System Of Nonlinear Equations Research Articles

Some Non-Linear Periodic Systems of Difference Equations

Some Non-Linear Periodic Systems of Difference Equations

Jacobi polynomials and the numerical solution of ray tracing through the crystalline lens

The human eye is a fascinating optical system, with the crystalline lens playing a significant role in focusing light onto the retina of the eye. The ray tracing through the crystalline lens problem is a challenging problem in optics. In this paper, the case of a non-homogeneous optical medium is investigated, and the ray equation is numerically solved to get the ray paths. The governing equation is an ODE with a fractional derivative given in the Caputo sense. A novel numerical scheme is based on the Jacobi polynomial collocation technique to tackle this problem. A fast and accurate Broyden’s Quasi-Newton algorithm is applied to solve the nonlinear system of equations obtained from the collocation process. Numerical results are stated in detail to show the efficiency of our technique and are compared with other analytical and numerical methods using tables and illustrated figures, which will be useful to corroborate the clinical and physical data. Ray tracing through the crystalline lens is not only fascinating from a scientific perspective but also has practical implications across various domains, and the proposed scheme is considered a promising and practically reliable method to address such types of applications.

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Abstract The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points.

A knowledge-learning-and-transfer-aided differential evolution for nonlinear equation systems

Nonlinear equation systems (NESs) are common in practical applications, and solving them is an important task in numerical computation. Evolutionary algorithms (EAs) for handling NESs have received considerable attention. EAs generate a large amount of data, which reflect the evolutionary behavior. However, there are still deficiencies in the mining and use of these data. Inspired by the ability of humans to acquire knowledge from past historical experience, this paper proposes a knowledge-learning-and-transfer-aided hybrid niching-based differential evolution (HNDE/KLT). HNDE/KLT aims to acquire knowledge from historical experiences and use them to guide the evolution. The HNDE/KLT include two main features: i) artificial neural networks are embedded in EAs, so the algorithm can learn from successful historical evolutionary information and obtain the relationship between the current individual’s position and the optimal evolutionary direction; ii) a knowledge-transfer-based technique is designed to perform information exchange between different subpopulations, enhancing exploitation efficiency. Experimental results show that the HNDE/KLT can better solve NESs, indicating that learning plus transfer can make problem solving more intelligent. Additionally, we employed our algorithm to solve two real-world problems, and obtained good results.

Haar Wavelet Approach for the Mathematical Model on Hepatitis B Virus

Abstract The Haar wavelet collocation method, a wavelet technique, is discussed in this article to examine the mathematical model of Hepatitis B virus infection. We took into account the HB virus, cytotoxic T lymphocytes (CTL) immune response, birth rate, death rate, and infected and uninfected hepatocytes to identify the dynamics of the hepatitis B virus infection. An ordinary differential equation (ODE) system that is nonlinear makes up this model. Using this method, the Hepatitis B Virus model can be solved by expressing each dependent variable as a Haar wavelet and then converting the system of ordinary differential equations into a system of nonlinear algebraic equations. The unknown coefficient values are thought to be extracted using the collocation procedure and the Newton–Raphson method. Tables and graphs are used to illustrate the characteristics of the Hepatitis B virus. The obtained results show that the current approach outperforms other approaches found in the literature in terms of accuracy. Mathematica software is utilized to obtain numerical results and nature.

Impact of chemical reaction on the Cattaneo–Christov heat flux model for viscoelastic flow over an exponentially stretching sheet

In this article, the numerical solutions for the heat transfer flow of an upper-convected Maxwell fluid across an exponentially stretched sheet with a chemical reaction on the Cattaneo–Christov heat flux model have been investigated. Using similarity transformation, the controlling system of nonlinear partial differential equations was transformed into a system of ordinary differential equations. The resulting converted equations were solved numerically by a successive linearization method with the help of MATLAB software. A graphic representation was created to analyze the physical insights of the relevant flow characteristics. The findings were presented in the form of velocity, temperature, and concentration profiles. As the relaxation time parameter varied, the local Nusselt number increased. The thermal relaxation time was shown to have an inverse relationship with fluid temperature. Furthermore, the concentration boundary layer becomes thinner as the levels of the reaction rate parameter increase. The results of this model can be applicable in biological fluids and industrial situations. Excellent agreement exists between the analysis's findings and those of the previous studies.

