System Of Nonlinear Equations Research Articles

Parameter Regression for Porous Electrodes Employed in Lithium-Ion Batteries and Application to Ni0.89Co0.05Mn0.05Al0.01O2

Abstract We developed a parameter regression scheme that can be used with battery models of interest to the battery-analysis community. We show that the recent reduced order model (ROM1, 2022 J. Electrochem. 169 070520, DOI: 10.1149/1945-7111/ac7c93), which is based on a perturbation solution, can be used in place of the full system of nonlinear partial differential equations with minimal loss of accuracy for the conditions of this work, which are relevant for electric vehicle applications. The use of the computationally efficient ROM1, cast in the Python programming language, along with a routine native to Python for the nonlinear regression of model parameters through the minimization of the squared differences between experimental results and model calculations, provides a fast method for the overall endeavor. We applied the procedure to examine Ni0.89Co0.05Mn0.05Al0.01O2, a high-capacity material that is of increasing interest with respect to electric vehicles and other products that rely on batteries of high energy density. Difficulties encountered in this work include the large number of parameters governing the battery model, parameter sensitivity in the regression analyses, and the potential for multiple solutions. We close this publication with a discussion of these challenges and open questions with respect to parameter identification.

ATEP: An advanced transport model for energetic particles

Abstract In this paper we report on the implementation and verification of a phase- space resolved energetic particle (EP) transport model. It is based on a first-principle theoretical framework, i.e. the system of non-linear gyrokinetic equations and the related transport equations. Its focus is primarily directed toward understanding the meso-scale character of EPs and its consequences. Compared to the conventional description of thermal radial transport via a one-dimensional radial diffusion equation, the newly developed model is three-dimensional using canonical constants-of-motion (CoM) variables. The model does not assume diffusive processes to be dominant a priori, instead the EP fluxes are self-consistently calculated and directly evolved in CoM space. We use the EP-Stability workflow and the HAGIS code to determine the phase space fluxes explicitly either in the limit of constant mode amplitudes or an energy-conserving quasi-linear model. As an application of the model the transport of neutral-beam-generated EPs due to a toroidal Alfv ́en eigenmode in an ITER plasma is investigated. As there are no sources and collisions taken into account so far (for an extension of the model see the companion paper ref. [1]), the results cannot be considered as an exhaustive study, but rather as a practical demonstration of the conceptual framework on the way to a comprehensive reduced description of burning plasmas.

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.

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.

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 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

AbstractIn 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.

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.

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.

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.

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.

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.