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

M. Sh. Shabozov ; M. K. Abdukarimzoda

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 250-261

Résumé

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 $T:=\left\{t_{k}:0\leq t_{1}$– 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.

Détail

Keywords: curvilinear integral, quadrature formula, error, rectangle formula, functions class, nodes.