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Chebyshevskii Sbornik, 2020, Volume 21, Issue 3, Pages 250–261
DOI: https://doi.org/10.22405/2226-8383-2018-21-3-250-261
(Mi cheb940)
 

BRIEF MESSAGE

Best quadrature formulas calculation of curvilinear integrals for some classes of functions and currves

M. Sh. Shabozov, M. K. Abdukarimzoda

Tajik National University (Dushanbe)
References:
Abstract: For an approximate calculation of a curvilinear integral
$$J(f;\Gamma):=\int\limits_{\Gamma}f(x_1,x_2,\ldots,x_m)dt$$
when the curve $\Gamma$ is given by parametric equations
$$x_{1}=\varphi_{1}(t), x_{2}=\varphi_{2}(t),\ldots,x_{m}=\varphi_{m}(t), 0\leq t\leq L$$
the quadrature formula is entered into consideration
$$J(f;\Gamma):\approx\sum_{k=1}^{N}p_{k} f\Bigl(\varphi_{1}(t_k), \varphi_{2}(t_k), \ldots, \varphi_{m}(t_k)\Bigr),$$
where $P=\left\{p_{k}\right\}_{k=1}^{N}$ and $T:=\left\{t_{k}:0\leq t_{1}<t_{2}<\cdots<t_{N}\leq L\right\}$– are arbitrary vector coefficients and nodes. Let $H^{\omega_{1},\ldots,\omega_{m}}[0,L]$– sets of curves $\Gamma$, whose coordinate functions $\varphi_{i}(t)\in H^{\omega_{i}}[0,L] \ (i=\overline{1,m}),$ where $\omega_{i}(t) \ (i=\overline{1,m})$– are given moduli of continuity $\mathfrak{M}_{\rho}^{\omega,p}$– functions class $f(M),$ defined in point $M\in\Gamma,$ such for any two points $M^{\prime}=M(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m}^{\prime}),$ $M^{\prime\prime}=M(x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{m}^{\prime\prime})$ belonging to a curve $\Gamma \in H^{\omega_{1},\ldots,\omega_{m}}[0,L]$ satsify the condition
$$\Bigl|f(M^{\prime})-f(M^{\prime\prime})\Bigr|\le\omega(\rho_{p}(M^{\prime}, M^{\prime\prime})),$$
where
$$\rho_{p}(M^{\prime}, M^{\prime\prime})=\left\{\sum_{i=1}^{m}|x^{\prime}_{i}-x_{i}^{\prime\prime}|^{p}\right\}^{1/p}, \ 1\leq p\leq \infty,$$
$\omega(t)$– given moduls of continuity. It is proved that among all quadrature formulas of the above from, the best for a class of functions $\mathfrak{M}_{\rho}^{\omega,p}$ and a class of curves $H^{\omega_{1},\ldots,\omega_{m}}[0,1]$, is the formula of average rectangles.
The exact error estimate of the best quadrature formula is calculated for all the functional classes under consideration and the curves are given a generalization for more general classes of functions.
Keywords: curvilinear integral, quadrature formula, error, rectangle formula, functions class, nodes.
Received: 21.02.2020
Accepted: 22.10.2020
Document Type: Article
UDC: 517.5
Language: Russian
Citation: M. Sh. Shabozov, M. K. Abdukarimzoda, “Best quadrature formulas calculation of curvilinear integrals for some classes of functions and currves”, Chebyshevskii Sb., 21:3 (2020), 250–261
Citation in format AMSBIB
\Bibitem{ShaAbd20}
\by M.~Sh.~Shabozov, M.~K.~Abdukarimzoda
\paper Best quadrature formulas calculation of curvilinear integrals for some classes of functions and currves
\jour Chebyshevskii Sb.
\yr 2020
\vol 21
\issue 3
\pages 250--261
\mathnet{http://mi.mathnet.ru/cheb940}
\crossref{https://doi.org/10.22405/2226-8383-2018-21-3-250-261}
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