Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.

A comparison of piecewise cubic Hermite interpolating polynomials,cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.

UV-Vis Spectrum and the Third-order Nonlinear Optical Properties of the Chiral Camphorderived β-diketonate Platinum Complexes

UV-Vis spectrum and the third-order nonlinear optical properties of the chiral camphor-derived β-diketonate have been studied at the B3LYP/6-31G＊ level. The results showed that the introduction of electron-drawing group -CF3 and -C3F7 on β-diketonate made the strongest absorption peak red-shift and the lowest energy absorption blue-shied. Introduction of -OC2H5 on the benzene or pyridine ring made the lowest energy absorption blue-shift. When the -C2H3 was introduced on the benzene or pyridine ring, the lowest energy absorption was red-shifted. Introduction of electron-donating group on β-diketonate can enlarge their nonlinear optical properties. On the contrary, the introduction of electron-drawing group dropped it down.

Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

CHARACTERISTIC THEOREMS OF TH ENATURAL TCHEBYSHEFF SPLINE FUNCTION

In this paper,we give four characteristic theorems of the natural Tchebysheff splint functionassociated with multiple knots.These theorems possess specific form,that arc convenient forapplicaton;In the case of with simple knots or polynomial splint,the corollaries of this paper’s the-orems give corresponding results.

THE APPLICATION OF SPLINE FUNCTION IN THE GEOMETRIC REPRODUCTION OF LOG SHAPE

The paper introduces a method to get three-dimensional reproduction of log shape by adopting Spline Function in fitting the curve of the finite log data. The method has ad-vantages of higher accuracy, less acquired data, easier to use, etc. Making use of high-precision drawing function of computer, the graphs of log geometric shape in different visual angles can be achieved easily with this method. It also provided a firm foundation for the determination of optimum saw cutting scheme.

The Property of a Special Type of Exponential Spline Function

Approximation theory experienced a long term history. Since 50’ last century, the rise of spline function as well as the advance of calculation promotes the growth of classical approximation theory and makes them develop a profound theory in maths, and application values have shown among the field of scientific calculation and engineering technology and etc. At present, the study of spline function had made a great progress and had a lot of fruits, as for that, the reader could look up the book [1] or [2]. Nevertheless, the research staff pays less attention to exponential spline function, since polynomial spline function is a special case of that, so it is much essential and meaningful for one to explore the nature of exponential spline function.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

Numerical Solution of System of Fractional Delay Differential Equations Using Polynomial Spline Functions

The aim of this paper is to approximate the solution of system of fractional delay differential equations. Our technique relies on the use of suitable spline functions of polynomial form. We introduce the description of the proposed approximation method. The error analysis and stability of the method are theoretically investigated. Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

THE SOLUTION OF RECTANGULAR PLATES WITH LARGE DEFLECTION BY SPLINE FUNCTIONS

In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.

Automatically Smoothing the Spectroscopic Data by Cubic B-Spline Basis Functions

In the present paper,a new criterion is derived to obtain the optimum fitting curve while using Cubic B-spline basis functions to remove the statistical noise in the spectroscopic data.In this criterion,firstly,smoothed fitting curves using Cubic B-spline basis functions are selected with the increasing knot number.Then,the best fitting curves are selected according to the value of the minimum residual sum of squares(RSS)of two adjacent fitting curves.In the case of more than one best fitting curves,the authors use Reinsch's first condition to find a better one.The minimum residual sum of squares(RSS)of fitting curve with noisy data is not recommended as the criterion to determine the best fitting curve,because this value decreases to zero as the number of selected channels increases and the minimum value gives no smoothing effect.Compared with Reinsch's method,the derived criterion is simple and enables the smoothing conditions to be determined automatically without any initial input parameter.With the derived criterion,the satisfactory result was obtained for the experimental spectroscopic data to remove the statistical noise using Cubic B-spline basis functions.

