Abdullo Rakhmonovich Hayotov

• Doctor of Sciences in Physics and Mathematics
• Researcher at Institute of Mathematics, Uzbekistan Academy of Sciences

In this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method. To do this, we first solve the boundary value problem for an extremal function. To solve the boundary value problem...

Our goal is to construct an approximation of the unknown function f by Sobolev’s method, we construct an approximation form of unknown function by interpolation splines minimizing the semi norm in K2(P3) Hilbert space. Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the h...

The numerical integration of definite integrals is essential in fundamental and applied sciences. The accuracy of approximate integral calculations is contingent upon the initial data and specific requirements, leading to the imposition of diverse conditions on the resultant computations. Classical methods for the numerical analysis of definite int...

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 discret...

The present work is devoted to construction of the optimal interpolation formula exact for trigonometric functions sin(ωx) and cos(ωx). Here the analytical representations of the coefficients of the optimal interpolation formula in a certain Hilbert space are obtained using the discrete analogue of the differential operator. Taking the coefficients...

This article delves into the construction of an optimal interpolation formula designed for approximating functions within the Hilbert space L (2) 2 (0, 1). This space encompasses functions that are square integrable with a second generalized derivative in the interval [0, 1]. The interpolation formula takes the form of a linear combination of funct...

We construct optimal quadrature formulas for numerical integration of the right Riemann–Liouville integrals in the Hilbert space W2m,m-1t,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

This paper is devoted to the construction of optimal quadrature formulas in the Hilbert space $\widetilde{W_2}^{(m,m-1)}$ of complex-valued, periodic functions for the numerical calculation of the integral $\int_{0}^{1}e^{2\pi i\omega x}\varphi(x)dx$ for $\omega\in\mathbb{Z}$. In the cases $m=1$ and $m=2$, the exponentially weighted integrals of so...

The article presents a software implementation of a parallel efficient and fast computational algorithm for solving the Cauchy problem for a nonlinear differential equation of a fractional variable order. The computational algorithm is based on a non-local explicit finite-difference scheme, taking into account the approximation of the Gerasimov-Cap...

This paper is devoted to constructing a new optimal quadrature formula in the Hilbert space of real-valued, periodic functions. Here, the norm of the error functional is calculated to obtain the upper bound for the absolute error of the considered quadrature formula. For this the extremal function of the quadrature formula is used. As well, optimal...

Eksponensial vaznli integrallarni taqribiy hisoblash uchun optimal kvadratur formulalar Ushbu maqolada davriy, kompleks qiymatli funksiyalarning W (m,m−1) 2 (0, 1] Hilbert fazosida eksponensial vaznli optimal kvadratur formulalar qurilgan. Shuningdek, W (m,m−1) 2 (0, 1] fazoda ω = 0 va m ≥ 2 bo'lganda to'g'ri to'rtburchaklar formulasi optimal ekanl...

This paper is devoted to the process of finding the upper bound for the absolute error of the optimal quadrature formula in the space W2 (m,m−1) of real-valued, periodic functions. For this the extremal function of the quadrature formula is used. In addition, it is shown that the norm of the error functional for the optimal quadrature formula const...

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...

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...

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...

This paper is devoted to the process of constructing an optimal quadrature formula in the Hilbert space of real-valued, periodic functions. Here, in order to obtain the upper bound for the absolute error of the considered quadrature formula, the norm of the error functional is calculated. In the calculation of the norm of the error functional the e...

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...

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.

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...

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...

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...

Eksponensial vaznli integrallarni taqribiy hisoblash uchun optimal kvadratur formula va uning tatbiqi Ushbu ishda SobolevningL (2) 2 (0, 1] davriy, kompleks qiymatli funsiyalar fazosida biz (2) ko'rinishdagi optimal kvadratur formulani quramiz. Ushbu optimal kvadratur formulalar asosida yaqinlashtirish formulasini olamiz. Bu formula ba'zi funksiyal...

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...

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...

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...

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...

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...

W (5,4) 2 Хаётов А. Р. 1 Расулов Р. Г. 2 Eyler-Makloren kvadratur formulasini W (5,4) 2 fazoda kengaytirish. Maqolada W (5,4) 2 (0, 1) fazosida kvadratur optimal formula qurish bilan Eyler-Makloren kvadratur formulasini kengaytirish masalasi qaralgan. Kalit so'zlar: Optimal kvadratur formula; Gilbert fazosi; xatolik funksionali; Sobolev usuli; disk...

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...

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...

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...

In the paper we consider an extension problem of the Euler-Maclaurin quadrature formula in the Hilbert space (3,2) 2 by constructing an optimal quadrature formula. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the second derivative of a integrand. Using the discrete analogue of the op...

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...

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...

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...

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...

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...

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...

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.

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...

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...

In the present paper optimal interpolation formulas are constructed in $W_2^{(m,m-1)}(0,1)$ space. Explicit formulas for coefficients of optimal interpolation formulas are obtained. Some numerical results are presented.

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.

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...

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}.

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...

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...

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.

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 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...

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...

In the present work in the space $W_2^{(2,1)}(0, 1)$ the coefficients of the optimal interpolation formula are found.

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...

In the present work in the space $W_2^{(2;1)}(0, 1)$ the coefficients of the optimal interpolation formula are found.

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...

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.

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...

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...

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 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.

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...

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...

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...

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...

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...

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...

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...

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.

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\...

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 $$ \int\limits_0^1\phi(x)dx\cong \sum\limits_{\beta=0}^NC_{\beta}\phi(x_{\beta}). $$