非线性抛物型方程的二次元有限体积元方法

In this paper, a fully discrete finite volume element method for a class of second order nonlinear parabolic equations is given. Piecewise quadratic trial functions and piecewise constant test functions are used to obtain error estimates. A numerical example is given at, the end to show the feasibility of the method.     

A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems

A family of fourth and second-order accurate numerical schemes is presented for the solution of fifth-order boundary-value problems with two-point-boundary conditions. The non-polynomial sextic spline functions are applied to construct the numerical algorithms. This approach generalizes polynomial spline algorithms, and provides solution at every point of range of integration. Convergence of the methods is discussed through standard convergence analysis. A numerical illustration is given to show the pertinent features of the technique.     

A basis function for approximation and the solutions of partial differential equations

In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A new computational method for solving two‐dimensional Stratonovich Volterra integral equation

This paper presents a method for computing numerical solutions of two‐dimensional Stratonovich Volterra integral equations using one‐dimensional modification of hat functions and two‐dimensional modification of hat functions. The problem is transformed to a linear system of algebraic equations using the operational matrix associated with one‐dimensional modification of hat functions and two‐dimensional modification of hat functions. The error analysis of the method is given. The method is computationally attractive, and applications are demonstrated by a numerical example. Copyright © 2017 John Wiley & Sons, Ltd.     

A remainder formula of numerical differentiation for the generalized Lagrange interpolation

Through exploiting the generalized Lagrange interpolation for continuous piecewise smooth functions, a remainder term of numerical differentiation is given. Also, the error estimates for the numerical differentiation formula are presented.     

On the discretisation error of the weighted Simpson rule

A method of estimating the discretisation error for the weighted Simpson rule is given. The results are used to obtain the convergence rates of a numerical method for solving integral equations with various singular kernel functions.     

QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach

In this paper we consider numerical integration of smooth functions lying in a particular reproducing kernel Hilbert space. We show that the worst-case error of numerical integration in this space converges at the optimal rate, up to some power of a log?N factor. A similar result is shown for the mean square worst-case error, where the bound for the latter is always better than the bound for the square worst-case error. Finally, bounds for integration errors of functions lying in the reproducing kernel Hilbert space are given. The paper concludes by illustrating the theory with numerical results.     

A method to calculate the distribution function when the characteristic function is known

A formula for numerical inversion of characteristic functions based on the Poisson formula is presented, and a numerical example is also given.     

Construction and optimization of predictions on the basis of first-degree recurrence splines

The problem of point and interval predictions of time series on the basis of first-degree recurrence splines with depth 1 is investigated. Optimality conditions for the coefficients of the calculation formula are found. Results of numerical experiments for analytically given functions with errors are presented. A comparison with classical methods of prediction of time series is given.     

非线性抛物型方程组的二次有限体积元方法及其误差估计

讨论基于三角形网格的二维非线性抛物型方程组的有限体积元方法,其中试探函数空间为二次Lagrange元,检验函数空间为分片常数函数空间,对问题的全离散格式证明了最优的能量模误差估计。最后给出一个相关数值算例以验证格式的有效性。     

On tests for the mean direction of the Langevin distribution

The asymptotic expansions of the distribution of a sum of independent random vectors with Langevin distribution are given. The power functions of the likelihood ratio criterion, Watson statistic, Rao statistic and the modified Wald statistic for testing the hypothesis of the mean direction are obtained asymptotically and a numerical comparison is made.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.     

Some remarks on the efficient algorithms of numerical statistical simulation

Some numerical algorithms for the simulation of random variables and functions, as well as for parametrical numerical statistical analysis are considered. Specifications and explanations of the algorithms important for practical purposes are given.     

Computational properties of three-term recurrence relations for Kummer functions

Several three-term recurrence relations for confluent hypergeometric functions are analyzed from a numerical point of view. Minimal and dominant solutions for complex values of the variable z are given, derived from asymptotic estimates of the Whittaker functions with large parameters. The Laguerre polynomials and the regular Coulomb wave functions are studied as particular cases, with numerical examples of their computation.     

An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy

In this paper, we introduce an algorithm and a computer code for numerical differentiation of discrete functions. The algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points. The algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas, in particular when the derivatives are required with high approximation accuracy. Moreover, the given Matlab computer code can be implemented to solve boundary-value ordinary and partial differential equations with high numerical accuracy. The numerical technique is based on the undetermined coefficient method in conjunction with Taylor’s expansion. To avoid the difficulty of solving a system of linear equations, an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions. This is done by using an explicit closed formula for the Vandermonde matrix inverse. Moreover, the code is designed to give a unified approximation order throughout the given domain. A numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code. It is found that the method and the code work properly for any degree of derivative and any order of accuracy.     

一种基于小波尺度函数变换的Laplace反演方法

该文基于Ｄａｕｂｅｃｈｉｅｓ小波尺度函数变换建立了关于Ｌａｐｌａｃｅ变换的一种反演数值方法．通过对小波尺度函数的低带通谱特性的定性与定量讨论,给出了这一反演方法所得原像函数的适用域．结果发现：其区域大小随着小波尺度函数的分辨指标（ｒｅｓｏｌｕｔｉｏｎｌｅｖｅｌ）选取的升高而增大．最后,以颤振曲线、具有指数增长的复函数、和一维振动弦的初边值问题等为例,定量给出了其反演方法的数值结果．通过与相应的原像精确结果对比发现：在反演的有效区域内,其数值反演的原像几乎与精确的原像图象重合．这表明这一Ｌａｐｌａｃｅ反演数值方法是有效和可靠的．     

Iterative variants of the Nyström method for the numerical solution of integral equations

Some iterative variants of the Nyström method for the numerical solution of linear and nonlinear integral equations are introduced and discussed. Numerical examples are given; some are for integral equations with singular kernel functions.     

Numerical solution of the nonlinear Fredholm integral equations by positive definite functions

A numerical method for solving the nonlinear Fredholom integral equations is presented. The method is based on interpolation by radial basis functions (RBF) to approximate the solution of the Fredholm nonlinear integral equations. Several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method.     

Modified quadrature formulas for functions with nearby poles

This paper is concerned with the numerical integration of functions with poles near the interval of integration. A method is given for modifying known quadrature rules, to obtain rules which are exact for certain classes of rational functions.