scholarly journals The order of convergence of an optimal quadrature formula with derivative in the space W(2,1)2

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3835-3844
Author(s):  
A.R. Hayotov ◽  
R.G. Rasulov
Keyword(s):  
Quadrature Formula ◽  
Order Of Convergence ◽  
Discrete Analog ◽  
Trapezoidal Rule ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
First Derivative ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The present work is devoted to extension of the trapezoidal rule in the space W(2,1)2. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of an integrand. Using the discrete analog of the operator d2/dx2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span{1,x,ex,e-x}. Finally, in the space W(2,1) 2 the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L(2)2 .

scholarly journals Optimal interpolation formulas with derivative in the space L(m)2(0,1)

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5661-5675
Author(s):  
M.Kh. Shadimetov ◽  
A.R. Hayotov ◽  
F.A. Nuraliev
Keyword(s):  
Order Of Convergence ◽  
Linear Equations ◽  
Interpolation Formula ◽  
Discrete Analog ◽  
Optimal Interpolation ◽  
System Of Linear Equations ◽  
Lagrange Method ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The paper studies the problem of construction of optimal interpolation formulas with derivative in the Sobolev space L(m)2 (0,1). Here the interpolation formula consists of the linear combination of values of the function at nodes and values of the first derivative of that function at the end points of the interval [0,1]. For any function of the space L(m)2 (0, 1) the error of the interpolation formulas is estimated by the norm of the error functional in the conjugate space L(m)* 2 (0,1). For this, the norm of the error functional is calculated. Further, in order to find the minimum of the norm of the error functional, the Lagrange method is applied and the system of linear equations for coefficients of optimal interpolation formulas is obtained. It is shown that the order of convergence of the obtained optimal interpolation formulas in the space L(m)2 (0,1) is O(hm). In order to solve the obtained system it is suggested to use the Sobolev method which is based on the discrete analog of the differential operator d2m= dx2m. Using this method in the cases m = 2 and m = 3 the optimal interpolation formulas are constructed. It is proved that the order of convergence of the optimal interpolation formula in the case m = 2 for functions of the space C4(0,1) is O(h4) while for functions of the space L(2)2 (0,1) is O(h2). Finally, some numerical results are presented.

An optimal quadrature formula in the Sobolev space

2021 ◽  
Vol 65 (3) ◽  
pp. 46-59
Keyword(s):  
Sobolev Space ◽  
Quadrature Formula ◽  
Approximate Calculation ◽  
Order Of Convergence ◽  
Quadrature Formulas ◽  
Explicit Formulas ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Quadrature Formulas

This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with trigonometric weight in the L(2m)(0, 1) space for any ω ൐= 0, ω ∈ R. Here explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formulas for the case m = 1, 2. The obtained optimal quadrature formulas are exact for Pm−1(x), where Pm−1(x) is a polynomial of degree (m − 1).

scholarly journals Optimal quadrature formula in the sense of Sard in K2(P3) space

2014 ◽  
Vol 95 (109) ◽  
pp. 29-47 ◽  
Author(s):  
Abdullo Hayotov ◽  
Gradimir Milovanovic ◽  
Kholmat Shadimetov
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Quadrature Formula ◽  
Asymptotic Optimality ◽  
Trigonometric Functions ◽  
Numerical Examples ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Formula

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using Sobolev?s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (3)2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

scholarly journals TOP EVALUATION FOR THE RA TION FOR THE RATE OF FUNC TE OF FUNCTIONAL OF ERROR AL OF ERROR WEIGHT CUBATURE FORMUL TURE FORMULA IN SP A IN SPACE

2019 ◽  
Vol 3 (4) ◽  
pp. 32-37
Author(s):  
Ozodjon Isomidinovich Jalolov ◽  
◽  
Khurshidzhon Usmanovich Khayatov
Keyword(s):  
Sobolev Space ◽  
Upper Bound ◽  
Cubature Formula ◽  
Integration Formulas ◽  
Error Functional ◽  
The Error Functional ◽  
Error Weight ◽  
Norm Of The Error ◽  
Approximate Integration

An upper bound is obtained for the norm of the error functional of the weight cubature formula in the Sobolev space . The modern formulation of the problem of optimization of approximate integration formulas is to minimize the norm of the error functional of the formula on the selected normalized spaces. In these works, the problem of optimality with respect to some definite space is investigated. Most of the problems are considered in the Sobolev space

scholarly journals The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jin Li ◽  
Xiuzhen Li
Keyword(s):  
Error Function ◽  
Quadrature Rule ◽  
Posteriori Error ◽  
Trapezoidal Rule ◽  
Boundary Element Methods ◽  
Posteriori Error Estimate ◽  
Order Of Accuracy ◽  
Hypersingular Integrals ◽  
Error Functional ◽  
The Error Functional

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

Optimal quadrature formula in space

2012 ◽  
Vol 62 (12) ◽  
pp. 1893-1909 ◽  
Author(s):  
Kh.M. Shadimetov ◽  
A.R. Hayotov ◽  
S.S. Azamov
Keyword(s):  
Quadrature Formula ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature

On an optimal quadrature formula in the sense of Sard

2010 ◽  
Vol 57 (4) ◽  
pp. 487-510 ◽  
Author(s):  
Abdullo Rakhmonovich Hayotov ◽  
Gradimir V. Milovanović ◽  
Kholmat Mahkambaevich Shadimetov
Keyword(s):  
Quadrature Formula ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature
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