Composite Simpson's Rule Research Articles

Numerical Treatment for the Distributed Order Fractional Optimal Control Coronavirus (2019-nCov) Mathematical Model

In this paper, we presented the distributed order fractional optimal control of the Coronavirus (2019-nCov) mathematical model. The distributed order fractional operator is defined in the Caputo sense. Control variables are considered to reduce the transmission of infection to healthy people. The discretization of the composite Simpson's rule and Grünwald-Letnikov nonstandard finite difference method is constructed to solve the obtained optimality system numerically. The stability analysis of the proposed method is studied. Numerical examples and comparative studies for testing the applicability of the utilized method and to show the simplicity of this approximation approach are presented. Moreover, by using the proposed method we can conclude that the model given in this paper describes well the confirmed real data given in Spain and Wuhan.

FRKVF: High-accuracy motion interpolation for polishing operations using fourth-order Runge-Kutta and velocity flexibility planning

Traditional interpolation algorithms often fail to meet the precision requirements of ultra-precision machining when applied to super-precision polishing machines. Moreover, the processing of fused silica glass optical components is frequently threatened by mechanical impacts due to their fragility. In order to address this issue, this paper proposes a high-precision motion interpolation method based on fourth-order Runge-Kutta and velocity-flexible planning. This method is designed for open-architecture small multi-axis optical polishing machines to polish quartz glass. The algorithm initially employs composite Simpson's rule to calculate the lengths of sub-paths within the polishing trajectory. Based on these length values, flexible velocity planning is executed to ensure the smooth continuity of velocity, acceleration, and jerk during motion interpolation. This reduces the risk of mechanical impacts that could damage the components during the machining process. The introduction of the adaptive fourth-order Runge-Kutta method significantly enhances the parameter point calculation accuracy of NURBS curves. The incorporation of adaptive principles also maintains a higher processing speed, thereby greatly improving processing efficiency. This method comprehensively addresses both the precision of curve interpolation and execution efficiency. Finally, experimental validation is conducted on an open-architecture small multi-axis optical polishing machine. The proposed method based on FRKVF not only mitigates mechanical impacts resulting from discontinuous acceleration, thereby ensuring the machining quality of optical components, but also satisfies the high-precision requirements for processing fused silica optical elements.

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton–Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one- and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

Choquet integration by Simpson’s rule with application in Hellinger distance

In non-additive measure theory, there are a few studies on the numerical Choquet integral in continuous case on real line. Recently, based on the statistics software R, Torra and Narukawa (Inf Fusion 31:137–145, 2016) considered the problem of computing a numerical Choquet integral and some algorithms for Hellinger distance between two monotone measures. In this paper, the composite Simpson rule for numerical Choquet integration is proposed. Then some algorithms in Mathematica software are given. As well as, CPU time of the Examples in terms of seconds is reported which shows, in computational view, the presented method has a high speed.

A novel two-step iterative scheme based on composite Simpson rule for solving nonlinear equations

This article proposes a novel two-step iterative scheme of order three based on fundamental theorem of calculus and composite Simpson rule for solving nonlinear equations. The efficiency of method is illustrated with the help of numerical examples and a comparative study is done with some well-known existing iterative methods.

Space-time finite element method for the distributed-order time fractional reaction diffusion equations

In this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t=0, the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t−α,α∈(0,1) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method.

Thermodynamic insights into membrane fouling in a membrane bioreactor: Evaluating thermodynamic interactions with Gaussian membrane surface

While membrane bioreactor (MBR) technology is generally considered as one of the most promising technologies for wastewater treatment and recovery, membrane fouling remains the major obstacle limiting its applications. Interfacial interactions, which critically determine adhesion process and membrane fouling, were investigated in this study. It was found that, natural membrane surface was of a Gaussian surface obeying Gaussian distribution. A Gaussian approach integrating Fourier transform technique, Gaussian distribution and spectrum method was deduced to simulate rough surface topography of membrane. Thereafter, surface element integral (SEI) method, together with composite Simpson rule and triangulation of Gaussian surface was proposed to calculate interfacial interactions. By using the unified method, quantification of interfacial interactions with a Gaussian membrane surface was realized for the first time to date. It was further found that, membrane surface topography had profound impacts on interfacial interactions and adhesive fouling in the MBR. The deduced method can be used to address impacts of various factors on interfacial interactions and adhesive fouling, posing in-depth thermodynamic insights into membrane fouling and pointing towards its widespread potential in fouling research in MBRs.

Membrane fouling in a membrane bioreactor: A novel method for membrane surface morphology construction and its application in interaction energy assessment

A novel method was developed to construct membrane surface morphology in this study. The method was based on fractal geometry theory, and involved a modified Weierstrass-Mandelbrot (WM) function for modeling surface topography. With MATLAB programming, membrane surface could be graphically modeled. It was found that the fractal dimension (Df) and cutoff frequency (Ls) were the two key parameters affecting the topography, and the membrane surface constructed by this method could be very close to the real situation. The method was then applied in interaction energy assessment regarding membrane fouling in a membrane bioreactor. The procedure which combines membrane surface topography construction, surface element integration (SEI) method and the composite Simpson's rule was theoretically proposed. The surface properties of foulant and membrane were experimentally measured. Accordingly, a membrane surface was reconstructed for interaction energy assessment. With these data, quantitatively assessing interfacial interactions between foulant particles and a nearly-real rough membrane surface is expected to be firstly realized. This study provided a complete solution for quantitative assessment of interaction energies between a foulant particle and rough membrane surface.

A semi-numerical model for near-critical angle scattering.

Numerous phenomena in the fields of physics and mathematics as seemingly different as seismology, ultrasonics, crystallography, photonics, relativistic quantum mechanics, and analytical number theory are described by integrals with oscillating integrands that contain three coalescing criticalities, a branch point, stationary phase point, and pole as well as accumulation points at which the speed of integrand oscillation is infinite. Evaluating such integrals is a challenge addressed in this paper. A fast and efficient numerical scheme based on the regularized composite Simpson's rule is proposed, and its efficacy is demonstrated by revisiting the scattering of an elastic plane wave by a stress-free half-plane crack embedded in an isotropic and homogeneous solid. In this canonical problem, the head wave, edge diffracted wave, and reflected (or compensating) wave each can be viewed as a respective contribution of a branch point, stationary phase point, and pole. The proposed scheme allows for a description of the non-classical diffraction effects near the "critical" rays (rays that separate regions irradiated by the head waves from their respective shadow zones). The effects include the spikes present in diffraction coefficients at the critical angles in the far field as well as related interference ripples in the near field.

Deriving the Composite Simpson Rule by Using Bernstein Polynomials for Solving Volterra Integral Equations

In this paper we use Bernstein polynomials for deriving the modified Simpson's 3/8 , and the composite modified Simpson's 3/8 to solve one dimensional linear Volterra integral equations of the second kind , and we find that the solution computed by this procedure is very close to exact solution.

Deriving the Composite Simpson Rule by Using Bernstein Polynomials for Solving Volterra Integral Equations

In this paper we use Bernstein polynomials for deriving the modified Simpson's 3/8 , and the composite modified Simpson's 3/8 to solve one dimensional linear Volterra integral equations of the second kind , and we find that the solution computed by this procedure is very close to exact solution.

