Relative extrema of Legendre functions of the second kind

Let μ _k,n denote the relative maxima of ?Q _n(x)?, the Legendre function of the second kind, ordered so that μ _k+1,n occurs to the left of μ _k,n . They by analogy with a theorem of Szegö for Legendre polynomials, μ _k,n +1<μ _k,n ,k=1,...,n,n=1,2,...     

Legendre spectral collocation method for volterra-hammerstein integral equation of the second kind

This paper is concerned with obtaining the approximate solution for VolterraHammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω(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 L~2 norm and L~∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.     

Ostrowski不等式的改进及其应用

通过对Ostrowski不等式的改进,扩大了Ostrowski积分公式的适用范围,将该积分公式应用于数值积分推广了经典的中点积分公式、梯形积分公式和Simpson积分公式,同时得到相应的最佳误差限,并给出了具体的数值应用.     

Cardinal functions for Legendre pseudo-spectral method for solving the integro-differential equations

In this article, we present a new numerical method to solve the integro-differential equations (IDEs). The proposed method uses the Legendre cardinal functions to express the approximate solution as a finite series. In our method the operational matrix of derivatives is used to reduce IDEs to a system of algebraic equations. To demonstrate the validity and applicability of the proposed method, we present some numerical examples. We compare the obtained numerical results from the proposed method with some other methods. The results show that the proposed algorithm is of high accuracy, more simple and effective.     

Index transforms with the product of the associated Legendre functions

Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a wedge for a third order partial differential equation.     

Fractional-order Legendre functions for solving fractional-order differential equations

In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Some properties of generalized associated Legendre functions of second kind

We investigate some new properties of generalized associated Legendre polynomials of the second kind, establish new relationships between these polynomials, construct differential operators with the functions P _k ^m,n (z), Q _k ^m,n (z), and describe some applications.Kiev Polytechnical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 34–45, 1989.     

数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.     

Szegö quadrature formulas for certain Jacobi-type weight functions

In this paper we are concerned with the estimation of integrals on the unit circle of the form by means of the so-called Szegö quadrature formulas, i.e., formulas of the type with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions related to the Jacobi functions for the interval nodes and weights in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

     

Gap functions for vector variational inequalities

In this article, we intend to study several scalar-valued gap functions for Stampacchia and Minty-type vector variational inequalities. We first introduce gap functions based on a scalarization technique and then develop a gap function without any scalarizing parameter. We then develop its regularized version and under mild conditions develop an error bound for vector variational inequalities with strongly monotone data. Further, we introduce the notion of a partial gap function which satisfies all, but one of the properties of the usual gap function. However, the partial gap function is convex and we provide upper and lower estimates of its directional derivative.     

一个关于广义Legendre关系猜测的简单证明与推广

该文通过揭示Gauss超几何函数的某些组合形式的单调性,给出了关于广义Legendre恒等式的猜测的一个简单证明,并将此猜测的结果作了进一步推广,有助于特殊函数理论的研究.     

The Legendre Galerkin-Chebyshev collocation method for Burgers-like equations

A Legendre Galerkin–Chebyshev collocation method for Burgers-likeequations is developed. This method is based on the Legendre–Galerkinvariational form, but the nonlinear term and the right-handterm are treated by Chebyshev–Gauss interpolation. Errorestimates of the semi-discrete scheme and the fully discretescheme are given in the L²-norm. Numerical results indicatethat our method is as stable and accurate as the standard Legendrecollocation method, and as efficient and easy to implement asthe standard Chebyshev collocation method.     

Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions

We evaluate explicitly the integrals , with the being any one of the four Chebyshev polynomials of degree . These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing in its interior.

     

Growth behavior of a class of merit functions for the nonlinear complementarity problem

When the nonlinear complementarity problem is reformulated as that of finding the zero of a self-mapping, the norm of the selfmapping serves naturally as a merit function for the problem. We study the growth behavior of such a merit function. In particular, we show that, for the linear complementarity problem, whether the merit function is coercive is intimately related to whether the underlying matrix is aP-matrix or a nondegenerate matrix or anR _o-matrix. We also show that, for the more popular choices of the merit function, the merit function is bounded below by the norm of the natural residual raised to a positive integral power. Thus, if the norm of the natural residual has positive order of growth, then so does the merit function.This work was partially supported by the National Science Foundation Grant No. CCR-93-11621.The author thanks Dr. Christian Kanzow for his many helpful comments on a preliminary version of this paper. He also thanks the referees for their helpful suggestions.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

On the Convolution of the Generalized Finite Legendre Transform

The translation operator and the convolution for the finite Legendre transformation are investigated in the space ??(?1,1) of testing-functions and its dual through an approach that emphasizes the close similarity existing between this transform and the infinite Mehler - Fock transformation. The theory developed is used in solving some distributional boundary-value problems.     

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m ⁴ n ^–6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday