Newton Interpolation Polynomial Research Articles

Application of Newton Polynomial Interpolation Method in Determining the Continuity of Functions Represented by Tabulated Discrete Points

Application of Newton Polynomial Interpolation Method in Determining the Continuity of Functions Represented by Tabulated Discrete Points

An analysis of fractional piecewise derivative models of dengue transmission using deep neural network

This manuscript investigates a fractional piecewise dengue transmission model using singular and non-singular kernels. The existence results and uniqueness of the solution are established by using the approach of fixed point and in the framework of piecewise derivative and integral. To obtain the approximate solution of the considered models we apply a piecewise numerical iteration scheme which is based on Newton interpolation polynomials. Furthermore, the numerical scheme for piecewise derivatives encompasses singular and non-singular kernels. This study aims to enhance our understanding of dengue internal transmission dynamics by using a novel piecewise derivative approach that considers both singular and non-singular kernels. This work contributes to clarifying the concept of piecewise derivatives and their significance in understanding crossover dynamics. Moreover, a deep neural network approach is employed with high accuracy in training, testing, and validation of data to investigate the specified disease problem. This methodology is employed to thoroughly investigate the intricacies of the specified disease problem.

Zero-Sequence Voltage Error Elimination-Based Offline VSI Nonlinearity Identification for PMSM Drives

The offline inverter nonlinearity identification is a useful method to deal with the inverter nonlinearity effect in machine drive. However, the conventional methods can only be achieved at particular initial rotor position, which reduces the application universality. Meanwhile, it is revealed in this paper that the zero-sequence voltage error exists in the zero-axis voltage, which reduces the identification accuracy and has rarely been investigated. To solve the above problems, an offline inverter nonlinearity identification scheme applicable at arbitrary initial rotor positions is proposed, where the zero-sequence voltage error is fully considered. In this paper, the characteristics of zero-sequence voltage error at different rotor position cases is fully investigated by the theoretical analysis for the first time. The offline identification approach at different rotor position cases is then introduced, with the zero-sequence voltage error eliminated. Finally, the second order Newton interpolation polynomial (NIP) based fitting strategy is proposed to reduce the calculation complexity. The proposed method is validated on a 2.2kW permanent magnet synchronous motor (PMSM) platform.

Investigation and application of a classical piecewise hybrid with a fractional derivative for the epidemic model: Dynamical transmission and modeling.

In this research, we developed an epidemic model with a combination of Atangana-Baleanu Caputo derivative and classical operators for the hybrid operator's memory effects, allowing us to observe the dynamics and treatment effects at different time phases of syphilis infection caused by sex. The developed model properties, which take into account linear growth and Lipschitz requirements relating the rate of effects within its many sub-compartments according to the equilibrium points, include positivity, unique solution, exitance, and boundedness in the feasible domain. After conducting sensitivity analysis with various parameters influencing the model for the piecewise fractional operator, the reproductive number R0 for the biological viability of the model is determined. Generalized Ulam-Hyers stability results are employed to preserve global stability. The investigated model thus has a unique solution in the specified subinterval in light of the Banach conclusion, and contraction as a consequence holds for the Atangana-Baleanu Caputo derivative with classical operators. The piecewise model that has been suggested has a maximum of one solution. For numerical solutions, piecewise fractional hybrid operators at various fractional order values are solved using the Newton polynomial interpolation method. A comparison is also made between Caputo operator and the piecewise derivative proposed operator. This work improves our knowledge of the dynamics of syphilis and offers a solid framework for assessing the effectiveness of interventions for planning and making decisions to manage the illness.

Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton's interpolation polynomials

This paper proposes some updated and improved numerical schemes based on Newton's interpolation polynomial. A Burke-Shaw system of the time-fractal fractional derivative with a power-law kernel is presented as well as some illustrative examples. To solve the model system, the fractal-fractional derivative operator is used. Under Caputo's fractal-fractional operator, fixed point theory proves Burke-Shaw's existence and uniqueness. Additionally, we have calculated the Lyapunov exponent (LE) of the proposed system. This method is illustrated with a numerical example to demonstrate the applicability and efficiency of the suggested method. To analyze this system numerically, we use a fractal- fractional numerical scheme with a power-law kernel to analyze the variable order fractal- fractional system. Furthermore, the Atangana-Seda numerical scheme, based on Newton polynomials, has been used to solve this problem. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.•The purpose of this section is to define a mathematical model to study the dynamic behavior of the Burke-Shaw system.•With the Danca algorithm [1,2] and Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying Dissipativity.•In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.