Study on Methods of Lateral Deviation of Track for Static Inspecting in High-Speed Railway

According to the critical factor of railway track geometric irregular, the deformations of track can be controlled. The lateral deviation is the key to determine the railway track irregularity. Currently, in the static inspected methods, the combined method of geodetic surveys and track surveying trolleys (inspecting instruments for static geometry parameter of track) is widely used in high-speed railway. Depending on some tests, the model of track irregularity in track surveying trolleys can be reconstructed by another method. According to the special features of track of high-speed railway, it is necessary to study on the accurate and effective lateral deviation algorithm which is suitable for the precise track inspection of high-speed railway. Based on some existing methods of construction layout in highway, the primary contents of this paper are: (1) reduces those methods to three algorithms, such as the Longitudinal Deviation Algorithm with Composite Simpson rule (LDACS), the Distance Function Algorithm of Newton’s method (DFAN) and Normal Perpendicular to Tangent Algorithm of Newton’s method (NPTAN), and (2) completes the algorithm steps of DFAN and NPTAN on circular curve, and proves the results of two algorithms on circular curve same, and (3) proposes the three algorithms to calculate lateral deviation and mileage of any rail detection points for inspecting the static geometric state of track in high-speed railway. Depending on some simulation data, the experimental results are: (1) the calculations of DFAN, NPTAN and LDACS, in which the number of subintervals of equal greater than or equal to five, meet the accuracy of the precise track detection of high-speed railway, and (2) the difference mileage and lateral deviation between DFAN and NPTAN are less than 0.001 mm, and (3) the efficiency of those algorithms is very considerable and the efficiency of DFAN is basically the same with NPTAN and higher than LDACS, and (4) the longer the transition curve is, the lower the efficiency and accuracy of DFAN and NPTAN are. The bigger the radius is, the higher the accuracy of LDACS is. According to the measurement data of the Chengdu Dujiangyan Railway Line (Cheng Guan Line), the above mentioned results of (1) and (2) can be proved correctly.

On the accuracy of estimating pest insect abundance from data with random error

Numerical integration is a popular technique that can be successfully applied to evaluating the pest insect abundance in an agricultural field. In this paper we apply numerical integration in the problem where data about insects obtained as a result of a trapping procedure have random error (noise). We compare several methods of numerical integration that have different accuracy of evaluation when precise data are considered. In particular, we consider the composite trapezoidal and composite Simpson's rules of integration, and compare them with a statistical approach to obtaining an estimate based on the sample mean. The comparison is first done in the case when the number of traps where the data are available is large. It will be shown in the paper that noise in the data badly affects the accuracy of evaluation on fine grids of traps, so the different methods of numerical integration no longer differ in terms of their accuracy. We then consider an ecologically relevant case of a small number of traps, i.e. when the data available for evaluation are sparse. It will be discussed in the paper that the impact of noise is negligible on coarse grids of traps and therefore we can keep the accuracy hierarchy of numerical integration methods established from the consideration of precise data. We are then able to give recommendations on how to use methods of numerical integration to evaluate pest abundance. Our results are illustrated by numerical experiments.

HETEROGENEOUS COMPUTATION OF RAINBOW OPTION PRICES USING FOURIER COSINE SERIES EXPANSION UNDER A MIXED CPU-GPU COMPUTATION FRAMEWORK

This paper focuses on comparing different heterogeneous computational designs for the calculation of rainbow options prices using the Fourier-cosine series expansion COS method. We also propose a simple enough way to automatically decide the ratio of load balancing at runtime. A general-purpose computing on graphic processing unit implementation of the two-dimensional composite Simpson rule free of conditional statements with some degree of loop unrolling is also introduced. We will also show how to reduce the integration domain of coefficients appearing in the option pricing and by doing so achieve a substantial speed-up and improve accuracy when compared with a straightforward implementation. Copyright © 2014 John Wiley & Sons, Ltd.

