ON THE ERROR OF QUADRATURE FORMULAE FOR CAUCHY PRINCIPAL VALUE INTEGRALS BASED ON PIECEWISE INTERPOLATION

We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.

ON QUADRATURE FORMULAE FOR SINGULAR INTEGRALS OF ARBITRARY ORDER

Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.

The Maximum Trigonometric Degrees of Quadrature Formulae with Prescribed Nodes

The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with m prescribed nodes and n unknown additional nodes in the interval（-π, π]. We show that for a fixed n, the quadrature formulae with m and m ＋ 1 prescribed nodes share the same maximum degree if m is odd. We also give necessary and sufficient conditions for all the additional nodes to be real, pairwise distinct and in the interval（-π, π] for even m, which can be obtained constructively. Some numerical examples are given by choosing the prescribed nodes to be the zeros of Chebyshev polynomials of the second kind or randomly for m ≥ 3.

Quadrature Formula of Singular Integral Based on Rational Interpolation

We construct a quadrature formula of the singular integral with the Chebyshev weight of the second kind by using Lagrange interpolation based on the rational system {1/(x?a 1), 1/(x?a 2), …}, and both the remainder and convergence of the quadrature formula established here are discussed. Our results extend some classical ones.

Generalized Gaussian Quadrature Formulas Based on Chebyshev Nodes with Explicit Coefficients

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the(n-1)st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].

OPTIMAL QUADRATURE OF THE SOBOLEV CLASS W_1~r(R) DEFINED ON WHOLE REAL AXIS

In this paper,we study the optimal quadrature problem with Hermite-Birkhoff type,on the Sobolev class(R)defined on whole red axis,and we give an optimal algorithm and determite its optimal error.

Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series

In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line are carefully studied. This research is a continuation of the results in the [1]-[3]. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]). In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series , where .

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

LEGENDRE SPECTRAL COLLOCATION METHOD FOR VOLTERRA-HAMMERSTEIN INTEGRAL EQUATION OF THE SECOND KIND

This paper is concerned with obtaining the approximate solution for Volterra- Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function w（x） = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L2 norm and L^∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.

Convergence analysis for delay Volterra integral equation

In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction.

Space Discretization of Time-Fractional Telegraph Equation with Mamadu-Njoseh Basis Functions

In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.

Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs

In this paper, a group of Gauss-Legendre iterative methods with cubic convergence for solving nonlinear systems are proposed. We construct the iterative schemes based on Gauss-Legendre quadrature formula. The cubic convergence and error equation are proved theoretically, and demonstrated numerically. Several numerical examples for solving the system of nonlinear equations and boundary-value problems of nonlinear ordinary differential equations (ODEs) are provided to illustrate the efficiency and performance of the suggested iterative methods.

On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

A random simulation method was used for treatment of systems of Volterra integral equations of the second kind. Firstly, a linear algebra system was obtained by discretization using quadrature formula. Secondly, this algebra system was solved by using relaxed Monte Carlo method with importance sampling and numerical approximation solutions of the integral equations system were achieved. It is theoretically proved that the validity of relaxed Monte Carlo method is based on importance sampling to solve the integral equations system. Finally, some numerical examples from literatures are given to show the efficiency of the method.

GENERALIZED GAUSSIAN QUADRATURE FORMULASWITH CHEBYSHEV NODES

Explicit expressions of the Cotes numbers of the generalized Gaussian quadrature formulas for the Chebyshev nodes (of the first kind and the second kind) and their asymptotic behavior are given.

Optimal Quadrature Problem on n-Information for Hardy-Sobolev Classes

For β 〉 0 and an integer r 〉 2, denote by H∞,β those 2π-periodic, real-valued functions f on R, which are analytic in Sβ = {z ∈ C： [ImzI 〈β} and satisfy the restriction If（r）（z）[ ≤ 1, z ∈ Sβ. The optimal quadrature formulae about information composed of the values of a function and its kth （k ： 1,..., r - 1） derivatives on free knots for the classes H∞,β are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.

ON SOME MULTI-DIMENSIONAL QUADRATURE FORMULAS WITH NUMBER-THEORETIC NETS

In the present paper some multi-dimensional quadrature formulas of periodic functions are established by means of the number-theoretic method.Some results of Hua and Wang are generalized or improved.

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

Discusses an extremal problem with Birkhoff interpolation constraints. Quadrature formula derived from an extremal problem; Overview of auxiliary lemmas; Derivation of some properties of polynomials.

A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS

We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation

The Iyengar Type Inequalities with Exact Estimations and the Chebyshev Central Algorithms of Integrals

In this paper, both low order and high order extensions of the Iyengar type inequality are obtained. Such extensions are the best possible in the same sense as that of the Iyengar inequality. hzrthermore, the Chebyshev central algorithms of integrals for some function classes and some related problems are also considered and investigated.

Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods.