用二次插值实现近似弧长参数化

分段二次Ｈｅｒｍｉｔｅ插值用来保单调地反插值参数曲线的弧长函数．所作近似弧长参数化曲线在插值节点处,近似弧长是精确的,并且具有与精确弧长参数曲线同方向的单位切矢．在整个近似弧长参数区间,近似弧长的误差可达到０（△ｔ）＾２（△ｔ为节点步长）．数值实例得到了很好的结果．     

Sparse shape functions for tetrahedral <Emphasis Type="Italic">p</Emphasis>-FEM using integrated Jacobi polynomials

Summary In this paper, we investigate the discretization of an elliptic boundary value problem in 3D by means of the hp-version of the finite element method using a mesh of tetrahedrons. We present several bases based on integrated Jacobi polynomials in which the element stiffness matrix has nonzero entries, where p denotes the polynomial degree. The proof of the sparsity requires the assistance of computer algebra software. Several numerical experiments show the efficiency of the proposed bases for higher polynomial degrees p.      

Gröbner bases with respect to several orderings and multivariable dimension polynomials

Let D=K[X]$D=K\left[X\right]$ be a ring of Ore polynomials over a field K$K$ and let a partition of the set of indeterminates into p$p$ disjoint subsets be fixed. Considering D$D$ as a filtered ring with the natural p$p$-dimensional filtration, we introduce a special type of reduction in a free D$D$-module and develop the corresponding Gröbner basis technique (in particular, we obtain a generalization of the Buchberger Algorithm). Using such a modification of the Gröbner basis method, we prove the existence of a Hilbert-type dimension polynomial in p$p$ variables associated with a finitely generated filtered D$D$-module, give a method of computation and describe invariants of such a polynomial. The results obtained are applied in differential algebra where the classical theorems on differential dimension polynomials are generalized to the case of differential structures with several basic sets of derivation operators.     

A note on the numerical solution of non-linear parabolic equations in chebyshev series

A new method for the numerical solution of non linear parabolic equations is presented. The method is an extension of an existing algorithm for linear equations. Solutions are obtained in the form of a Chebyshev series, which is produced by approximating the partial differential equation by a set of ordinary differential equations over a small time interval. The method appears to be both accurate and economical.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.