Electronic Absorption Spectra and Third-Order Nonlinear Optical Property of Dinaphtho[2,3-b:2’,3’-d]Thiophene-5,7,12,13- Tetraone (DNTTRA) and Its Phenyldiazenyl Derivatives: DFT Calculations

Third-order nonlinear optical (NLO) materials have broad application prospects in high-density data storage, optical computer, modern laser technology, and other high-tech industries. The structures and frequencies of Dinaphtho[2,3-b:2’,3’-d]thiophene-5,7,12,13-tetraone (DNTTRA) and its 36 derivatives containing azobenzene were calculated by using density functional theory B3LYP and M06-2X methods at 6-311++g(d, p) level, respectively. Besides, the atomic charges of natural bond orbitals (NBO) were analyzed. The frontier orbitals and electron absorption spectra of A-G5 molecule were calculated by TD-DFT (TD-B3LYP/6-311++g(d, p) and TD-M06-2X/6-311++g(d, p)). The NLO properties were calculated by effective finite field FF method and self-compiled program. The results show that 36 molecules of these six series are D-π-A-π-D structures. The third-order NLO coefficients γ (second-order hyperpolarizability) of the D series molecules are the largest among the six series, reaching 107 atomic units (10-33 esu) of order of magnitude, showing good third-order NLO properties. Last, the third-order NLO properties of the azobenzene ring can be improved by introducing strong electron donor groups (e.g. -N(CH3)2 or -NHCH3) in the azobenzene ring, so that the third-order NLO materials with good performance can be obtained.

The G^3 spline basis functions

The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.

Numerical Solution of Functional Integral and Integro-Differential Equations by Using B-Splines

This paper describes an approximating solution, based on Lagrange interpolation and spline functions, to treat functional integral equations of Fredholm type and Volterra type. This method extended to functional integral and integro-differential equations. For showing efficiency of the method we give some numerical examples.

APPLICATIONS OF SURFACE SPLINE FUNCTIONS TO STRUCTURAL ANALYSIS IN COAL GEOLOGY

A surface spline function is used to fit a coal seam surface in structural anal ysis in coal geology. From the surface spline function, the first and second partial derivatives can also be derived and used to structural analysis, especially for recogni tion of the concealed structures. The detection of structures related to faulting is em phasized.

Analysis of Stability and Large Deflection Buckle of Rolled Strip by Third Power B-Spline Function Method

A new method——the third power B-spline function method is developed to analyse the stability and the buckle of rolled strip under residual stress.The large deflection theory of thin plate is used to calculate the buckle of rolled strip and criterion of critical buckle is given.The computed results tally with those of experiment well,which provides theoretical basis and method for developing the mathematical model of flatness control.

Optimization of Quantizer’s Segment Threshold Using Spline Approximations for Optimal Compressor Function

In this paper, the optimization of quantizer’s segment threshold is done. The quantizer is designed on the basis of approximative spline functions. Coefficients on which we form approximative spline functions are calculated by minimization mean square error (MSE). For coefficients determined in this way, spline functions by which optimal compressor function is approximated are obtained. For the quantizer designed on the basis of approximative spline functions, segment threshold is numerically determined depending on maximal value of the signal to quantization noise ratio (SQNR). Thus, quantizer with optimized segment threshold is achieved. It is shown that by quantizer model designed in this way and proposed in this paper, the SQNR that is very close to SQNR of nonlinear optimal companding quantizer is achieved.

基于B-Spline的位移监测数据动态优化拟合

为更好地进行工程位移的监测,对基坑施工过程钢支撑水平位移激光监测数据特征、误差类型及其成因进行了分析,研究了B-Spline在海量监测数据去噪中的适用性,并通过引入待估参数的B样条基函数修正,建立最小二乘控制下位移-时间关系的高精度拟合优化。结果表明:待估参数ζ_(i)的修正重构,把监测数据序列转化为参数估计问题,使位移时变关系的最优化拟合成为可能;移动时间坐标方法,可实现等时距监测数据的动态处理,优化计算时间;B-Spline融合最小二乘法,在最小方差控制下实现监测数据-时间关系最优拟合,形成光滑、连续的高精度拟合曲线,有利于提高监测反馈控制的精度和可靠性。实例应用验证了成果的可靠性和适用性。