# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 8 Conclusions  
This work focuses on solving the optimization problem for weighted quadrature formulas with derivatives in Sobolev spaces. Through examples involving linear Fredholm integral equations of the second kind, the theoretical soundness of the constructed quadrature formulas is demonstrated by comparing results with those obtained by other researchers.  
The main results of this study are as follows:  
* The squared norm of the error functional for the quadrature formula under consideration was computed using the extremal function;
* This norm was minimized with respect to the quadrature formula coefficients, leading to a system of linear algebraic equations for determining the optimal coefficients;
* The uniqueness of the solutions to this system was established;
* An algorithm was developed to solve this system, enabling the determination of optimal coefficients for weighted quadrature formulas with derivatives;
* A procedure was provided for applying these weighted quadrature formulas with derivatives to approximate solutions of linear Fredholm integral equations of the second kind;
* Simplified expressions for the optimal coefficients of quadrature formulas with derivatives were derived for the spaces \(L_{2}^{(1)}(0,1)\), \(L_{2}^{(2)}(0,1)\) and \(L_{2}^{(3)}(0,1)\);
* Finally, numerical results were compared against those reported by other authors to validate the approach.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(x_{i}\)} & \multirow{2}{*}{**ES**} & \multicolumn{3}{c}{**OQF**} & \multicolumn{3}{c}{**Absolute Error**} \\ \cline{3-8}  & & \(m=2\) & \(m=4\) & \(m=8\) & \(m=2\) & \(m=4\) & \(m=8\) \\ \hline
0.0 & 0.361 & 0.361 & 0.361 & 0.361 & 6.26\(E-09\) & \(3.73E-13\) & \(1.47E-21\) \\ \hline
0.1 & 0.479 & 0.479 & 0.479 & 0.479 & \(6.59E-09\) & \(3.92E-13\) & \(1.55E-21\) \\ \hline
0.2 & 0.599 & 0.599 & 0.599 & 0.599 & \(6.92E-09\) & \(4.12E-13\) & \(1.63E-21\) \\ \hline
0.3 & 0.719 & 0.719 & 0.719 & 0.719 & \(7.28E-09\) & \(4.33E-13\) & \(1.71E-21\) \\ \hline
0.4 & 0.841 & 0.841 & 0.841 & 0.841 & \(7.65E-09\) & \(4.55E-13\) & \(1.80E-21\) \\ \hline
0.5 & 0.963 & 0.963 & 0.963 & 0.963 & \(8.04E-09\) & \(4.79E-13\) & \(1.89E-21\) \\ \hline
0.6 & 1.087 & 1.087 & 1.087 & 1.087 & \(8.46E-09\) & \(5.03E-13\) & \(1.99E-21\) \\ \hline
0.7 & 1.212 & 1.212 & 1.212 & 1.212 & \(8.89E-09\) & \(5.29E-13\) & \(2.09E-21\) \\ \hline
0.8 & 1.338 & 1.338 & 1.338 & 1.338 & \(9.35E-09\) & \(5.56E-13\) & \(2.19E-21\) \\ \hline
0.9 & 1.466 & 1.466 & 1.466 & 1.466 & \(9.83E-09\) & \(5.85E-13\) & \(2.31E-21\) \\ \hline
1.0 & 1.595 & 1.595 & 1.595 & 1.595 & \(1.03E-08\) & \(6.15E-13\) & \(2.43E-21\) \\ \hline  & & & **MAE** & \(1.03E-08\) & \(6.15E-13\) & \(2.43E-21\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Results of solving the integral equation in Example 10.  
**Author Contributions:** Conceptualization, K.S. and K.U.; methodology, K.S. and K.U.; validation, K.U.; formal analysis, K.U.; investigation, K.S. and K.U.; writing--original draft preparation, K.U.; writing--review and editing, K.S.; supervision, K.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** There is no any funding.  
**Data Availability Statement:** Data is contained within the article.  
**Conflicts of Interest:** The authors declare no conflict of interest.