also, graphical representations are presented and discussed here. The new analytical results contribute to optimizing the consistency of this model. The ensuring outcomes have been verified utilizing the existing numerical data with prior findings, and we are then presented with an adequate level of agreement.

Non-Similar Analysis of Mixed Convection Biomagnetic Boundary Layer Flow Over a Vertical Plate with Magnetization and Localized Heating/Cooling

Theoretical and numerical investigation of an applied magnetic field on mixed convection flow of a biofluid through a vertical plate using contained heating or cooling is observed in this study. The mathematical formulation is that of the full Biomagnetic Fluid Dynamics (BFD) model which deals with on the ferrohydrodynamics (FHD) and magnetohydrodynamics (MHD) principle. In this work, the study is performed on a specific biofluid, viz. human blood. Assume that the magnetization very linearly with magnetic field strength, temperature dependency of dynamic viscosity and thermal conductivity is noticed. A system of non-linear equations with appropriate boundary condition is obtained by familiarizing suitable non-dimensional variables in the physical problem. For the numerical solution, we used finite difference method which is based on an efficient technique is applied in the problem. Computations for flow profiles, local skin friction coefficient and local heat transfer coefficient are performed with the magnetic parameter Mn, the viscosity/temperature parameter θr and the thermal/conductivity parameter S∗. The effect of the localized heating or cooling is examined. The computational results presented graphically and have been validated in an appropriate manner. The study reveals that the impact of a magnetic field for blood flow in arteries is found significantly. The results presented bear the promise of valuable applications in physiology, medicine and bioengineering.

Numerical solution of generalized fractional Lane–Emden-Fowler equation using Haar wavelet method

This paper presents a numerical technique to solve the generalized fractional Lane–Emden–Fowler equation. We use the Haar wavelet technique to approximate the solution of the generalized fractional Lane–Emden–Fowler equation. In this method, a differential equation transforms into a system of nonlinear equations, and this system is further solved to obtain Haar coefficients. We also showed the convergence of this method by establishing an upper bound. The rate of convergence is also discussed in our algorithm. Many numerical examples have been taken, and the results of those numerical experiments have been written in tabular form. We also documented the experiments graphically to show the efficiency and accuracy of this method.

Mathematical modeling of unemployment dynamics with skills development and cyclical effects

In the present work, a new mathematical model is introduced to investigate the effects of skills development on the governance of cyclical unemployment. A system of non-linear differential equations is formed and analyzed. In the modeling approach, we categorize the considered workforce into three distinct classes: the fundamentally unemployed Us, employed individuals E and cyclically unemployed people Uc. We also make the assumption that some government programs can qualify individuals who have been out of work for a period to reintegrate and prevent them from becoming long-term unemployed. The qualitative analysis of differential equations is used to analyze the proposed model. It has been demonstrated that the suggested model has only one equilibrium that is globally asymptotically stable under particular conditions. The impact of key parameters is analyzed by examining their sensitivity. In order to verify the important theoretical results, numerical illustrations of the model have been performed. The obtained results demonstrate that the development of skills among the cyclically unemployed is effective in reducing unemployment.

Hybrid Nanofluid Flow over a Shrinking Darcy-Forchheimer Porous Medium with Shape Factor and Solar Radiation: A Stability Analysis

This research aimed to develop a numerical solution to analyze the effects of solar radiation and nanoparticle shape factors on the flow of a hybrid nanofluid past a shrinking Darcy-Forchheimer porous medium. The base fluid chosen for this study is water (H2O), and the hybrid nanofluid consists of nanoparticles of silver (Ag) and titanium dioxide (TiO2) in four different shapes: bricks, cylinders, platelets, and blades. To account for solar radiation, the energy model incorporated a radiative heat flux, while the momentum problem considers the influence of a magnetic field. The application of an appropriate similarity transformation method converts the partial differential equations (PDEs) model into a system of nonlinear ordinary differential equations (ODEs). The mathematical model is solved using the shooting technique method and the bvp4c solver. The obtained results, along with the effects of the nanoparticle shape factor, solar radiation parameter, shrinking parameter, Darcy-Forchheimer number, and nanofluid volume fraction, are visually presented through figures and tables. It is worth noting that, in our numerical results, we observed the presence of dual solutions when λ < 0. Our findings indicate that the thermal transmittance increases with an increase in the nanoparticle shape factor and solar radiative parameter. Additionally, we observed an escalation in the velocity distribution in relation to the shrinking parameter and nanofluid volume fraction. Before reaching the two solutions, a flow stability analysis revealed that the first branch appears to be the most stable. Overall, these findings provide valuable insights into the behaviour of hybrid nanofluid flow in the presence of solar radiation and porous media.

Applications of a variable anchoring iterative method to equation and inclusion problems on Hadamard manifolds

In this paper, we introduce a new iterative technique with a variable anchoring operator for reckoning the solution of a variational inequality problem over the set of the common fixed points of a nearly nonexpansive sequence of operators in the framework of Hadamard manifolds. We also establish a convergence result on the proposed algorithm for approximating a solution of the problem, under suitable assumptions. We apply our results for finding the solutions of a system of nonlinear equations, and of inclusion problems to support their utility. Our work improves results in the recent literature. Numerical simulations are given for a better understanding of the effectiveness of our outcomes.

Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization

Orifice Hollow Cathodes are electric devices necessary for the functioning of common plasma thrusters for space applications. Their reliability mainly depends on the success of a spacecraft’s mission equipped with electric propulsion. The development of plasma models is crucial in the evaluation of plasma properties within the cathodes that are difficult to measure due to the small dimensions. Many models, based on non-linear systems of plasma equations, have been proposed in the openiterature. These are solved commonly by means of iterative procedures. This paper investigates the possibility of solving them by means of the Particle Swarm Optimization method. The results of the validation tests confirm the expected trends for all the unknowns; the confidence bound of the discharge current as a function of mass flow rate is very narrow (2 ÷ 5 V); moreover, the results match very well the experimental data except at theowest mass flow rate (0.08 mg/s) and discharge current (1A), where the computations underpredict the discharge current to the utmost by 40%. The highest data dispersion regards the plasma density in the emitter region (±20% of the average value) and the wall temperatures (±50 K with respect to the average values) of the orifice and insert; those of the others variables are very tiny.

E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.

Comparison of Genetic Algorithm and Particle Swarm Optimization in Determining the Solution of Nonlinear System of Equations

Nonlinear systems of equations often appear in various fields of science and engineering, but their analytical solutions are difficult to find, so numerical methods are needed to solve them. Optimization algorithms are very effective in finding solutions to nonlinear systems of equations especially when traditional analytical and numerical methods are difficult to apply. Two popular optimization methods used for this purpose are Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). This study aims to compare the effectiveness of GA and PSO in finding solutions to nonlinear systems of equations. The criteria used for comparison include accuracy and speed of convergence. This research uses several examples of nonsmooth nonlinear systems of equations for experimentation and comparison. The results provide insight into when and how to effectively use these two algorithms to solve nonlinear systems of equations as well as their potential combinations

Similarity solution for the magnetogasdynamic shock wave in a self-gravitating and rotating ideal gas under the influence of radiation heat flux

The Lie invariance method is used to analyze the one-dimensional, unsteady flow of a cylindrical shock wave in a rotating, self-gravitating, radiating ideal gas under the influence of an axial or azimuthal magnetic field, with an emphasis on adiabatic conditions. The analysis assumes a stationary environment just ahead of the shock wave and considers variations in fluid velocity, magnetic field, and density within the perturbed medium just behind the shock front. In the governing equations, the impact of thermal radiation under an optically thin limit is integrated into the energy equation. Utilizing the Lie invariance method, the set of partial differential equations governing the flow in this medium is transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity variables. Two distinct cases of similarity solutions are obtained by selecting different values for the arbitrary constants associated with the generators. Among these cases, one yields similarity solutions assuming a power-law shock path and the other an exponential-law shock path. For both cases, the resulting set of nonlinear ODEs are numerically solved using the 4th-order Runge–Kutta method in MATLAB software. The article thoroughly explores the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), σ (ambient density exponent), l1 (rotational parameter), and G0 (gravitational parameter) on the flow properties. The findings are visually presented to offer a comprehensive insight into the effects of these parameters.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

Implementation and Linearization of a Coupled Panel and Vortex Particle Method in State-Space Form

This paper describes the implementation and linearization of a coupled panel and vortex particle method in a state-variable form. More specifically, the coupled panel and vortex particle dynamics are formulated as a nonlinear system of ordinary differential equations in first-order form to be self-contained and inherently linearizable. Linearization of this coupled panel and vortex particle method is demonstrated via finite differencing and a novel analytical linearization technique to yield a linear time-invariant representation. The code is implemented in MATLAB® and validated against an open-source aerodynamic solver for a fixed wing in both stationary and unsteady conditions. The linearized models are verified against the nonlinear dynamics both in the time and frequency domains. Linearized models accurately represent the wake dynamics about the equilibrium condition for moderate amplitude inputs and for input frequencies covering the typical frequency range of flight dynamics and flight controls, i.e., 0.3–30 rad/s. Analytical linearization is shown to abate the cost of linearization over perturbation methods by O(n2), where n is the total number of states of the system. Linearized models of the coupled panel and vortex particle dynamics have applications in the flight dynamics of both fixed- and rotary-wing vehicles, where these dynamics can be used to augment the rigid-body dynamics. The coupled rigid-body and wake dynamics can be leveraged to assess the stability, response characteristics, and handling qualities of vehicles experiencing aerodynamic interactions between rotors, wings, and/or obstacles.

