Asymptotic representation of the weights in the Gauss quadrature formula

1990 ◽  
Vol 47 (4) ◽  
pp. 354-358
Author(s):  
A. L. Marshak
Keyword(s):  
Quadrature Formula ◽  
Asymptotic Representation ◽  
Gauss Quadrature ◽  
Gauss Quadrature Formula

scholarly journals Wavelet-weighted Gauss quadrature formula for reduction of computational work in wavelet BEM

2001 ◽  
Vol 4 ◽  
pp. 127-136
Author(s):  
Kazuhiro KORO ◽  
Kazuhisa ABE ◽  
Hideyuki HIRABAYASHI
Keyword(s):  
Quadrature Formula ◽  
Gauss Quadrature ◽  
Computational Work ◽  
Gauss Quadrature Formula

scholarly journals Asymptotic error estimates for the Gauss quadrature formula

1968 ◽  
Vol 22 (101) ◽  
pp. 91-91 ◽  
Author(s):  
M. M. Chawla ◽  
M. K. Jain
Keyword(s):  
Error Estimates ◽  
Quadrature Formula ◽  
Gauss Quadrature ◽  
Asymptotic Error ◽  
Gauss Quadrature Formula

scholarly journals Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
M. El-Kady ◽  
S. M. El-Sayed ◽  
H. E. Fathy
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Quadrature Formula ◽  
Numerical Results ◽  
Basis Functions ◽  
Gauss Quadrature ◽  
Gauss Quadrature Formula ◽  
Huxley Equation ◽  
Numerical Treatments

Numerical treatments for the generalized Burger's—Huxley GBH equation are presented. The treatments are based on cardinal Chebyshev and Legendre basis functions with Galerkin method. Gauss quadrature formula and El-gendi method are used to convert the problem into a system of ordinary differential equations. The numerical results are compared with the literatures to show efficiency of the proposed methods.

scholarly journals A p-version of the collocation method for solving the Fredholm integral equations of the second kind in the Mathematica environment

2019 ◽  
pp. 1-11
Author(s):  
Е.В. Ворожцов ◽  
В.П. Шапеев
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Quadrature Formula ◽  
Gauss Quadrature ◽  
Fredholm Integral Equations ◽  
Rounding Errors ◽  
Solution Of Equations ◽  
Gauss Quadrature Formula

Предложена и реализована p-версия метода коллокации численного решения интегральных уравнений Фредгольма второго рода. В данной реализации осуществлены возможности варьирования степени полинома в полиномиальном представлении приближенного решения уравнений и варьирования количества узлов используемой квадратурной формулы Гаусса для влияния на точность решения. Исследовано влияние числа точек коллокации, использованных для аппроксимации решения, и количества узлов квадратурной формулы Гаусса на число обусловленности системы линейных алгебраических уравнений, к решению которой сводится построение приближенного решения, и на его точность путем численного решения примеров, в том числе приведенных в известных изданиях. Предложенный алгоритм реализован на языке программного пакета Mathematica. Во всех рассмотренных примерах предложенная версия метода коллокации позволила достичь точности решения уравнений, близкой к уровню машинных ошибок округления. Программный продукт, реализующий предложенную p-версию, получился достаточно компактным, а метод оказался экономичным: машинное время, необходимое для решения рассмотренных в работе задач, не превышало 3 секунды работы персонального компьютера. Описан алгоритм, позволяющий оценить точность приближенного решения по предложенной p-версии метода в тех случаях, когда точное решение интегрального уравнения неизвестно. A p-version of the collocation method for the numerical solution of Fredholm integral equations of the second kind is proposed and implemented. In the considered implementation, the possibilities are realized for the variation of the polynomial degree in the polynomial representation of the approximate solution of equations and the variation of the number of nodes of the employed Gauss quadrature formula to affect the solution accuracy. The influence of the number of collocation points used for the solution approximation and of the number of nodes of the Gauss quadrature formula on the condition number of the system of linear algebraic equations to the solution of which the construction of the approximate solution is reduced and on its accuracy are studied by the numerical solution of examples, including some examples presented in well-known publications. The proposed algorithm is implemented in the language of the program package Mathematica. In all considered examples, the proposed version of the collocation method has enabled us to reach the accuracy of the solution of equations, which is close to the level of the machine rounding errors. The program product implementing the proposed p-version has proved to be compact and the method turned out to be economical: the machine time required for the solution of problems considered in the paper did not exceed 3 seconds of the CPU time of a personal computer. We describe an algorithm allowing us to estimate the accuracy of the approximate solution obtained by the proposed p-version of the method in the cases where the exact solution of the integral equation is unknown.

A Gauss quadrature formula for Cauchy type integrals

1988 ◽  
Vol 4 (2) ◽  
pp. 137-148 ◽  
Author(s):  
G. Tsamasphyros ◽  
G. Dimou
Keyword(s):  
Quadrature Formula ◽  
Gauss Quadrature ◽  
Cauchy Type ◽  
Cauchy Type Integrals ◽  
Gauss Quadrature Formula

scholarly journals Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

2020 ◽  
Vol 146 (4) ◽  
pp. 699-728
Author(s):  
Patricia Díaz de Alba ◽  
Luisa Fermo ◽  
Giuseppe Rodriguez
Keyword(s):  
Integral Equations ◽  
Numerical Approximation ◽  
Gauss Quadrature ◽  
Upper And Lower Bounds ◽  
Stability And Convergence ◽  
Quadrature Rules ◽  
Fredholm Integral Equations ◽  
Numerical Tests ◽  
Gauss Quadrature Formula ◽  
Gauss Quadrature Rules

AbstractThis paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.

Sign in / Sign up

Export Citation Format

Share Document