Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

General Interpolation Formulae for Barycentric Blending Interpolation

General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.

Efficient object location determination and error analysis based on barycentric coordinates

In this paper,we propose an efficient computational method for converting local coordinates to world coordinates using specially structured coordinate data.The problem in question is the computation of world coordinates of an object throughout a motion,assuming that we only know the changing coordinates of some fixed surrounding reference points in the local coordinate system of the object.The proposed method is based on barycentric coordinates;by taking the aforementioned static positions as the vertices of a polyhedron,we can specify the coordinates of the object in each step with the help of barycentric coordinates.This approach can significantly help us to achieve more accurate results than by using other possible methods.In the paper,we describe the problem and barycentric coordinate-based solution in detail.We then compare the barycentric method with a technique based on transformation matrices,which we also tested for solving our problem.We also present various diagrams that demonstrate the efficiency of our proposed approach in terms of precision and performance.

Numerical Solution of Two Dimensional Fredholm Integral Equations of the Second Kind by the Barycentric Lagrange Function

This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

Spatiotemporal characteristics of cultivated land use eco-efficiency and its influencing factors in China from 2000 to 2020

Improving cultivated land use eco-efficiency(CLUE)can effectively promote agricultural sustainability,particularly in developing countries where CLUE is generally low.This study used provincial-level data from China to evaluate the spatiotemporal evolution of CLUE from 2000 to 2020 and identified the influencing factors of CLUE by using a panel Tobit model.In addition,given the undesirable outputs of agricultural production,we incorporated carbon emissions and nonpoint source pollution into the global benchmark-undesirable output-super efficiency-slacks-based measure(GB-US-SBM)model,which combines global benchmark technology,undesirable output,super efficiency,and slacks-based measure.The results indicated that there was an upward trend in CLUE in China from 2000 to 2020,with an increase rate of 2.62%.The temporal evolution of CLUE in China could be classified into three distinct stages:a period of fluctuating decrease(2000-2007),a phase of gradual increase(2008-2014),and a period of rapid growth(2015-2020).The major grain-producing areas(MPAs)had a lower CLUE than their counterparts,namely,non-major grain-production areas(non-MPAs).The spatial agglomeration effect followed a northeast-southwest strip distribution;and the movement path of barycentre revealed a"P"shape,with Luoyang City,Henan Province,as the centre.In terms of influencing factors of CLUE,investment in science and technology played the most vital role in improving CLUE,while irrigation index had the most negative effect.It should be noted that these two influencing factors had different impacts on MPAs and non-MPAs.Therefore,relevant departments should formulate policies to enhance the level of science and technology,improve irrigation condition,and promote sustainable utilization of cultivated land.

Distributed planar formation maneuvering of leader-follower networked systems via a barycentric coordinate-based approach

This paper presents a distributed planar leader-follower formation maneuver control strategy for multi-agent systems with different agent dynamic models.This method is based on the barycentric coordinate-based(BCB)control,which can be performed in the local coordinate frame of each agent with required local measurements.By exploring the properties of BCB Laplacians,a time-varying target formation can be BCB localizable by a sufficient number of leaders uniquely,and this formation is converted from a given nominal formation with geometrical similarity transformation.The proposed control laws can continuously maneuver collective single-and double-integrator agents to achieve a translation,scale,rotation,or even their compositions in various directions.For the formation shape control problem of multi-car systems with/without saturation constraints,the obtained control performance can preserve good robustness.Global stability is also proven by mathematical derivations and verified by numerical simulations.

On the Metric Dimension of Barycentric Subdivision of Cayley Graphs

In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W） of v with respect to W is the k-tuple （d（v, w1）, d（v, w2）,…, d（v, wk））. The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β（G）. Metric dimension is a generalization of affine dimension to arbitrary metric spaces （provided a resolving set exists）. In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay （Zn Z2）. We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay （Zn Z2）.

Is the Growth of the Astronomical Unit Caused by the Allais Eclipse Effect?

In addition to the Pioneer anomaly and the Earth flyby anomaly for spacecraft, other unexplained anomalies disrupt the solar system dynamics, like the astronomical unit. We show in this paper that the Allais eclipse effect causes the major part of the growth of the length scale for the entire solar system. It is the rough disturbance on the barycenter Earth-Moon implying the Sun that was recorded in the movement of the paraconical pendulum. Earth and Moon revolve around their common center of gravity, which in turn orbits the Sun, and the perturbation of the eclipse hits this double, coupled Kepler’s movements. The thesis of the tidal friction supports that oceanic tidal friction transfers the angular momentum of the Earth to the Moon and slows down the rotation of the Earth while taking away the Moon. However, we think that there are not enough shallow seas to sanction this interpretation. The Earth-Moon tidal system might be inaccurate or unreliable in determining the Earth’s actual rotational spin-down rate. Our assertion is that the change in the Earth’s rotation is caused by a repulsive gravitational interaction during solar eclipse. The perturbation would submit to variations and distortions the region of the barycenter of the Earth-Moon system which revolves around the Sun, with the dual secular effects that the Moon spirals outwards and that the Earth-Moon system goes away from the Sun.

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.

