System Of Nonlinear Equations Research Articles

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

Rate Lifting for Stochastic Process Algebra by Transition Context Augmentation

This paper 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 paper, 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.

A space-time Caputo fractional order and modified homotopy perturbation method for evaluating the pathological response of tumor-immune cells

Tumors result from genetic mutations or environmental factors that prompt cells to divide uncontrollably. This study aims to examine the behavior of tumor-immune cell growth in the presence of chemotherapy drug diffusion at a Caputo fractional order. To accomplish this, we employed the modified homotopy perturbation method to solve a proposed system of nonlinear differential equations. We obtained the analytical solutions to study the spatiotemporal pathological response of tumor-immune cell growth. Our analysis also considered the impact of the Caputo-fractional order on the system's dynamics and compared the results with the classical integer-order scenario. Our findings demonstrated that the proposed method is an effective and precise technique for understanding the intricate interactions of tumor-immune cell growth. Additionally, we revealed that the Caputo-fractional order plays a significant role in the system's behavior and should not be overlooked in future analyses of such systems. In conclusion, this study holds important implications for cancer research by providing insights into the behavior of tumor-immune cell growth in the presence of time-fractional administration of chemotherapy drugs.

Correction: A Bregman–Kaczmarz method for nonlinear systems of equations

Correction: A Bregman–Kaczmarz method for nonlinear systems of equations

Quasilinearization variational iteration method for system of nonlinear ODEs

In this manuscript, we discuss a new technique for solving system of nonlinear differential equations, which is a modification of the variation iteration method (VIM) implemented using the quasilinearization method and Adomian’s polynomial. The quasilinearization variational iteration method (QVIM) is the name given to this proposed method. The proposed method’s convergence analysis in Banach space is also discussed here. Three application problems, including the Genesio-Tesi system, are considered to test the applicability of our approach. We also discuss the case study of the chaotic and non-chaotic solutions of the Genesio-Tesi system (GTS). The convergence behaviour of the method is studied for various values of parameter x. To assess the viability and efficacy of QVIM, we compare it to the existing well-known Adomian decomposition method. The results show that the proposed method is highly efficient and simple to implement.

Exploration of thermal radiation and stagnation point in MHD micropolar nanofluid flow over a stretching sheet with Navier slip

The quest to strengthen heat conduction of thermal science base fluid for effective industrial outputs and engineering derives has recently increased. Thus, this study aims to determine how thermal radiation and slip effects affect the flow of a micropolar nanofluid near a stagnation point over an extending sheet. Using similarity transformations, the flow-controlling partial differential equations (PDEs) are turned into a set of non-linear ordinary differential equations (ODEs). The non-linear system of equations has been solved by the numerical technique Runge-Kutta-Fehlberg integration scheme implementing the shooting technique with suitable conditions to generate a numerical solution. The essential factors affecting the flow are depicted graphically and tabularly. Additionally, a comparison is conducted between the present result and previously published data on the Nusselt and Sherwood numbers; it claims that thermophoresis and Brownian motion vary under some restrictive conditions. An increase in the magnetic field parameter was found to boost the velocity of the micropolar nanofluid. In contrast, a rise in the micropolar parameter reduces the angular velocity.

Mathematical Model of Closed Microecosystem “Algae – Heterotrophic Bacteria”

Model of closed microecosystem “algae - heterotrophic bacteria” is proposed in this paper. Mathematical model is the Cauchy problem for system of nonlinear ordinary differential equations. To develop the model the Liebig’s law of the minimum is consistently used for both specific rate of biomass growth and specific death rate of algae and bacteria cells. To describe the specific rate of substrate utilization by algae and bacteria the Andrew’s model (substrate inhibition) is used. It is assumed that specific death rate of algae and bacteria cells increases with decreasing substrate concentration. It is also assumed that carbon and nitrogen are main biogenic elements, and in the system they are in the form of mineral substrate (CO2, NO2, NO3, NH4) and biological substrate (proteins, lipids and carbohydrates). Mathematical model describing time variations in concentration of elements of microecosystem is formulated under the following assumptions: 1) stoichiometric coefficients of algae and bacteria cells are constant in the development of microecosystem; 2) utilization of carbon and nitrogen by algae and bacteria occurs independently; 3) oxygen produced by algae cells during photosynthesis completely covers the demand for oxygen for algae and bacteria cells. To verify the proposed model experimental data for microecosystems «Clorella vulgaris – Pseudomonas sp.» и «Scenedesmus obliquus – Pseudomonas sp.» are used. These systems were studied in laboratory conditions, and concentrations of elements of microecosystems in stationary state were obtained. Parameters of functions describing specific rate of utilization of biogenic elements were derived from experimental data for growth kinetics of algae and bacteria. Concentration of the biomass in stationary state obtained with the use of the proposed model is in reasonable agreement with experimental data.

