B‐spline spectral method for constrained fractional optimal control problems

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.     

Splines and Linear Control Theory

In this work, the relationship between splines and the linear control theory has been analyzed. We show that spline functions can be constructed naturally from the control theory. By establishing a framework based on control theory, we provide a simple and systematic way to construct splines. We have constructed the traditional spline functions including polynomial splines and the classical exponential spline. We have also discovered some new spline functions such as the combination of polynomial, exponential and trigonometric splines. The method proposed in this paper is easy to implement. Some numerical experiments are performed to investigate properties of different spline approximations.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

Smoothing noisy data for irregular regions using penalized bivariate splines on triangulations

The penalized spline method has been widely used for estimating univariate smooth functions based on noisy data. This paper studies its extension to the two-dimensional case. To accommodate the need of handling data distributed on irregular regions, we consider bivariate splines defined on triangulations. Penalty functions based on the second-order derivatives are employed to regularize the spline fit and generalized cross-validation is used to select the penalty parameters. A simulation study shows that the penalized bivariate spline method is competitive to some well-established two-dimensional smoothers. The method is also illustrated using a real dataset on Texas temperature.     

NEW APPROACH TO THE LIMITER FUNCTIONS

1.IntroductionSince1980's,differenceschemeswithTVDorTVBpropertieshavebeenusedformoreandmoreCFDproblems,especiallythefollowingsystemofconservationlaws:ThereasonisthattheTVBpropertywillguaranteetheconvergenceofanysubsequenceofthedifferencesolutionsequencetoaweeksolutionofthedifferentialequation.Obviouslyiftheweeksolutionisunique,thenthewholesequencewillconvergetothatsolution.OneofthefrequentlyusedTVDschemeisthesecondorderfive--pointconservativeone:HereHi 1/2~H(Ull,,Uln,Uz71,U17,),iscons…     

Spline approximant in Hilbertian subspaces ofC(Ω)

Summary In this paper we give a new approach of approximation by spline functions. We define and study approximant spline functions which can be easly calculated without solving a linear system. We investigate also the error in using approximant spline functions.     

A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form

In this article, we report an efficient high order numerical method based on cubic spline approximation and application of alternating group explicit method for the solution of two point non-linear boundary value problems, whose forcing functions are in integral form, on a non-uniform mesh. The proposed method is applicable when the internal grid points of solution interval are odd in number. The proposed cubic spline method is also applicable to integro-differential equations having singularities. Computational results are given to demonstrate the utility of the method.     

Multivariate approximation by PDE splines

This work deals with an approximation method for multivariate functions from data constituted by a given data point set and a partial differential equation (PDE). The solution of our problem is called a PDE spline. We establish a variational characterization of the PDE spline and a convergence result of it to the function which the data are obtained. We estimate the order of the approximation error and finally, we present an example to illustrate the fitting method.     

构造非均匀剖分上局部支集样条函数的积分方法

１　引　　言 给定递增数列｛ｘｉ｝,｛ｙｉ｝,｛ｚｋ｝．对于Ｒ２,由直线｛ｘ＝ｘｉ）和｛ｙ＝ｙｊ｝形成的剖分称为非均匀矩形剖分;再连接每个矩形［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］的正斜率对角线形成的剖分称为非均匀Ｉ－型三角剖分,记为　１;若连接每个矩形的两条对角线,则所得剖分为非均匀Ⅱ－型三角剖分,记为　２．对于Ｒ３,由平面｛ｘ＝ｘｉ｝,｛ｙ＝ｙｊ｝,｛ｚ＝ｚｋ｝形成的剖分称为非均匀长方体剖分;若再连接每个小长方体［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］×［ｚｋ,ｚｋ＋１］的过点（ｘｉ,ｙｊ,ｚ…     

ANTIPERIODIC WAVELETS

1.PreliminaryAswekn0w,manyauth0rshavemadegreateff0rtsinconstructingthe0rthonormalorbiorthon0rmalbasis0nthewholereallinem10ronthewh0len-dimensionalspaceMnl"7],butinmanypracticalpr0blems0neneedst0construct0rthon0rmalbasisonsomefiniteintervawiths0meboundaryconditi0ns.HerewepresentamethodofconstructingtheantiperiodicorthonormalwaveletsbasisontheintervaI=[0,2x1.Maindifficultyintheaboveproblemisthec0nstructi0noftheorthonormalbasisofWm-l-theorth0g0nalc0mplement0fVm-1inVm-thekeystepisthatwehavetocon…     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

基于再生核三次样条基底的构造及其应用

将三次样条理论与再生核理论相结合,利用再生核函数巧妙地构造了三次样条函数空间的一组基底.基于三次样条插值的高收敛特点,得到了微分方程边值问题近似解的一种新的求解方法.数值算例展现出算法简单、有效.     

WEB‐spline based mesh‐free finite element analysis for the heat equation and the time‐dependent Navier–Stokes equation: A survey

In this article, we give some numerical techniques and error estimates using web‐spline based mesh‐free finite element method for the heat equation and the time‐dependent Navier–Stokes equations on bounded domains. The web‐spline method uses weighted extended B‐splines on a regular grid as basis functions and does not require any grid generation. We demonstrate the method by providing numerical results for the Poisson's and stationary Stokes equation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Dimensions of spline spaces over T-meshes

A T-mesh is basically a rectangular grid that allows T-junctions. In this paper, we propose a method based on Bézier nets to calculate the dimension of a spline function space over a T-mesh. When the order of the smoothness is less than half of the degree of the spline functions, a dimension formula is derived which involves only the topological quantities of the T-mesh. The construction of basis functions is briefly discussed. Furthermore, the dimension formulae for T-meshes after mesh operations, such as edge insertion and mesh merging, are also obtained.     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.     

Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C¹ quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on the tensor products of spline quadratures and classical composite Simpson’s rules.     

混合体积、对数凹序列和B样条函数

本文主要通过样条函数方法研究与之相关的离散几何学和组合学问题.在离散几何学方面主要考虑超立方体切面(cube slicing)体积和混合体(mixed volume)的样条表示,利用B样条函数的几何解释,将超立方体切面问题转化为与之等价的样条函数问题,分别给出Laplace和P′olya关于超立方体切面定理的样条证明,将样条函数与混合体积联系起来,给出一类混合体积的样条解释.利用这种解释可以得到一类具有对数凹性质的组合序列,从而部分地回答了Schmidt和Simion所提出的关于混合体积的公开问题.在组合数学方面主要考虑多种组合多项式与样条函数的关联以及组合序列对数凹性质的样条方法研究.本文借助丰富的样条函数理论,不但验证了离散几何学和组合数学中很多现有的结果,而且得到了一系列离散数学对象的新性质,建立了离散数学问题与具有连续性特质的样条函数之间的内在联系.     

Spline Smoothing on Surfaces

This article presents a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a finite element method. The article has a substantial tutorial component: we start by reviewing smoothness measures for functions defined on surfaces, simplicial surfaces and differentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, we show results of an experiment comparing finite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an effective way of determining the optimal degree of smoothing for function estimation on surfaces.     

Extrema of matrix functionals

The problem of determining conditional extrema of functionals with matrix arguments is considered. We derive the necessary and sufficient mathematical conditions for the existence of extrema of functionals satisfying constraints of the form of matrix equalities on the arguments. The construction of extrema is based on functions and matrices of indeterminate Lagrange multipliers. As applications we consider an example of determining the optimal strength coefficient matrix in a dynamical system with an adaptive Carleman filter and an example, from the theory of statistical decisions, of minimizing the volume of the dispersion error ellipsoid. Our approach has wide applications not only in optimization problems from automatic control theory but also in mathematical statistics and the theory of material strength and plasticity.Translated from Dinamicheskie Sistemy, No. 5, pp. 103–106, 1986.