Modified virtual internal bond model based on deformable Voronoi particles

In last time,the series of virtual internal bond model was proposed for solving rock mechanics problems.In these models,the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds).It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio,namely,0.25 for threedimensional cases.So the shear springs used to represent the diversity of the Poisson ratio.However,the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements.In this letter,we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles.The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds).The movements of particle’s centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells.For each bond,there are the corresponded dual face of some Voronoi cell.We can consider the normal bond as some beam and in this case,the appropriate face of Voronoi cell will be a cross section of this beam.If during deformation the Voronoi face was expand,then,according Poisson effect,the length of bond should be decrees.The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio.Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson’s ratio in interval from–0.5 to–0.1.

High-precision stress determination in photoelasticity

Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics.The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization.To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM)is proposed.The derivatives of the shear stress on the calculation path are determined by using the differential matrix which converts the differential form of the equations of equilibrium into a series of algebraic equations.The advantage of the proposed method is that the auxiliary lines,grids,and error accumulation which are commonly used in traditional shear difference methods(SDMs)are not required.Simulation and experimental results indicate that the proposed meshless method is able to provide high computational accuracy in the full-field stress determination.

New Approach to Approximate Circular Arc by Quartic Bezier Curve

This paper presents a result of approximation an arc circles by using a quartic Bezier curve. Based on the barycentric coordinates of two and three combination of control points, the interior control points are determined by forcing the curvature at median point as similar as the given curvature at end points. Hausdorff distance is used to investigate the order of accuracy compare to the actual arc circles through central angle of . We found that the optimal approximation order is eight which is somewhat similar to preceding methods in the literatures.

Real-Time Tool for Affine Transformations of Two Dimensional IFS Fractal

This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the points that constitute the image ofa fractal, and thus enables any affine transformation of the affine basis, done by click-and-drag, to be immediately followed by the same affine transformation of the fractal. The barycentric coordinates can be relative to an arbitrary affine basis of ~2, but in order to have a better control over the fractal, a kind of minimal simplex that contains the fractal attractor is used.

Are Uranus &Neptune Responsible for Solar Grand Minima and Solar Cycle Modulation?

Detailed solar Angular Momentum (AM) graphs produced from the Jet Propulsion Laboratory (JPL) DE405 ephemeris display cyclic perturbations that show a very strong correlation with prior solar activity slowdowns. These same AM perturbations also occur simultaneously with known solar path changes about the Solar System Barycentre (SSB). The AM perturbations can be measured and quantified allowing analysis of past solar cycle modulations along with the 11,500 year solar proxy records (14C & 10Be). The detailed AM information also displays a recurring wave of modulation that aligns very closely with the observed sunspot record since 1650. The AM perturbation and modulation is a direct product of the outer gas giants (Uranus & Neptune). This information gives the opportunity to predict future grand minima along with normal solar cycle strength with some confidence. A proposed mechanical link between solar activity and planetary influence via a discrepancy found in solar/planet AM along with current AM perturbations indicate solar cycle 24 & 25 will be heavily reduced in sunspot activity resembling a similar pattern to solar cycles 5 & 6 during the Dalton Minimum (1790-1830).

Earth-Moon Barycentre Excursions and Anomalous Quaternary Sea Level Highstands

Plate tectonics is driven by Earth-Moon barycentre shifts in the lower mantle. The eastern Canary Islands have geographic and geological conditions derived from the movements of the Central American plates. Some features of these islands are influenced by the rotation of the Earth from west to east in the evolution of the marine currents that surround them and the opening of the North Atlantic to the North Pole with little dependence of the glacial isostatic adjustment (GIA). In addition, their position with respect to the Tropic of Cancer and the African continent affect the north-south and east-west climatic change dynamics and their tectonic stability respectively. Dated lavas contain marine and aeolian deposits and some of the Pleistocene marine deposits indicate higher sea level in cooler circumstances, which is anomalous. Relating those marine deposits produced during the warmest interglacial, the last interglacial and the Holocene with their equivalents in the Southern Hemisphere, they reflect shifts in the barycentre. Thanks to Holocene radiocarbon, topographic and day length data and alkenone temperature, we describe a mechanism by which the oscillation of the Moon’s inclination (and declination) reaches extreme values (14ºand 34ºabout 4.9ºmore than current values) approximately every 1450 years. These values occur when there is a harmonic distortion in surface areas of the Earth’s crust as response associated with oscillations in the displacements of the barycentre of the Earth-Moon system. As the declination influences the movement of oceanic waters, there is also a relationship with the Bond Events of the North Atlantic, of unknown cause until now.

A New Economy Forecasting Method Based on Data Barycentre Forecasting Method

A new and useful method of technology economics, parameter estimation method, was presented in light of the stability of gravity center of object in this paper. This method could deal with the fitting and forecasting of economy volume and could greatly decrease the errors of the fitting and forecasting results. Moreover, the strict hypothetical conditions in least squares method were not necessary in the method presented in this paper, which overcame the shortcomings of least squares method and expanded the application of data barycentre method. Application to the steel consumption volume forecasting was presented in this paper. It was shown that the result of fitting and forecasting was satisfactory. From the comparison between data barycentre forecasting method and least squares method, we could conclude that the fitting and forecasting results using data barycentre method were more stable than those of using least squares regression forecasting method, and the computation of data barycentre forecasting method was simpler than that of least squares method. As a result, the data barycentre method was convenient to use in technical economy.

Chebfun and numerical quadrature

Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.

ON TRIANGULAR C^1 SCHEMES: A NOVEL CONSTRUCTION

In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.