Overview of the Application of Physically Informed Neural Networks to the Problems of Nonlinear Fluid Flow in Porous Media

To describe unsteady multiphase flows in porous media, it is important to consider the non-Newtonian properties of fluids by including rheological laws in the hydrodynamic model. This leads to the formation of a nonlinear system of partial differential equations. To solve this direct problem, it is necessary to linearize the equation system. Algorithm construction for inverse problem solution is problematic since the numerical solution is unstable. The application of implicit methods is reduced to matrix equations with a high rank of the coefficient matrix, which requires significant computational resources. The authors of this paper investigated the possibility of parameterized function (physics-informed neural networks) application to solve direct and inverse problems of non-Newtonian fluid flows in porous media. The results of laboratory experiments to process core samples and field data from a real oil field were selected as examples of application of this method. Due to the lack of analytical solutions, the results obtained via the finite difference method and via real experiments were proposed for validation.

The effects of spatially-constrained treatment regions upon a model of wombat mange

The use of therapeutic agents is a critical option to manage wildlife disease, but their implementation is usually spatially constrained. We seek to expand knowledge around the effectiveness of management of environmentally-transmitted Sarcoptes scabiei on a host population, by studying the effect of a spatially constrained treatment regime on disease dynamics in the bare-nosed wombat Vombatus ursinus. A host population of wombats is modelled using a system of non-linear partial differential equations, a spatially-varying treatment regime is applied to this population and the dynamics are studied over a period of several years. Treatment could result in mite decrease within the treatment region, extending to a lesser degree outside, with significant increases in wombat population. However, the benefits of targeted treatment regions within an environment are shown to be dependent on conditions at the start (endemic vs. disease free), as well as on the locations of these special regions (centre of the wombat population or against a geographical boundary). This research demonstrates the importance of understanding the state of the environment and populations before treatment commences, the effects of re-treatment schedules within the treatment region, and the transient large-scale changes in mite numbers that can be brought about by sudden changes to the environment. It also demonstrates that, with good knowledge of the host-pathogen dynamics and the spatial terrain, it is possible to achieve substantial reduction in mite numbers within the target region, with increases in wombat numbers throughout the environment.

Existence of common fuzzy fixed points via fuzzy F-contractions in b-metric spaces

The main goal of this study is to establish common fuzzy fixed points in the context of complete b-metric spaces for a pair of fuzzy mappings that satisfy F-contractions. To strengthen the validity of the derived results, non-trivial examples are provided to substantiate the conclusions. Moreover, prior discoveries have been drawn as logical extensions from pertinent literature. Our findings are further reinforced and integrated by the numerous implications that this technique has in the literature. Using fixed point techniques to approximate the solutions of differential and integral equations is very useful. Specifically, in order to enhance the validity of our findings, the existence result of the system of non-linear Fredholm integral equations of second-kind is incorporated as an application.

Transport phenomena in pasta drying: a dough-air double domain advanced modeling

In the realm of pasta drying, where precise prediction of temperature and water-content distribution is crucial, the application of modeling and simulation emerges as highly significant. This study endeavors to evaluate operational conditions ensuring the production of a safe, high-quality pasta product. An advanced two-domain model was devised and solved to estimate transport phenomena in both the food and air domains. Unlike conventional models relying on specified interfacial heat and mass transfer coefficients, the proposed model serves as a versatile tool applicable across a broad spectrum of process and fluid-dynamic conditions in real pasta driers. The system of non-linear unsteady-state partial differential equations governing the behavior of a “Rigatone" pasta sample in a drier was solved using the finite elements method to estimate how air characteristics influence drying performance. A pilot-scale drying chamber was used for model validation for inlet air temperature Ta,in=80°C at relative humidity values (RHin) of 50 % and 60 %, with error consistently below 10 %.