Parametrically excited shape distortion of a submillimeter bubble

The existence of finite amplitude shape distortion caused by parametrically excited surface instabilities for a gas bubble in water driven by a temporally periodic, spatially uniform pressure field in an axisymmetric geometry is investigated. Employing a nonlinear coupled system of equations which includes shape mode interactions to third order, the resultant spherical oscillations, translation, and shape distortion of the bubble are modelled, placing no restriction on the size of the spherical oscillations. The model accounts for viscous and thermal damping with compressibility effects. The existence of synchronous and higher order parametrically induced sustained, finite amplitude, periodic shape deformation is demonstrated. The excitement of an odd shape mode via the synchronous mechanism is shown to give rise to linear bubble self-propulsion. For larger driving amplitudes, it is shown that more than one shape mode can be parametrically excited at the same driving frequency but by different resonance mechanisms, leading to more involved shape deformation and the increased possibility of bubble self-propulsion.

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Converting estimated breeding values from the observed to probability scale for health traits

Dairy cattle health traits are paramount from a welfare and economic viewpoint; therefore, modern breeding programs prioritize the genetic improvement of these traits. Estimated breeding values for health traits are published as the probability of animals staying healthy. They are obtained using threshold models, which assume that the observed binary phenotype (i.e., healthy or sick) is dictated by an underlying normally distributed liability exceeding or not a threshold. This methodology requires significant computing time and faces convergence challenges as it implies a nonlinear system of equations. Linear models have more straightforward computations and provide a robust approximation to threshold models; thus, they could be used to overcome the mentioned challenges. However, linear models yield estimated breeding values on the observed scale, requiring an approximation to the liability scale analogous to that from threshold models to later obtain the estimated breeding values on the probability scale. In addition, the robustness of the approximation of linear to threshold models depends on the amount of information and the incidence of the trait, with extreme incidence (i.e., ≤ 5%) deviating from optimal approximation. Our objective was to test a transformation from the observed to the liability and then to the probability scale in the genetic evaluation of health traits with moderate and very low (extreme) incidence. Data comprised displaced abomasum (5.1M), ketosis (3.6M), lameness (5M), and mastitis (6.3M) records from a Holstein population with a pedigree of 6M animals, of which 1.7M were genotyped. Univariate threshold and linear models were performed to predict breeding values. The agreement between estimated breeding values on the probability scale derived from threshold and linear models was assessed using Spearman rank correlations and comparison of estimated breeding values distributions. Correlations were at least 0.95, and estimated breeding value distributions almost entirely overlapped for all the traits but displaced abomasum, the trait with the lowest incidence (2%). Computing time was ∼3x longer for threshold than for linear models. In this Holstein population, the approximation was suboptimal for a trait with extreme incidence (2%). However, when the incidence was ≥6%, the approximation was robust, and its use is recommended along with linear models for analyzing categorical traits in large populations to ease the computational burden.

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

On the Role of Early Case Detection and Treatment Failure in Controlling Tuberculosis Transmission: A Mathematical Modeling Study

Tuberculosis (TB) remains a pressing global health concern, demanding urgent attention to mitigate its spread and impact. In this study, we present a rigorous mathematical model of TB transmission that incorporates early case detection and addresses the critical issue of treatment failure. Through the development of a system of nonlinear ordinary differential equations, we conduct comprehensive analyses to assess the dynamics of TB transmission and the efficacy of intervention strategies. Our findings underscore the urgent need for effective TB control measures. Mathematical analyses reveal that the model exhibits a TB-free equilibrium, which is globally asymptotically stable only if the control reproduction number falls below one. However, we identify a concerning phenomenon: the model demonstrates a forward bifurcation when the control reproduction number equals one, suggesting that the disease-free equilibrium loses its stability, while simultaneously, the stable unique endemic equilibrium begins to emerge. Moreover, sensitivity analysis highlights the complex interplay between case detection rates, treatment failure probabilities, and TB transmission dynamics. Contrary to expectations, increasing case detection rates and minimizing treatment failure probabilities may not consistently reduce the basic reproduction number or the size of the infected population. Instead, there exists a critical threshold for intervention effectiveness, beyond which TB transmission can be significantly curtailed. Biologically, this phenomenon may occur if there is no balance between case detection and treatment efforts. If treatment quality does not improve, then case detection will not have a significant impact, and in the worst case scenario, it can exacerbate the intervention’s negative effects. These findings underscore the urgency of implementing targeted intervention strategies to combat TB transmission effectively. Failure to meet the critical intervention threshold risks undermining TB elimination efforts and exacerbating the global TB burden. Through numerical simulations, we elucidate potential intervention scenarios necessary for achieving TB elimination goals in human populations. In conclusion, our study highlights the urgent imperative for coordinated action to control TB transmission effectively. By elucidating the dynamics of TB spread and intervention efficacy, we provide valuable insights to inform evidence-based policy decisions and accelerate progress towards TB elimination on a global scale.

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