Improved Fluorescence Excitation‐Emission Matrix Regional Integration to Quantify Spectra for Fluorescent Dissolved Organic Matter

The purpose of this short communication is to demonstrate the importance of numerical analysis and wavelength increment selection when characterizing fluorescent dissolved organic matter (FDOM) using fluorescence excitation-emission matrix (EEM) regional integration. A variety of water samples, representing a landscape gradient and different types of FDOM, were analyzed for their percentage distribution of five operationally defined FDOM fractions (aromatic protein I, aromatic protein II, fulvic acid-like, soluble microbial byproduct-like, and humic acid-like) using three numerical methods in integrating volume under the surface of the fluorescence EEMs: Riemann summation, composite trapezoidal rule, and composite Simpson's rule. The influence of wavelength increment was also examined for the precision of the percentage distribution of each fraction. Our results show that the FDOM fraction estimated by Riemann summation with a 10- or 5-nm excitation wavelength can cause >40% or >5% errors, respectively, when compared with the best estimated values obtained by averaging results from composite trapezoidal rule and composite Simpson's rule with 1-nm excitation wavelength at the same emission increment. Also, our experiments show that fluorescence matrix regional integration could underestimate the two aromatic protein fractions but could overestimate the soluble microbial byproduct-like and humic acid-like fractions if improper increment and integral methods are used. The error can be reduced if a smaller wavelength increment is used. The smallest increment in a spectrofluorometer and composite Simpson's rule should be used for scanning fluorescence EEMs and calculating the percentage distribution of each FDOM fraction. Alternatively, 5-nm wavelength increments with composite Simpson's rule could be cost effective, and the error of each FDOM fraction commonly falls within 5% compared with those estimated by 1-nm increments.

Thickness effects for a penny-shaped dielectric crack in a magnetoelectroelastic layer

The problem of a penny-shaped dielectric crack in a magnetoelectroelastic layer is considered within the theory of linear magnetoelectroelasticity. Under the assumption of mode-I magnetoelectromechanical loadings, the semipermeable crack-face electromagnetic boundary conditions are applied to describe the case of an opening penny-shaped crack. In order to solve the boundary-value problem, the Hankel transform technique is utilized and three coupled Fredholm integral equations are further derived. The intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) are further determined by the composite Simpson's rule. Thickness effects of a magnetoelectroelastic layer on the electric displacement and magnetic induction of the crack interior, and the field intensity factors are illustrated through numerical computations for a BaTiO3-CoFe2O4 composite. The obtained results reveal that an increase of the ratio of the layer thickness and the crack radius h/a increases the electric displacement and magnetic induction of a dielectric crack interior, and decreases the field intensity factors regardless of the permeability of the crack interior. Based on the COD intensity factor, the influences of applied electric and magnetic fields on the growth of a dielectric crack are further investigated and presented graphically.

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

Abstract The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO 3 -CoFe 2 O 4 composite with a penny-shaped vacuum crack are discussed in detail.

Value-at-risk in US stock indices with skewed generalized error distribution

This investigation proposes a composite Simpson's rule, a numerical integral method, for estimating quantiles on the skewed generalized error distribution (SGED). Daily spot prices of S&P500 and Dow-Jones stock indices are used as data to examine the one-day-ahead VaR (Value at Risk) forecasting performance of the GARCH-N and GARCH-SGED models. Empirical results show that the GARCH-SGED models provide more accurate VaR forecasts than the GARCH-N models for both low and high confidence levels. These findings demonstrate that the use of SGED distribution, which explicitly accommodates both skewness and kurtosis, is essential for out-of-sample VaR forecasting in US stock markets.

MODEL-BASED SOIL PARAMETER IDENTIFICATION FOR WHEEL-TERRAIN INTERACTION DYNAMICS

This paper presents an algorithm for identifying soil parameters for wheel-terrain interaction dynamics. The soil parameters are useful for traversability prediction, traction control, and performance optimization of a wheeled vehicle traveling on unknown terrain. The Composite Simpson's Rule (CSR) is employed to approximate integrals of the full wheel-terrain interaction dynamic model. This is to facilitate the implementation of soil parameter identification on this model and allow fast identification speed. The 2-stage iterative Newton Raphson (NR) method is used for soil parameter identification. Simulation results show successful identification of a complete set of soil parameters with relatively fast speed. The approach in this paper has great potential to be applied for real off-road wheeled vehicle.

A High-Order Difference Scheme for a Nonlocal Boundary-Value Problem for the Heat Equation

Abstract This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.