Generalization of Bernoulli polynomials to find optimal solution of fractional hematopoietic stem cells model

The study introduces a fractional mathematical model in the Caputo sense for hematopoietic stem cell-based therapy, utilizing generalized Bernoulli polynomials (GBPs) and operational matrices to solve a system of nonlinear equations. The significance of the study lies in the potential therapeutic applications of hematopoietic stem cells (HSCs), particularly in the context of HIV infection treatment, and the innovative use of GBPs and Lagrange multipliers in solving the fractional hematopoietic stem cells model (FHSCM). The aim of the study is to introduce an optimization algorithm for approximating the solution of the FHSCM using GBPs and Lagrange multipliers and to provide a comprehensive exploration of the mathematical techniques employed in this context. The research methodology involves formulating operational matrices for fractional derivatives of GBPs, conducting a convergence analysis of the proposed method, and demonstrating the accuracy of the method through numerical simulations. The major conclusion is the successful introduction of GBPs in the context of the FHSCM, featuring innovative control parameters and a novel optimization technique. The study also highlights the significance of the proposed method in providing accurate solutions for the FHSCM, thus contributing to the field of mathematical modeling in biological and medical research.

A study using a semi-analytical approach to the reaction kinetics that describes the behavior of heterogeneous reaction–diffusion process with a glyoxyl-agarose immobilized enzyme (penicillin G acylase) in a batch reactor

This research article examines the reaction–diffusion process in a batch reactor by means of immobilized penicillin G acylase (PGA) with a glyoxyl-agarose matrix support. This study employs the Akbari-Ganji’s Method (AGM) when the system is under the internal (intra-particle) diffusional restrictions, and the external (film) diffusional restrictions is negligible. The semi-analytical approximations by the AGM technique are employed for computing the dimensionless steady-state solutions for a system of nonlinear differential equations to examine the impact of internal diffusional restrictions (IDR) as a function of catalyst enzyme loading and particle size. The model entails highly nonlinear components that adhere to the standard Michaelis-Menten kinetics. In addition to the proposed model, the dynamics for the mean integrated effectiveness factor (MIEF) of the dimensionless substrate concentration phenylglycine methyl ester (PGME) is presented. In light of this, there is a satisfactory level of agreement on the comparison of the semi-analytical result with the simulated numerical results across the whole concentration range where the dimensionless initial substrate and product concentrations are S1b=30,S2b=15, and P1b=60,P2b=100. The behavior of concentration profiles under the steady/unsteady state conditions for the effect of IDR on the overall reaction rates by immobilized biocatalyst PGA were examined. A sensitivity analysis is carried out to identify the effective parameter which can influence the diffusional restrictions on MIEF. An error analysis is performed on the dimensionless concentration equations to assess the amount of accuracy and the convergence of a solution.

1D drift-kinetic numerical model based on semi-implicit particle-in-cell method

The paper presents a new one-dimensional drift-kinetic electrostatic model based on the particle-in-cell method and capable of simulating the processes of plasma heating and confinement in mirror traps. The most of particle and energy losses in these traps occur along the magnetic field lines. The key role in limiting these losses is played by the ambipolar electric potential which creates a potential barrier for electrons and significantly reduces the heat flux that, without this barrier, would go to wall due to the classical electron thermal conductivity. However, modeling the formation of such a potential on real spatial and temporal scales of experiments is a challenging problem, since it requires a detailed description of not only ion, but also electron kinetics. In this work, we propose to solve the problem of taking into account electron kinetic effects on the time scale of plasma confinement in a mirror trap using the particle-in-cell method adapted to the approximate drift-kinetic equations of plasma motion. Unlike other electrostatic particle-in-cell models, which use fully implicit schemes to solve the nonlinear system of Vlasov-Poisson and Vlasov-Ampere equations, we propose a semi-implicit approach. By analogy with the Energy Conserving Semi-Implicit Method (ECSIM), it allows for precise conservation of energy and reduces the procedure for finding the electric field to inverting a tridiagonal matrix without multiple nonlinear iterations. Such a model will be useful for simulating not only collisional losses of hot plasma in fusion experiments, but also for studying the features of creating cold starting plasma in mirror traps using plasma or electron guns.

A numerical inquiry of the MHD radiative thin film nanofluid flow and heat transfer augmentation for Ni and Ta nanoparticles of different shapes dispersed in C 12 H 26–C 15 H 32 over an unsteady, curved stretching surface

Thin films are a versatile technology used in various industries, including light-emitting diodes, magnetic recording media, electronic device manufacturing, and optical coating. They are crucial for studying quantum phenomena, superlattices, and multiferroic materials. Their small size allows for better thermal transport studies due to the combination of thin film flow and nanoparticles. Therefore, this study aims to investigate thin film flow and heat transfer with the effects of thermal radiations, magnetic field, and velocity slip in association with tantalum ( Ta ) and nickel ( Ni ) nanoparticles that emerged in kerosene oil ( C 12 H 26 − C 15 H 32 ) toward a curved stretching surface. A review of the pertinent literature indicates that no research has previously been done on the present problem of an unsteady thin film flow across a curved stretching surface. The problem under consideration is modeled by choosing the curvilinear coordinate system. The controlling nonlinear system of partial differential equations is transformed into the system of ordinary differential equations by using an appropriate similarity transformation and then solved numerically using the BVP 4 C technique in MATLAB . The impact of significant physical parameters on velocity, temperature, skin friction, and Nusselt numbers are depicted through graphs and radar charts. The investigation highlights the significant influence of magnetohydrodynamics ( MHD ) , thermal radiation, volumetric fraction, and velocity slip on the enhancement of nanofluid temperature. It is also noted that cylindrical-shaped Ta nanoparticles exhibited the maximum velocity, while blade-shaped Ni nanoparticles showed the lowest. Moreover, the maximum and minimum temperatures were observed for blade-shaped Ni nanoparticles and cylindrical Ta nanoparticles, respectively. Additionally, the maximum and minimum values of skin friction are found for cylindrical Ni nanoparticles and blade-shaped Ta nanoparticles, respectively. Similarly, the maximum and minimum Nusselt numbers are recorded for cylindrical Ta nanoparticles and blade-shaped Ni nanoparticles, respectively.

Analytical Solution of Concentrated Mixtures of Hydrogen Sulfide and Methanol in Steady State in Biofilm Model

The mathematical model of steady-state biofiltration of hydrogen sulfide and methanol mixtures is explored in this work. The non-linear term associated with monoid kinetics is a part of the system of non-linear reaction-diffusion equations on which this model is built. By applying the Akbari Ganji Method and the Homotopy Perturbation Method to resolve the non-linear equations, an approximate analytical solution of the concentrate mixtures in the biofilm model is obtained. Furthermore, this paper reports numerically by exploiting the Matlab software. By comparing the analytical solution with numerical findings, the accuracy of the method is presented. The new analytical results contribute to optimizing the consistency of this model. These novel approaches produce a compact set of analytical approximations that are straightforward to compute and verify as well.

Solving an electrically conducting nanofluid over an impermeable stretching cylinder problem with a spectral reproducing kernel method

In this paper, a nonlinear mechanical system of ordinary differential equations (ODEs) with multi-point boundary conditions is considered by a novel type of reproducing kernel Hilbert space method (RKHSM). To begin, we define the unknown variables in terms of the reproducing kernel function. The roots of the Shifted Chebyshev polynomials (SCPs) are then utilized to collocate the resulting system. Finally, Newton’s iterative method is employed to find the unknown expansion coefficients. The solutions of this system of equations, which arise from the flow of an electrically conducting nanofluid over an impermeable stretching cylinder, are numerically analyzed, and convergence analysis is discussed to demonstrate the reliability of the presented method (PM). Tables and figures are provided to further discuss the solutions and assess the effectiveness of the method in comparison to other techniques in the literature.

Effect of slip velocity on Newtonian fluid flow induced by a stretching surface within a porous medium

This article aims to examine an unsteady 2-D laminar flow of magnetohydrodynamic fluid caused by an elastic surface immersed in a permeable medium under the influence of thermal radiation and extended heat flux. Thermal conductivity and viscosity both are supposed to vary with temperature. This flow model also includes velocity slip, heat source, and joule heating. The governing equations of the fluid, including momentum and energy equations, of the proposed problem are transfigured into a system of interconnected non-linear ordinary differential equations through similarity transformations. The resultant equations are solved efficiently by employing the shooting technique in combination with the fourth-order Runge-Kutta method. Numerical values and the effect of numerous governing factors on the flow field, temperature distribution, local skin friction coefficient, and Nusselt number are showcased via graphs and tables. The investigation reveals that velocity slip, heat source, and porosity parameters enhance the temperature field while diminishing the velocity field. Furthermore, the velocity slip parameter notably reduces both the coefficient of skin friction and the Nusselt number.

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.