The computer application for the mathematical modeling of optical fibers used in metrology based on Newton polynomial interpolation

The computer application for the mathematical modeling of optical fibers used in metrology based on Newton polynomial interpolation

Performance of low-cost solar radiation logger

<p>In solar power systems, irradiance value data are among the most important parameters. Such data can be used in installing photovoltaic (PV) modules, such as determining the exact location, tilt angle, and required area, for optimal power efficiency. In this study, the comprehensive simulation and implementation of a solar radiation meter with a PV cell and temperature sensor are presented. The irradiance measurement value is based on the power reading generated by the small capacity of the PV cell at a specific load converted into a digital value in the microcontroller using the implicit Newton polynomial interpolation (NPI) equation as a low-cost alternative method. The effect of temperature is included in the conversion to obtain precise measurement results. Firstly, the structure and characteristics of the PV cell are discussed. Secondly, the parameters, measuring method, and conversion of the measurement reading data using the NPI equation are presented to assess the results. Finally, the simulation of the solar radiation meter using the PSIM and implementation of the hardware are conducted to validate the concepts and compare their results. The proposed hardware has an average error of 2.72% in the implementation of the measurement test.</p>

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

A Hybrid Full-Discretization Method of Multiple Interpolation Polynomials and Precise Integration for Milling Stability Prediction

As milling chatter can lead to poor machining quality and limit the efficiency of productivity, it is of great significance to learn about milling chatter stability and research the effective and fast prediction of milling stability. In this study, a hybrid full-discretization method of multiple interpolation polynomials and precise integration (HFDM) is proposed for milling stability prediction. Firstly, the third-order Newton interpolation polynomial, third-order Hermite interpolation polynomial and linear interpolation are applied to approximate the state term, delay term and periodic coefficient matrix, respectively. Meanwhile, the matrix exponentials can be calculated based on the precise integration algorithm, which can improve computational accuracy and efficiency. The numerical simulation results indicate that the proposed method can not only effectively generate a stability lobe diagram (SLD) but also obtain better prediction accuracy and computation efficiency. A milling experiment is offered to demonstrate the feasibility of the method.

A fractional order Monkeypox model with protected travelers using the fixed point theorem and Newton polynomial interpolation

This study formulates a Monkeypox model with protected travelers. The fixed point theorem is used to obtain the existence and uniqueness of the solution with Ulam–Hyers stability for the analysis of the solution to the model. The Newton polynomial interpolation scheme is employed to solve an approximate solution of the fractional Monkeypox model. The numerical simulations and the graphical representations suggest that the fractional order affects the dynamics of the Monkeypox. The fractional order shows other underlining transmission trends of the Monkeypox disease. We conclude that the result obtained for each compartment conforms to reality as the fractional order approaches unity.

Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel.

Chemical kinetics is a branch of chemistry that is founded on understanding chemical reaction rates. Chemical kinetics relates many aspects of cosmology, geology, and even in some cases of, psychology. There is a need for mathematical modelling of these chemical reactions. Therefore, the present research is based on chemical kinetics-based modelling and dynamics of enzyme processes. This research looks at the two-step substrate-enzyme reversible response. In the two step-reversible reactions, substrate combines with enzymes which is further converted into products with two steps. The model is displayed through the flow chart, which is then transformed into ODEs. The Atangana-Baleanu time-fractional operator and the Mittag-Leffler kernel are used to convert the original set of highly nonlinear coupled integer order ordinary differential equations into a fractional-order model. Additionally, it is shown that the solution to the investigated fractional model is unique, limited, and may be represented by its response velocity. A numerical scheme, also known as the Atangana-Toufik method, based on Newton polynomial interpolation technique via MATLAB software, is adopted to find the graphical results. The dynamics of reaction against different reaction rates are presented through various figures. It is observed that the forward reaction rates increase the reaction speed while backward reaction rates reduce it.

Dynamical study of a novel 4D hyperchaotic system: An integer and fractional order analysis

In this article, a new nonlinear four-dimensional hyperchaotic model is presented. The dynamical aspects of the complex system are analyzed covering equilibrium points, linear stability, dissipation, bifurcations, Lyapunov exponent, phase portraits, Poincaré mapping, attractor projection, sensitivity and time series analysis. To analyze hidden attractors, the proposed system is investigated through nonlocal operator in Caputo sense. The existence of solution of the system in fractional sense is studied by fixed point theory. The stability of fractional order system is demonstrated via Matignon stability criteria. The fractional order system is numerically studied via newly developed numerical method which is based on Newton polynomial interpolation. The evolution of the attractors are depicted with different fractional orders. For few fractional orders, some hidden strange chaotic attractors are observed through graphs. Theoretical and numerical studies demonstrate that this model has complex dynamics with some stimulating physical characteristics. To verify and validate the results, we implement Field Programmable Analog Arrays (FPAA).

Models and algorithms for managing the emotional state of customers in commercial banks using deep convolutional neural networks

In this research, a model for managing the emotional state of customers in a commercial bank has been developed using a deep convolutional neural network (DCNN) and algorithms for distributing conflicting customers along the routes to the certain operator, depending on this emotional state. In order to route a customer to the certain operator, it was necessary to develop a mathematical model of emotional target routing based on the Newton interpolation polynomial. The developed model has four classes [angry, happy, neutral, and sad], trained and tested on the well-known FER2013 dataset using machine learning and computer vision. Finally, the model validation accuracy of 70.35% has been achieved.

NUMERICAL ANALYSIS FOR HIDDEN CHAOTIC BEHAVIOR OF A COUPLED MEMRISTIVE DYNAMICAL SYSTEM VIA FRACTAL–FRACTIONAL OPERATOR BASED ON NEWTON POLYNOMIAL INTERPOLATION

Dynamical features of a coupled memristive chaotic system have been studied using a fractal–fractional derivative in the sense of Atangana–Baleanu. Dissipation, Poincaré section, phase portraits, and time-series behaviors are all examined. The dissipation property shows that the suggested system is dissipative as long as the parameter [Formula: see text]. Similarly, from the Poincaré section it is observed that, lowering the value of the fractal dimension, an asymmetric attractor emerges in the system. In addition, fixed point notions are used to analyze the existence and uniqueness of the solution from a fractal–fractional perspective. Numerical analysis using the Adams–Bashforth method which is based on Newton’s Polynomial Interpolation is performed. Furthermore, multiple projections of the system with different fractional orders and fractal dimensions are quantitatively demonstrated, revealing new characteristics in the proposed model. The coupled memristive system exhibits certain novel, strange attractors and behaviors that are not observable by the local operators.

Mathematical Modeling of COVID-19 Transmission Using a Fractional Order Derivative

In this article, the mathematical model of COVID-19 is analyzed in the sense of a fractional order Caputo operator with the consideration of an asymptomatic class. The suggested model is comprised of four compartments. The results from fixed point theory are used to theoretically analyze the existence and uniqueness of solution of the model in fractional perspective. For the numerical approximation of the suggested problem, a numerical iterative scheme is used, which is based on the Newton polynomial interpolation. For the efficiency and applicability of the suggested technique with a fractional Caputo operator, we simulate the results for various fractional orders.