Thermal and solutal transport by Cattaneo-Christov model for the magnetohydrodynamic Williamson fluid with joule heating and heat source/sink

This article scrutinizes the 2-dimensional and boundary layer flow of magnetohydrodynamic Williamson fluid flowing on a stretchable surface with variable viscosity. The thermal and solutal rates are examined through the Cattaneo-Christov model with Joule heating, heat source/sink, and chemical reaction. The authors are motivated to conduct this study because of its practical and scientific significance in various processes, including polymer processing, textile industries, food industries, solar energy, biomedical science, wind turbine blades, oil spill clean-up, metal rolling, and forging. With the mentioned assumptions, the partial differential equations are achieved by using the basic governing laws, including momentum law, energy law, and concentration law. This non-linear system of equations is transmuted into ordinary differential equations by taking similarity transformations. The main novelty behind the conduction of this work is the numerical technique, namely the ‘Adams-Milne (Predictor-Corrector)’ method along with the Runge-Kutta technique on Matlab software, which has not previously been studied by any researcher in the literature. The analytical solution of the determined equations is not possible due to their highly non-linear nature; therefore the multistep numerical method namely the ‘Adams-Milne (Predictor-Corrector)’ method, along with the Runge-Kutta technique is used to determine the numerical results. The outcomes are noted due to numerous parameters for velocity, temperature, and concentration profiles. The explanation of graphical and numerical results is discussed here. The graphical impression of the Williamson parameter reveals that the velocity and temperature curves diminish for higher inputs of this parameter. The movement of fluid shows the declining behavior for the Hartmann number and viscosity parameter. The solutal and thermal findings due to Cattaneo-Christov heat and mass relaxation coefficients mark the reducing behaviour in respective field. The rise in reaction coefficient decreases the mass distribution. The analyses of comparison of results are also presented here.

Thermal energy transport in stratified 2D-Casson fluid flow over an inclined exponentially stretching surface with Soret/Dufour effects: Numerical simulations and applications in energy harvesting

Significance: The thermal energy transfer in nanofluid flow over an exponentially stretching surface has crucial practical configurations in various industrial processes, and it has potential applications in heat exchangers, chemical engineering, energy harvesting, and material processing. Purpose: This study is devoted to exploring the features of free convection in the thermally stratified, unsteady flow of Casson fluid over an inclined, exponentially stretching surface. Moreover, the implications of nonlinear thermal radiation, activation energy, and thermal/salute stratification effects are examined over the thermal energy transport distributions. Diffusion-thermo and thermo diffusion impressions are also taken into consideration. Methodology: By introducing reasonable transformations, partial differential equations are altered into ordinary differential equations. A nonlinear system of differential equations is solved numerically by employing the Midrich numerical technique. Findings: The impacts of diverse fluid parameters like the Soret/Dufour number, temperature difference parameter, radiation parameter, thermal/salute stratification parameter, magnetic parameter, and Prandtal number are assessed and depicted in plots by explaining the physical justifications of each parameter. Also, numerical values of sink friction and local Nusselt and Sherwood numbers are computed and examined for different values of pertinent variables involved in the problems. It is found that the rate of thermal energy transport is significantly enhanced by the larger estimation of the radiation parameter. Furthermore, it is perceived that the escalation in the temperature ratio constant leads to increased thermal convection in the fluid, while the larger thermal stratification constant decays the rate of heat transport in the fluid. Additionally, the rate of thermal transport is de-escalated due to the escalation in the intensity of thermal stratification parameter.

Convergent scheme for a non-local transport system modeling dislocations dynamics

In this paper, we consider a diagonal hyperbolic, not necessarily strictly hyperbolic, nonlinear and non-local system of transport equations that arises in the theory of dislocations densities dynamics. Under certain monotony assumptions on the initial data, global existence and uniqueness results of Hloc1 solutions for this system was proved in Hajj (2007). We propose a natural implicit scheme verifying a gradient L2 estimate at the discrete level compatible with that obtained in the continuous case in Hajj (2007). Which leads us to prove the convergence of the numerical solution to the continuous one. To our knowledge, this is the first convergent scheme proposed for this model.

A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion

This paper presents the computational solutions of a time-dependent nonlinear system of partial differential equations (PDEs) known as the Lotka-Volterra competition system with diffusion. We propose a combined semi-discretized spectral matrix collocation algorithm to solve this system of PDEs. The first part of the algorithm deals with the time-marching procedure, which is performed using the well-known Taylor series formula. The resulting linear systems of ordinary differential equations (ODEs) are then solved using the spectral matrix collocation technique based on the novel Touchard family of polynomials. We discuss and establish the error analysis and convergence of the proposed method. Additionally, we examine the stability analysis and the equilibrium points of the model to determine the stability condition for the system. We perform numerical simulations using diverse model parameters and with different Dirichlet and Neumann boundary conditions to demonstrate the utility and applicability of our combined Taylor-Touchard spectral collocation algorithm.

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

Using the State Space Method for Calculating Electromechanical Transients

The state space method is widely used for the analysis and synthesis of automatic control systems in a wide variety of industries. It was revealed that the analysis of electromechanical processes in electrical systems is associated with the solution of a system of nonlinear differential equations by very laborious mathematical methods. It is shown how to represent a system of differential equations describing electromechanical transients in synchronous machines in d, q, 0 – coordinates, in matrix form. An algorithm for calculating electromechanical transients by the state space method is considered. The transit matrix of states is represented by a decomposition formula, the terms of which are determined by matrices with constant elements. Comparison with known numerical calculation methods allows us to conclude that the state space method is a fairly effective way to analyze electromechanical transients in synchronous machines.

Two-Point Localization Algorithm of a Magnetic Target Based on Tensor Geometric Invariant.

Currently, magnetic gradient tensor-based localization methods face challenges such as significant errors in geomagnetic field estimation, susceptibility to local optima in optimization algorithms, and inefficient performance. In addressing these issues, this article propose a two-point localization method under the constraint of overlaying geometric invariants. This method initially establishes the relationship between the target position and the magnetic gradient tensor by substituting an intermediate variable for the magnetic moment. Exploiting the property of the eigenvector corresponding to the minimum absolute eigenvalue being perpendicular to the target position vector, this constraint is superimposed to formulate a nonlinear system of equations of the target's position. In the process of determining the target position, the Nara method is employed for obtaining the initial values, followed by the utilization of the Levenberg-Marquardt algorithm to derive a precise solution. Experimental validation through both simulations and experiments confirms the effectiveness of the proposed method. The results demonstrate its capability to overcome the challenges faced by a single-point localization method in the presence of some errors in geomagnetic field estimation. In comparison to traditional two-point localization methods, the proposed method exhibits the highest precision. The localization outcomes under different noise conditions underscore the robust noise resistance and resilience of the proposed method.

Optimal Temperature Calculation for Multicriteria Optimization of the Hydrogenation of Polycyclic Aromatic Hydrocarbons by NSGA-II Method

Introduction. Multicriteria optimization, taking into account contradicting criteria, is used to improve production efficiency, reduce costs, improve product quality and environmental safety of processes. The literature describes the application of multicriteria optimization for production purposes, including the selection of reaction conditions and improvement of technological processes. In the presented paper, the object of research is the process of hydrogenation of polycyclic aromatic hydrocarbons (PAH) in the production of high-density fuels. To determine the optimal conditions of the process, the problem of multicriteria optimization based on the kinetic model is solved. The criteria include maximizing the yield of targeted naphthenes and conversion of feedstock. The research objective is to create a program implementing the multicriteria optimization non-dominated sorting genetic algorithm-II (NSGA-II). Due to this, it is possible to calculate the optimal temperature for the PAH hydrogenation process on the basis of the kinetic model.Materials and Methods. The NSGA-II genetic algorithm was used to solve the multicriteria optimization problem. Modified parental and survival selection within the Pareto front was also used. If it was necessary to divide the front, solutions based on the Manhattan distance between them were selected. The program was implemented in Python.Results. In the system of ordinary nonlinear differential equations of chemical kinetics, the concentration was designated yi, the conditional contact time of the reaction mixture with the catalyst — τ. The system was solved for the hydrogenation reaction of polycyclic aromatic hydrocarbons. The calculations showed that at τ = 0 y1(0) = 0.025; y2(0) = 0.9; y6(0) = 0.067; y9(0) = 0.008; yi(0) = 0, i = 3–5, 7, 8, 10–20; Q(0) = 1. The process temperature was considered as a control parameter according to two optimality criteria: maximizing the yield of target naphthenes (f1) at the end of the reaction, and maximizing the conversion of feedstock (f2). Values f1 were in the range of 0.43–0.79; conversion — 0.01–0.03; temperature — 200–300 K. The growth of temperature was accompanied by an increase in the yield of target naphthenes and a decrease in the conversion of feedstock. Each solution obtained was not an unimprovable one. When modeling the process of hydrogenation of PAH, an algorithm was launched with a population size of 100 and a number of generations of 100. A program implementing the NSGA-II algorithm was developed. The optimal set of values of the PAH hydrogenation reaction temperature was calculated, which made it possible to obtain unimprovable values of the optimality criteria — maximizing the yield of target naphthenes and conversion of feedstock.Discussion and Conclusion. The NSGA-II algorithm is effective for solving the problem of non-dominance, and deriving the optimal solution for all criteria. Future research should be devoted to the selection of optimal algorithm parameters to increase the speed of the solution. Based on the obtained theoretical optimal conditions of the PAH hydrogenation reaction, it is possible to implement the process in industry

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.