# Kholmat M. ShadimetovInstitute of mathematics, National University of Uzbekistan · Computational methods

Kholmat M. Shadimetov

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28

Publications

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155

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Citations since 2016

## Publications

Publications (28)

The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals ∫02πeiωxφ(x)dx in the Sobolev space H˜2m. Here, H˜2m is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound...

This work studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_{2}^{(m)}(0,1)$ L 2 ( m ) ( 0 , 1 ) for numerical calculation of Fourier coefficients. Using Sobolev’s method, we obtain new sine and cosine weighted optimal quadrature formulas of such type for $N + 1\geq m$ N + 1 ≥ m , where $N + 1$ N...

This article discusses the development of a new algorithm, which is based on optimal quadrature formulas for obtaining solutions to the generalized Abel integral equations. Based on the study, it was found that the proposed scheme is very effective, extremely accurate and can be extended to other special tasks. The quadrature formulas presented in...

"In the present paper in $L_2^{(m)}(-1,1)$ space the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution of the singular integral equation is obtained applying the optimal quadrature formulas. Explicit forms of coefficients f...

In this work, an extremal function is found and the square of the norm of the error functional of composite quadrature formulas is calculated, the optimal coefficients of the quadrature formulas are also calculated. The uniqueness of optimal quadrature formulas is proved.

The paper is devoted to investigation of optimal formulas for approximate integration with derivatives in the Sobolev space L2(m)˜(0,1) of periodic functions. Here the extremal function for quadrature formulas of Hermite type is obtained. Applying this extremal function the square of the norm for the error functional of the considered Hermite type...

In the present paper in L2(m) (0,1) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Hyper-singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented to show the validity and accuracy of the proposed method.

In the present paper, it is considered the problem of construction of optimal difference formulas for approximate solution of the Cauchy problem in the Sobolev space of functions which are square integrable with m-th order derivatives (in generalized sense). Here, using the discrete analogue of the differential operator of 2m-th order the represent...

The discrete analogues of certain differential operators of order 2m play important role in construction of optimal formulas of approximate integration, interpolation formulas and difference schemes. In the present paper, in the Sobolev space H2(m)(0,1), the discrete analogue Dm[β] of the differential operator L=∑k=0m(km)(−1)kd2(m−k)dx2(m−k)
is co...

The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W
(m,0)2
by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with t...

In the present paper, the construction process of the optimal quadrature formulas for weighted integrals is presented in the Sobolev space of complex-valued periodic functions which are square integrable with m th order derivative. In particular, optimal quadrature formulas are given for Fourier coefficients. Here, using these optimal quadrature fo...

Abstract In the present paper, using the discrete analogue of the operator d 6 / d x 6 − 1 $\mathrm{d} ^{6}/\mathrm{d} x^{6}-1$ , we construct an interpolation spline that minimizes the quantity ∫ 0 1 ( φ ‴ ( x ) + φ ( x ) ) 2 d x $\int _{0}^{1}(\varphi {'''}(x)+\varphi (x))^{2}\,\mathrm{d}x$ in the Hilbert space W 2 ( 3 , 0 ) $W_{2}^{(3,0)}$ . We...

he work is devoted to onstrution of optiml qudrture formuls for pproximte solution of the eel generlized integrl equtionF rere for optiml oe0ients the system of liner equtions is otinedF he existene nd uniqueness of the solution for the otined system re studiedF Keywords: eel integrl equtionD qudrture formulD oolev speD the vgrnge methodF Mathemati...

In this paper, the coefficients of optima quadrature formulas for the approximate
solution of the generalized Abel equation is found. In addition, the square of the norm
of the error functional of quadrature formulas is calculate in the Sobolev space.

In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2\pi i\omega x}\varphi(x)dx$ with $\omega\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of complex-valued functions which are square integrable with $m$-th order derivative. Here, using the discrete an...

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W _2^(m,m−1) [0, 1] space for calculating Fourier coefficients. Using S. L. Sobolev’s method we obtain new optimal quadrature formulas of such type for N + 1 ≥ m, where N + 1 is the number of the nodes. Moreover, explicit formulas for the optim...

An optimal quadrature formula in the sense of Sard in the Hilbert space K2ðPmÞ is constructed.
New optimal quadrature formula of such a type and explicit expressions for the
corresponding optimal coefficients are obtained using S.L. Sobolev’s method. The obtained
optimal quadrature formula is exact for the trigonometric functions $sin\omega x$, $co...

In the present paper, using S.L. Sobolev’s method, interpolation Dm-splines that minimizes
the expression $\int_0^1(\varphi^{(m)}(x))^2dx$ in the $L_2^{(m)}(0,1)$ space are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation spline
is exact for polynomials of degree m � 1. Some n...

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K-2(P-3). 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 opti...

In this paper we construct the optimal quadrature formulas in the sense of Sard, as well as interpolation splines minimizing the semi-norm in the space \(K_{2}(P_{2})\), where \(K_{2}(P_{2})\) is a space of functions \(\varphi\) which \(\varphi ^{\prime}\) is absolutely continuous and \(\varphi ^{\prime\prime}\) belongs to L
2(0, 1) and \(\int _{0}...

In the present paper in $L_2^{(2)}(0,1)$ S.L.Sobolev space the optimal quadrature formula is constructed for approximate calculation of Cauchy type singular integral.

Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in $K_2(P2)$, where $K_2(P_2)$ is the space of functions $\varphi$ such that $\varphi'$ is absolutely continuous, $\varphi''$ belongs to $L_2(0, 1)$. Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation s...

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m−2 and e
−x
. Also some numerical results are presented.

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K
2(P
2). Using S.L. 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...