A method of construction of weight optimal quadrature formulas with derivatives in the Sobolev space

Shadimetov, Kh. M.

doi:10.29229/uzmj.2018-3-14

Cited by 6 publications

(5 citation statements)

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“…i.e., in finding the minimum value of the norm (28) for the error functional ℓ by the coefficients C[β].…”

Section: Optimal Quadrature Formula In the Spacementioning

confidence: 99%

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The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Shadimetov,

Davronov

2023

Preprint

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The discrete analogue of the differential operator plays an important role in the construction of interpolation, quadrature and cubature formulas. In this work, we consider a discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers $m$. The effectiveness of this operator by constructing an optimal quadrature formula in the space $L_2^{(2,0)}(0,1)$ is shown. The errors of the optimal quadrature formula in $W_2^{(2,1)}(0,1)$ space and the optimal quadrature formula in $L_2^{(2,0)}(0,1)$ space are numerically compared. Numerical results showed that the error of the optimal quadrature formula what constructed in this work is smaller than the error of the quadrature formula constructed in $W_2^{(2,1)}(0,1)$ space. MSC: 41A05, 41A15, 65D30, 65D32, 65M60, 65L60.

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“…i.e., in finding the minimum value of the norm (28) for the error functional ℓ by the coefficients C[β].…”

Section: Optimal Quadrature Formula In the Spacementioning

confidence: 99%

“…and d k (k = 1, 4) are defined in Theorem 2.4 of [28]. Now, we calculate square of the norm of the error functional (27) of the optimal quadrature formula (25).…”

Section: Optimal Quadrature Formula In the Spacementioning

confidence: 99%

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Shadimetov,

Davronov

2023

Preprint

View full text Add to dashboard Cite

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“…Koman [7] constructed an optimal quadrature formula using  − function method in the (2) 2 (0,1) L space; d) Kh.M. Shadimetov [8] constructed the optimal quadrature formula in the () 2 (0,1) m L space when 0  = and calculated the norm of the error function; e) Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev [9] constructed the optimal quadrature formula for 1  = and estimated its error.…”

Section: Introduction: Statement Of the Problemmentioning

confidence: 99%

Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

Shadimetov,

Nuraliev,

Kuziev

2024

Int. J. Anal. Appl.

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In this research work, a new derived optimal quadrature formula is discussed, which includes the sum of the values of the function and its first and second order derivatives at the points located at the same distance on the interval [0,1] in the L2(m)(0,1) space. we first obtain an analytical representation of the error function norm, and a system of equations of the Wiener-Hopf type construct using the method of Lagrange unknown multipliers for finding the conditional extremum of multivariable functions. Optimal coefficients found by solving the system. Using the exact form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula for m=3 and m=4 calculate and the order of approximation was shown to be O(hm). The obtain theoretical conclusions confirmed by numerical experiments.

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“…It should be noted that the same result was independently obtained by P. Köhler in [15] using the spline method. Further, in [23], the weighted optimal quadrature formulas of the form (1.1) were constructed in the spaces L (m) 2 (0, 1) and L (m) 2 (0, N) for all natural m. F. Lanzara [16] gave a procedure to construct quadrature formulae which are exact for solutions of linear differential equations and are optimal in the sense of Sard. She presented the necessary and sufficient conditions under which such formulae do exist.…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

Shadimetov,

Hayotov,

Akhmadaliev

2024

Int. J. Anal. Appl.

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It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

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A method of construction of weight optimal quadrature formulas with derivatives in the Sobolev space

Cited by 6 publications

References 0 publications

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

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