DEVELOPMENT OF AN AUTOMATED INFORMATION SYSTEM FOR CALCULATING THE MOVEMENT OF A HIGH-SPEED SMALL-SIZED BODY

The article presents a variant of the development of an automated information system for mathematical modeling of the movement of a high-speed small-sized body based on differential equations and approximation of the trajectory of a high-speed small-sized body by numerical methods, as well as a comparative analysis of various options for numerical solutions. The automated information system allows calculating the flight trajectory coordinates of a high-speed small-sized body based on a differential equation, as well as determining the flight trajectory coordinates by the Euler method, using the Newton interpolation polynomial and the Lagrange interpolation polynomial. In the article, a search for a suitable variant of numerical solutions is carried out, as well as calculations of the movement of a high-speed small-sized body under conditions of limited time for decision-making. The substantiation of the use of an automated information system to automate the calculation and analysis of the movement of a high-speed small-sized body is carried out. While designing an automated information system for calculating and analyzing the flight of a high-speed small-sized body, it is proposed to apply four modules for calculating the flight trajectory of a high-speed small-sized body by numerical methods and a module for comparative analysis and making the results of the optimal variant. The proposed automated information system should provide users with uninterrupted operation, a convenient interface, the required system response time to a request, and the creation of necessary reports. The reports should reflect all types of calculations, as well as a comparative analysis with the choice of the optimal option for calculating the flight path of a high-speed small-sized body. An automated information system is needed to improve the quality of calculations and analysis of movement in the development of new and modernization of existing small arms.

PIECEWISE MODELING OF THE TRANSMISSION DYNAMICS OF CONTAGIOUS BOVINE PLEUROPNEUMONIA DEPENDING ON VACCINATION AND ANTIBIOTIC TREATMENT

Atangana and Araz have introduced a new term called piecewise differentiation and integration which has been successfully applied to epidemiology in these days. This concept paves the way of seeing different patterns of modeling real-world problems. In this study, a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) is considered and modified based on vaccination and antibiotic therapy by using these operators. This model is classified into six classes namely susceptible, exposed, vaccinated, infectious, continuously infected, and treated classes. The model is solved by a numerical scheme based on Newton interpolation polynomial. The existence and uniqueness of the model are investigated. Various patterns of piecewise concepts have been presented through the numerical figures.

Investigation of robotic milling chatter stability prediction under different cutter orientations by an updated full-discretization method

Chatter can easily occur during robotic milling owing to the low rigidity of serial robots. Chatter affects the quality of the workpiece surface and damages the robot equipment, which is a bottleneck that hinders the application of robotic milling. An effective way to ensure milling stability is to rationally select the cutting parameters using high-precision stability lobe diagrams (SLDs). In this study, an updated full-discretization chatter stability prediction method based on the fourth-order Hermite and third-order Newton interpolation polynomial was proposed. A dynamical model of robotic milling was established by considering the regenerative effect, cutter structure modal coupling effect, and the influence of cutter orientations. Based on the established robotic milling dynamic model and the proposed full-discretization method, SLDs related to the axial depth of cut, spindle speed, lead angle, and tilt angle were generated. The dominant influencing factors of the robotic milling chatter stability under different robot postures and cutter orientations were analysed. The robotic milling experiment results show that the dynamic characteristics of the robot are different in each direction. When the milling cutter feeding direction is parallel to the x-axis direction of the global coordinate system, the regenerative effect is dominant. When the feed direction of the milling cutter is parallel to the y-axis of the global coordinate system, the cutter structure modal coupling effect is more obvious. This is possibly because the dynamic characteristics of the tool tip change during the milling process under different robot postures. However, the milling process becomes more stable with an increase in the lead or tilt angle. This is mainly because when the cutting depth is constant, the cutter–workpiece engagement area gradually decreases with an increase in the tool angle.

A NUMERICAL SCHEME BASED ON TWO- AND THREE-STEP NEWTON INTERPOLATION POLYNOMIALS FOR FRACTAL–FRACTIONAL VARIABLE ORDERS CHAOTIC ATTRACTORS

The terms fractional differentiation and fractal differentiation are recently combined to form a new fractional differentiation operator. Several kernels, such as the power-law and the Mittag-Leffler function, are employed to investigate these novel operators. Both fractional-order and fractal dimensions are found in these developed operators. In the present investigation, we have applied these new operators to study certain chaotic attractors with Mittag-Leffler and power-law kernels. These chaotic models are solved using an effective numerical procedure called the Adams–Bashforth technique, based on two- and three-step Newton interpolation polynomials. In this analysis, the order of the fractional derivative is assumed to be the exponential, trigonometric, and hyperbolic forms of variables. The solutions obtained by the numerical procedure with two different kernels are portrayed in terms of chaotic plots.

Improvement of the WENO-NIP Scheme for Hyperbolic Conservation Laws

The WENO-NIP scheme was obtained by developing a class of L1-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks.