Polynomial Approximation Method Research Articles

Bifurcation analysis and control study of improved full-speed differential model in connected vehicle environment

Abstract In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear, so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.

Response analysis of asymmetric monostable energy harvester with an uncertain parameter

The nonlinear piezoelectric vibration energy harvesting system offers a solution to the power supply challenge for low-power sensors. However, In practical engineering, structural parameter uncertainties arise from manufacturing errors, environmental temperature fluctuations, and changes in material properties. Besides, due to inherent flaws in the structure and materials, asymmetry may occur in the potential well, which leads to the complexity of the dynamic behavior exhibited in energy harvesting system. In this paper, in order to investigate the dynamics of the asymmetric monostable piezoelectric energy harvesting system under an uncertain parameter, the orthogonal polynomial approximation method is applied to transform the stochastic system into an equivalent deterministic system. Then, the responses of the equivalent deterministic system are used to reveal the stochastic behavior of the uncertainty system. The results indicate that, when the uncertainty intensity of the cubic nonlinear coefficient is 0, an increase in the asymmetry level of the potential well leads to an increase in the output voltage. Increasing the other parameters, the output voltage is reduced. In addition, as the uncertainty intensity increases, the parameter intervals of chaotic response become wider and the energy harvesting efficiency decreases. Moreover, as the potential well asymmetry increases, the sensitivity of energy harvesting system to uncertain parameter increases.

ПОВЫШЕНИЕ ТОЧНОСТИ РЕГРЕССИОННЫХ МОДЕЛЕЙ МЕТОДАМИ ПОЛИНОМИАЛЬНОЙ АППРОКСИМАЦИИ

The study objective is to assess the influence of the approximation method on the accuracy of the regression model.
 The task is to increase the accuracy of regression models using polynomial approximation methods.
 Research methods: the least square method, the polynomial approximation method.
 The novelty of the work: dependencies for the regression analysis of a multifactor experiment using polynomial approximation is developed, the influence of the approximation method on the accuracy of the regression model is analyzed.
 The study results: as a result of applying the developed dependencies, the accuracy of single-factor regression equations increased by 2.4 times, two-factor equations - by 1.7 times.
 Conclusions: the use of polynomial approximation makes it possible to increase the accuracy of single-factor and multifactor regression models, which increases their adequacy, the significance of coefficients and expands the possibilities of their application.

A novel computational fractional modeling approach for the global dynamics and optimal control strategies in mitigating Marburg infection

<abstract><p>Marburg virus disease poses a significant risk to global health, impacting both humans and non-human primates. This study has yielded an optimal control model for potentially mitigating the transmission of the Marburg infection. The proposed mathematical model includes fractional-order derivatives in the Caputo sense. Initially, we analyzed the model without control measures, examining its key characteristics regarding local and global stabilities. Subsequently, we extended the model by incorporating suitable time-dependent optimal control variables. We have also introduced two time-dependent control measures: $ \Psi_1 $ for the prevention of human-to-human Marburg transmission, and $ \Psi_2 $ to enhance the rate of quarantine of exposed individuals. We performed simulation analysis for both cases i.e., with and without optimal controls using the two-step Newton polynomial approximation method, considering both fractional and classical orders. The numerical findings of the comparative study between classical and fractional cases validate the biological significance of the fractional operator and effectiveness of the proposed optimal control strategies.</p></abstract>

Unsteady MHD Free Convection Flow of Fluid in Micro Channel in the Presence of Viscous Dissipation Between Vertical Parallel Two Walls

Convection is a process by which heat is transferred by movement of a heated water. The phenomenon where heat transfer process is driven by natural fluid motion due to temperature and density gradients is called free convection. The effect of slip velocity on thermal behaviours of MHD free convection flow over a vertical parallel two walls is the focus of this research. The mathematical equations which described the phenomenom includes momentum, magnetic field and energy equations. Condition for the existence and uniqueness of solution of the model equations were established using the approach of Lipschitz continuity. The model equations in dimensionless form were solved using Olayiwola’s generalized polynomial approximation method (OGPAM). The results obtained which contains the dimensionless parameters were presented graphically and discussed. It was observed that increase in Reynolds number and Knudsen number both enhanced the velocity distribution at the two walls. Also, magnetic field and temperature field were enhanced by magnetic prandtl number and Eckert number respectively.

