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Abdullo Rakhmonovich Hayotov

Abdullo Rakhmonovich Hayotov
Institute of Mathematics, Uzbekistan Academy of Sciences · Computational methods

Doctor of Sciences in Physics and Mathematics
Head of the Computational Mathematics Laboratory, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Scienc

About

68
Publications
2,887
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477
Citations
Citations since 2016
35 Research Items
373 Citations
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
Additional affiliations
October 2019 - December 2021
V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Scinces, Tashkent, Uzbekistan
Position
  • Head of Department
September 2018 - August 2019
Korea Advanced Institute of Science and Technology
Position
  • Professor
January 2017 - September 2019
Institute of Mathematics, Uzbekistan Academy of Scinces, Tashkent, Uzbekistan
Position
  • Researcher
Education
September 2012 - June 2013
University of Santiago de Compostela
Field of study
  • Numerical Analysis
May 1997 - April 2000
Institute of Mathematics, Uzbek Academy of Sciences, Tashkent, Uzbekistan
Field of study
  • Computational Mathematics
September 1989 - June 1994
Bukhara State University, Bukhara, Uzbekistan
Field of study
  • Mathematics, Informatics and Computer Sciences

Publications

Publications (68)
Article
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
In the present paper, using the discrete analogue of the differential operator d2mdx2m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{d^{2m}}{dx^{2m}} $$\end{doc...
Article
Full-text available
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ?ba e2?i?x?(x)dx with ? ? R in the Sobolev space L(m)2 [a,b] of complexvalued functions which are square integrable with m-th order derivative. Here, using the discrete analogue of the differential operator d2m/dx2m, the...
Article
Full-text available
This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with cosine weight in the Sobolev space. Here explicit formulas for the optimal coefficients are obtained. The obtained optimal quadrature formulas are exact for a polynomial of degree (m-1). We study the order of convergence of th...
Conference Paper
In this paper there is considered the problem of construction of a new optimal quadrature formula in the sense of Sard in L2(m) 0,1] Hilbert space, using S.L. Sobolev’s method. There are given explicit formulas for coefficients of the optimal quadrature formula. Furthermore, some numerical results are presented.
Conference Paper
The present work is devoted to extension of the Euler-Maclaurin formula in the Hilbert space W2(2k,2k−1). The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the (2k – 1)-th derivative of a function. Using the discrete analogue of the operator d2dx2−1 the explicit formulas for the coefficie...
Conference Paper
Full-text available
The present paper is devoted to construction of an optimal quadrature formulas for approximation of Fourier integrals in the Hilbert space W2(m,m−1) of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on the uniform grid. The difference between integral and quadrature sum is...
Article
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...
Preprint
Full-text available
The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature su...
Preprint
Full-text available
In the present paper the optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2\pi i\omega x}\varphi(x)d x$ with $\omega\in \mathbb{R}$ in the Hilbert space $W_2^{(2,1)}[a,b]$ of complex-valued functions. Furthermore, the explicit expressions for coefficients of the constructed opti...
Article
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...
Article
Full-text available
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal...
Article
Full-text available
In the present paper the optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ∫︀ 2 ()d with ∈ R in the Hilbert space (2,1) 2 [ , ] of complex-valued functions. Furthermore, the explicit expressions for coefficients of the constructed optimal quadrature formulas are obtained. At the end of the pa...
Preprint
Full-text available
he present pper is devoted to onstrution of n optiml qudrture formul for pproximtion of pourier integrls in the rilert spe W (1,0) 2 [a, b] of nonEperiodiD omplex vlued funtionsF rere the qudrture sum onsists of liner omintion of the given funtion vlues on uniform gridF he di'erene etween integrl nd qudrture sum is estimted y the norm of the error...
Article
This paper deals with the construction of an optimal quadrature formula for approximation of Fourier integrals in the Sobolev space L2(1)[a,b] of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform grid. The d...
Article
Full-text available
The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature...
Preprint
Full-text available
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...
Article
Full-text available
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 quadr...
Preprint
Full-text available
The present work is devoted to extension of the trapezoidal rule in the space $W_2^{(2,1)}$. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of a integrand. Using the discrete analog of the operator $\frac{d^2}{dx^{2}}-1$ the explicit formulas for the coefficients o...
Preprint
Full-text available
The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ 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 w...
Preprint
Full-text available
This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of non-periodic, complex valued functions which are square integrable with first order derivative. Here the quadrature sum consists of linear combination of the given function values in a uniform g...
Article
Full-text available
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 sp...
Article
In the present paper, using the discrete analog of the differential operator d2m/dx2m, optimal interpolation formulas are constructed in L2(4)(0, 1) space. The explicit formulas for coefficients of optimal interpolation formulas are obtained.
Article
Full-text available
In the present paper, using S.L. Sobolev’s method interpolation splines that minimize the expression (Formula Presented) in the space K2(Pm) are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation splines are exact for monomials 1, x, x²,…,xm-3 and for trigonometric functions sin...
