NAOct 25, 2010
Optimal Quadrature Formulas in the Sense of Sard in $W_2^{(m,m-1)}$ SpaceKh. M. Shadimetov, A. R. Hayotov
In this paper in $W_2^{(m,m-1)}(0,1)$ space the problem of construction of optimal quadrature formula in the sense of Sard is considered and using S.L. Sobolev's method it is obtained new optimal quadrature formula of such type. For the optimal coefficients explicit formulas are obtained. Furthermore, the numerical results which confirm the theoretical results of this work is given.
NAFeb 2, 2018
Optimal interpolation formulas in $W_2^{(m,m-1)}$ spaceS. S. Babaev, A. R. Hayotov
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.
NADec 11, 2008
On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$Kh. M. Shadimetov, A. R. Hayotov, F. A. Nuraliev
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 calculated for arbitrary fixed $N$ and for any $m\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal.
NAOct 30, 2008
Calculation of Coefficients of the Optimal Quadrature Formulas in the $W_2^{m,m-1}(0,1)$ SpaceKh. M. Shadimetov, A. R. Hayotov
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 $$ \int\limits_0^1ϕ(x)dx\cong \sum\limits_{β=0}^NC_βϕ(x_β). $$
NADec 15, 2012
Construction of discrete analogue of the differential operator $\frac{d^4}{dx^4}+2\frac{d^2}{dx^2}+1$ and its propertiesA. R. Hayotov
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.
NADec 9, 2010
Optimal interpolation formulas in the periodic function space of S.L. SobolevKh. M. Shadimetov, A. R. Hayotov, N. H. Mamatova
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 calculated. Moreover, connection between optimal interpolation formula in the space $\widetilde{L_2^{(m)}} (0,1)$ and optimal quadrature formula in this space is shown. Finally, numerical results are given.
NAOct 30, 2008
Properties of Discrete Analogue of the Differential Operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$Kh. M. Shadimetov, A. R. Hayotov
In the paper properties of the discrete analogue $D_m(hβ)$ 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 $D_m(hβ)$ of this differential operator also have similar properties.
NAOct 30, 2014
Optimal quadrature formulas with derivatives in Sobolev spaceKh. M. Shadimetov, A. R. Hayotov, F. A. Nuraliev
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 optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number $N$ and for any $m\geq 4$ using S.L. Sobolev method which is based on discrete analogue of the differential operator $d^{2m}/dx^{2m}$. In particular, for $m=4,\ 5$ optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from $m=6$ new optimal quadrature formulas are obtained.
NANov 16, 2009
Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ SpaceKh. M. Shadimetov, A. R. Hayotov
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 $η_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with well known optimal formulas.
NANov 1, 2008
Calculation of the Norm of the Error Functional of Optimal Quadrature Formulas in the Space $W_2^{(2,1)}(0,1)$A. R. Hayotov
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.