Epidemiology of digestive diseases in children in modern conditions: regional data as a reflection of Russian trends

Introduction. Pathology of the digestive organs is one of the factors influencing the health indicators in children.
 Purpose of the work: to characterize current trends in the epidemiology of digestive diseases in children in the Udmurt Republic (UR).
 Materials and methods. The epidemiological features of digestive diseases in 0–14 and 15–17 years children were studied on the base on official statistics for the period from 2012 to 2022. The polynomial approximation method was used for forecasting.
 Results. There is a decrease in the prevalence of pathology of the digestive organs in 0–14 years children from 153.0 in 2012 to 75.0 in 2022 per 1000 children of the corresponding age, the average annual rate of decline was –6.3%. The predicted indicator will be 32.3 per 1000 children of the corresponding age in 2025. Similar trend in the prevalence of gastroenterological pathology were noted in the age group of 15–17 years: since 2012, this indicator has decreased by 1.9 times from 187.4 to 93.8 per 1000 children of the corresponding age with an average annual rate of decline of –6.3%. The predicted prevalence of gastroenterological pathology in 15–17 years children in the region will be 69.8 per 1000 children of the corresponding age in 2025. Since 2012, the trend in morbidity is characterized by a decrease in rates in both age groups with a higher rate of decline in the UR: from –10.6% in 0–14 years (Russia: –5.5%) to –7.7% in 15–17 years old (Russia: –3.0%). The predicted incidence rate in the UR will be 19.2 in 2025 for children 0–14 years old; 18.6 for children 15–17 years old.
 Conclusion. The established patterns indicate a decrease in the prevalence and morbidity of digestive organs among the child population of the UR.

Hopf bifurcation control of macroscopic traffic flow model considering vehicle braking effect.

Traffic congestion not only has a great impact on people's travel, but also increases energy consumption and air pollution. The control analysis of the macroscopic traffic flow model considering the vehicle braking effect is particularly important, reflecting the impact on the actual traffic flow density wave, so as to better solve the actual traffic problems. In this paper, based on a speed difference optimization speed model, the micro-macro-variables are transformed into a high-order continuous traffic flow model. Then, a random function considering the physical correlation of random components is added to the high-order continuous traffic flow model to establish a random traffic flow model that can reflect the uncertain behavior of traffic flow acceleration or deceleration. Based on this stochastic traffic model, the existence of Hopf bifurcation and bifurcation control of the traffic flow system model considering stochastic characteristics are derived by using Hopf bifurcation theorem. By Chebyshev polynomial approximation method, the stochastic problem of the system is transformed into the bifurcation control problem of its equivalent deterministic system. A feedback controller is designed to delay the occurrence of Hopf bifurcation and control the amplitude of the limit cycle. Without changing the equilibrium point of the system, the complete elimination of Hopf bifurcation can be achieved by controlling the amplitude of the limit cycle. That is, the feedback controller is used to modify the bifurcation characteristics of the system, such as the bifurcation appearing at the equilibrium point in the control system moves forward, moves backward or disappears, so as to achieve the effect of preventing or alleviating traffic congestion.

Response Analysis of the Three-Degree-of-Freedom Vibroimpact System with an Uncertain Parameter.

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

Spectral method of electrical circuits accelerated simulation with thyristors

Purpose. The development of transient processes calculation method in electric circuits with thyristors based on the use of functions approximation by orthogonal polynomials.
 Methodology. Functions approximation by orthogonal polynomials, numerical methods of differential equations integration, matrix methods, programming, theory of electric circuits.
 Obtained results. The method of solution function polynomial approximation of integro-differential equations of state, which describes the transient processes of an electric circuit with thyristors, is used in this paper. The used method showed the advantages over other known methods in increasing the accuracy and reducing the simulation time of transient electrical processes by more than 6 times.
 Findings. The solution is approximated by a series of Chebyshev polynomials. The integro-differential equations of state are transformed into linear algebraic equations for special depiction of the solution functions. The depiction of functions of true currents in the equivalent circuit is interpreted as direct currents. Such a schematic model creates visibility for a researcher performing simulation of transient electrical processes.
 Practical value. The proposed methods discover the possibility of using the apparatus of direct current electric circuits’ theory for transient processes in complex schemes modeling with thyristors.

Solving high order of non-linear Volterra-fredholm Integro-differential equation by using bou-baker Polynomials approximation Method

In this paper, the Bou-baker polynomial method is used to evaluate an approximate so1ution initia1 value problem of high-order of non1inear Vo1terra-Fredholm Integro-Differential equation of the second kind. Three different examp1es and their graphics are displayed.

Electrochemical investigation on calcium ion sensing and steady-state diffusion analysis using zinc oxide modified glassy carbon electrode

The concentration of calcium ions in human blood serum typically falls within a narrow range of 2 and 2.5 mmol/L. However, hyperthyroidism, resulting from an overactive parathyroid gland, can lead to an elevation of calcium ion concentration beyond 2.5 mmol/L, necessitating the development of a highly sensitive and selective electrochemical sensor for calcium ion detection. To address this issue, a highly sensitive and selective electrochemical sensor based on zinc oxide nanoparticles-chitosan composite modified glassy carbon electrode (GC/ZnO-Chitosan) was fabricated for the detection of calcium ions. Cyclic voltammetric analysis revealed that the fabricated electrode possesses excellent electrocatalytic activity for calcium ion oxidation and reduction. The fabricated electrode showed high sensitivity (449.887 µA mM−1) towards calcium ions over the wide detection range of 2.14–3.5 mM with a low detection limit of 2.26 µM, and a low quantification limit of 7.45 µM. Moreover, the fabricated electrode showed fast response time (<25 s) due to the synergistic effect of enhanced adsorption of calcium ions (Γ=2.17×10-7 mol cm−2), efficient electron shuttling capacity of ZnO nanoparticles between GCE (ks=6.01 s−1) and the adsorbed calcium ions, and enhanced diffusion of calcium ions (DCa2+=1.2788×10-6cm2s-1) within the ZnO-Chitosan composite film. Genocchi polynomial approximation method was employed for the establishment of a precise mathematical relationship between the steady-state concentration of calcium ions and the distance from the electrode's surface. This newly developed GC/ZnO-Chitosan electrode presents a significant improvement over previously developed electrodes for calcium ions, as it enables the precise analysis of calcium ion diffusion profiles at steady-state.

A 〖(2n+9)〗^thDegree Polynomial Approximation Method for Solving Eight Order Linear Boundary Value Problems

Galerking method, Collocation method and least squired method have been used for solving boundary value problems (BVPs) and from practical point of view almost any success of above mention method depends on the selection of basis functions. Also, Adomain decomposition and variational iteration methods have been applied for the solution of BVPs and required respectively calculating the Adomian’s polynomials and Lagrange multipliers. In this paper a simple method of Degree polynomial approximation solution were n is number of successive derivatives of the governing equation is suggested as an alternative of the above mentioned methods. The method consists of first obtaining from the governing equation its n successive derivatives and boundary conditions a linear differential system of equations evaluated at the boundary points. Next, an approximate solution in the form of polynomial of degree with unknown coefficients is assumed. To determine the unknown coefficients, the assumed solution is incorporate into the linear differential system which turns to be a linear system of equation with unknowns which is salvable uniquely. The method is tested for n=5 to four examples. It clear that from tables 3.1 to 3.4 and figures 3.1 to 3.4 the numerical outcomes agree with the exact solution and also better than some existing results in the literatures. it should be also noted that the accuracy of the method can be increased by increased n. All Numerical computation were performed using maple 2016 software.

Modelling Stage-discharge Relationships for Hydrological Stations in the Da River Basin using First-kind Chebyshev Polynomial Approximation

Observed discharge time series is essential data for hydrological analysis, streamflow forecasting, as well as water resources planning and management. Estimation of continuous discharge time series is generally based on the curves (or empirical models) that can properly simulate the relationships between water levels (stages) and concurrent water discharges observed at a particular hydrological station. For stations with complex hydraulic conditions, in addition to gauged stages at the discharge gauging section, another variable such as stage measured at an auxiliary station (to calculate the slope) or rate of change in stage is added to the estimation process. This paper presents the results of applying the first-kind Chebyshev polynomial approximation method in constructing rating curves for three hydrological stations in the Da river basin, each having different basin characteristics and various influencing factors. The analysis results using five evaluation indices (MAE, σ, Pbias, KGE, and MAPE) show that the Chebyshev polynomials are highly effective in modelling both simple and complex stage-discharge relationships. The Chebyshev polynomial approximation method is suggested to be used for establishing stage-discharge relationships in hydrometry.&#x0D;

Wild Bootstrap-Based Bias Correction for Spatial Quantile Panel Data Models with Varying Coefficients

This paper studies quantile regression for spatial panel data models with varying coefficients, taking the time and location effects of the impacts of the covariates into account, i.e., the implications of covariates may change over time and location. Smoothing methods are employed for approximating varying coefficients, including B-spline and local polynomial approximation. A fixed-effects quantile regression (FEQR) estimator is typically biased in the presence of the spatial lag variable. The wild bootstrap method is employed to attenuate the estimation bias. Simulations are conducted to study the performance of the proposed method and show that the proposed methods are stable and efficient. Further, the estimators based on the B-spline method perform much better than those of the local polynomial approximation method, especially for location-varying coefficients. Real data about economic development in China are also analyzed to illustrate application of the proposed procedure.

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of “pathological” behavior, e.g. “nearly” singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates.

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

Discussion on applicability of the complex fractional moment method in Hamiltonian system

In this paper, the complex fractional moment (CFM) method is extended from the Langevin system to the Hamiltonian system for the first time, the applicability and advantages of the CFM method in Hamiltonian systems are investigated. According to the quasi-periodic property, the stochastic averaging technique is applied to the stochastic quasi-nonintegrable Hamiltonian system, and the one-dimensional Hamiltonian stochastic differential equation is obtained. By Mellin transformation, the associated Fokker Planck Kolmogorov (FPK) equation governing the Hamilton function is transformed into a set of Ordinary Differential Equations (ODEs) of CFM. The semi-analytical solutions of the FPK equation are obtained by solving the ODEs. Furthermore, by a polynomial approximation method for the Hamiltonian case proposed in this paper, the numerical results show the influence of parameters and initial values on the evolution of the transient probability density functions (PDF) of the system. The accuracy of CFM method in the Hamiltonian case is verified by error analysis.

Numerical simulation acceleration of flat-chip solid oxide cell stacks by data-driven surrogate cell submodels

Three-dimensional (3D) multiphysics models are powerful tools for investigating the distributions of physical quantities such as temperature inside solid oxide cell (SOC) stacks, but their high computational cost remains an obstacle to their application in simulating industrial-scale stacks with tens of cells. To accelerate the simulation for the 3D model of a novel flat-chip SOC (FCSOC) stack, this study proposes a simplification method that replaces part of the governing equations with data-driven surrogate cell submodels. The submodels, built with the adaptive polynomial approximation (APA) method, take the form of polynomials and are easy to integrate into commercial CFD software such as COMSOL. Simulation shows that the simplified stack model can predict the temperatures and voltages accurately compared with the original stack model. At the same time, the time and memory required for computation are reduced by approximately 60% for a short stack model containing seven cells, owing to the simplified fuel-side mass transfer and charge transfer processes. For a large stack model with 21 cells, the reduction in computation time can even exceed 70%. The reduced computational cost makes it possible to simulate the models of industrial-scale FCSOC stacks with up to 61 cells.

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

In this work, a new efficient solution algorithm named Combined Matrix Iterative Algorithm based on the Mini-Max Polynomial Approximation (MMPA) method is proposed for burnup calculations. As a new matrix exponential method in recent years, MMPA method has good numerical stability and computational accuracy, and all calculation are real number operations. As the same with Chebyshev Rational Approximation Method (CRAM) which is widely and successfully used in solving burnup equations, the method can also directly deal with rigid and sparse burnup matrices. Compared with the traditional solution algorithms based on MMPA method, the Combined Matrix Iterative Algorithm (MMPA-CMIA) proposed in this paper can effectively reduce the computational complexity and improve the computational efficiency after equivalently transforming the expression of the MMPA method. Then the MMPA-CMIA is used to theoretically demonstrate and analyze the solution efficiency of random matrices with different orders which has the similar distribution characteristics to burnup matrices. Compared with the 16-order CRAM method and the other two traditional solution algorithms based on MMPA method, the algorithm shows good acceleration effect and computational precision. Finally, the algorithm is used in the actual burnup benchmarks to verify and validate the accuracy and efficiency. The burnup matrices of different orders are constructed through detailed burnup chain and simplified burnup chain. The results show that the algorithm has good accuracy and numerical stability. And comparing with 16-order CRAM, the algorithm can save about 30% of computing time for solving burnup equations. It shows that the algorithm is a potentially powerful method for accelerating the solution of large-scale burnup calculations.

Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.