Article
In the present paper we investigate the problem of construction of the optimal interpolation formulas in the space \(W_2^{(m,m-1)}(0,1)\). We find the norm of the error functional which gives the upper bound for the error of the interpolation formulas in the space \(W_2^{(m,m-1)}(0,1)\). Further we get the system of linear equations for coefficient...
Article
In the present paper in L2(m)(0,1) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented.
Article
Full-text available
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...
Article
Full-text available
In the present paper we give the method of construction of optimal interpolation formulas in the space $L_2^{(m)}(0; 1)$ which based on the discrete analogue of the differential operator $d^{2m}=dx^{2m}.
Article
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the 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 nodes. Moreover, explicit formulas for th...
Article
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space is considered. Here the quadrature sum consists of values of the integrand at nodes and values of derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and...
Article
In the present paper in L(m)2(−1,1) space the optimal quadrature formula is constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained.
Article
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...
Article
Full-text available
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of op...
Article
Full-text available
In the present paper, using S.L. Sobolev's method, interpolation spline that minimizes the expression $\int_0^1(\varphi^{(m)}(x)+\omega^2\varphi^{(m-2)}(x))^2dx$ in the $K_2(P_m)$ space are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation spline is exact for monomials $1,x,x^2...
Article
Full-text available
In the present work in the space $W_2^{(2,1)}(0, 1)$ the coefficients of the optimal interpolation formula are found.
Article
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...
Article
Full-text available
In the present work in the space $W_2^{(2;1)}(0, 1)$ the coefficients of the optimal interpolation formula are found.
Article
Full-text available
We construct a discrete analogue D_m (hβ) of the differential operator d^{2m} /dx^{2m} + 2ω^2d^{2m−2} /dx^{2m−2} + ω^4d^{2m−4} /dx^{2m−4} for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K_2(P_2) the discrete analogue D_2(hβ) for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are...
Article
Full-text available
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...
Chapter
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}...
Article
Full-text available
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.
Article
Full-text available
In the present paper we construct the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{\d^{2m}}{\d x^{2m}}+2\omega^2\frac{\d^{2m-2}}{\d x^{2m-2}}+\omega^4\frac{\d^{2m-4}}{\d x^{2m-4}}$. The discrete analogue $D_m(h\beta)$ plays the main role in construction of optimal quadrature formulas and interpolation splines minimizing the s...
Article
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1). In this paper the quadrature sum consists of values of the integrand at nodes and values of the first derivative of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas a...
Article
Full-text available
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...
Article
Full-text available
In the present paper the discrete analogue of the differential operator $\frac{d^4}{dx^4}+2\frac{d^2}{dx^2}+1$ is constructed and its some properties are proved.
Article
Full-text available
In this paper there is considered the problem of construction of a new optimal quadrature formula in the sense of Sard in K2(P2)K2(P2) Hilbert space, using S.L. Sobolevʼs method. There are given explicit formulas for coefficients of the optimal quadrature formula. Furthermore some numerical results are presented. The constructed optimal quadrature...
Article
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.
Article
Full-text available
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...
Article
In the Sobolev space L2(m)(0,1) optimal quadrature formulas of the form ∫01φ(x)dx≅∑β=0NCβφ(xβ) with the nodes xi=ηih,xN−i=1−ηih,i=0,t−1¯,0≤η0<η1<⋯<ηt−1<t,t∈N,xβ=hβ,t≤β≤N−t,h=1N are investigated. For optimal coefficients CβCβ explicit forms are obtained and the norm of the error functional is calculated for any natural numbers mm and NN. In particul...
Article
Full-text available
In this paper the problem of construction of lattice optimal interpolation formulas in the space $\widetilde{L_2^{(m)}} (0,1)$ is considered. Using S.L. Sobolev's method explicit formulas for the coefficients of lattice optimal interpolation formulas are given and the norm of the error functional of lattice optimal interpolation formulas is calcula...
Article
Full-text available
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W2(m, m-1)(0, 1) space. Using the 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 of the optimal coefficients are obtained. We investigat...
Article
Full-text available
In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing parameter $\eta_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with...
Article
Full-text available
In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculate...
Article
In this paper in the space $W_2^{(2,1)}(0,1)$ square of the norm of the error functional of a optimal quadrature formula is calculated.
Article
In the paper properties of the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are studied. It is known, that zeros of differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are functions $e^x$, $e^{-x}$ and $P_{2m-3}(x)$. It is proved that discrete analogue $D_m(h\...
Article
Full-text available
In this paper problem of construction of optimal quadrature formulas in $W_2^{(m,m-1)}(0,1)$ space is considered. Here by using Sobolev's algorithm when $m=1,2$ we find the optimal coefficients of the quadrature formulas of the form $$ \int\limits_0^1\phi(x)dx\cong \sum\limits_{\beta=0}^NC_{\beta}\phi(x_{\beta}). $$

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