# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 7 Numerical Results
Then integral Equation (50) will take the following form:_  
\[y(x)+\int_{0}^{1}x\times(e^{x\times s}-1)\times y(s)ds=e^{x}-x,x\in[0,1] \tag{54}\]0  
_The exact solution of integral Equation (57) is as follows:_  
\[y(x)+\int_{0}^{1}x\times(e^{x\times s}-1)\times y(s)ds=e^{x}-x,x\in[0,1] \tag{54}\]1  
The results for this example are given in \[y(x)+\int_{0}^{1}x\times(e^{x\times s}-1)\times y(s)ds=e^{x}-x,x\in[0,1] \tag{54}\]2  
\begin{table}
\begin{tabular}{c c c c c c c c} \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 & 1.000 & 1.000 & 1.000 & 1.000 & 0.00\(E+00\) & 0.00\(E+00\) & 0.00\(E+00\) \\ \hline
0  
To solve integral Equation (57), the author of [32] applied the polynomial method using Boubaker polynomials and obtained the result with the maximum absolute error \(E_{m}=7.1\times 10^{-3}\) at \(N=11\). The authors of [36] also applied the polynomial method. They used the Tushar polynomial and obtained the result with the maximum absolute error \(E_{m}=2.0\times 10^{-3}\) at \(N=11\) for \(n=2\). Using the Gauss-Lobatto quadrature formula, the author of [41] obtained good results. The maximum absolute error was \(E_{m}=1.28\times 10^{-15}\) at \(N=11\) and \(n=6\). The authors of [44] applied the polynomial method using Bernstein polynomials to solve integral Equation (57) and obtained the result with the maximum absolute error \(E_{m}=3.5\times 10^{-3}\) at \(N=11\) and \(n=6\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{x}_{i}\)} & \multirow{2}{*}{**ES**} & \multicolumn{4}{c}{**QQF**} & \multicolumn{4}{c}{**Absolute Error**} \\ \cline{3-8}  & & \(\mathbf{m=2}\) & \(\mathbf{m=4}\) & \(\mathbf{m=8}\) & \(\mathbf{m=2}\) & \(\mathbf{m=4}\) & \(\mathbf{m=8}\) \\ \hline
0.0 & 1.000 & 1.000 & 1.000 & 1.000 & \(0.00E+00\) & \(0.00E+00\) & \(0.00E+00\) \\ \hline
0.1 & 0.855 & 0.855 & 0.855 & 1.39\(E-08\) & \(3.31E-12\) & \(2.09E-19\) \\ \hline
0.2 & 0.719 & 0.719 & 0.719 & 0.719 & \(2.78E-08\) & \(6.62E-12\) & \(4.18E-19\) \\ \hline
0.3 & 0.591 & 0.591 & 0.591 & 0.591 & \(4.17E-08\) & \(9.92E-12\) & \(6.27E-19\) \\ \hline
0.4 & 0.470 & 0.470 & 0.470 & 0.470 & \(5.56E-08\) & \(1.32E-11\) & \(8.35E-19\) \\ \hline
0.5 & 0.357 & 0.357 & 0.357 & 0.357 & \(6.95E-08\) & \(1.65E-11\) & \(1.04E-18\) \\ \hline
0.6 & 0.249 & 0.249 & 0.249 & 0.249 & \(8.34E-08\) & \(1.98E-11\) & \(1.25E-18\) \\ \hline
0.7 & 0.147 & 0.147 & 0.147 & 0.147 & \(9.73E-08\) & \(2.32E-11\) & \(1.46E-18\) \\ \hline
0.8 & 0.049 & 0.049 & 0.049 & \(0.049\) & \(1.11E-07\) & \(2.65E-11\) & \(1.67E-18\) \\ \hline
0.9 & \(-0.043\) & \(-0.043\) & \(-0.043\) & \(-0.043\) & \(1.25E-07\) & \(2.98E-11\) & \(1.88E-18\) \\ \hline
1.0 & \(-0.132\) & \(-0.132\) & \(-0.132\) & \(-0.132\) & \(1.39E-07\) & \(3.31E-11\) & \(2.09E-18\) \\ \hline \end{tabular}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{x}_{i}\)} & \multirow{2}{*}{**ES**} & \multicolumn{4}{c}{**MAE**} & \multicolumn{4}{c}{\(1.39E-07\)} & \multicolumn{4}{c}{\(3.31E-11\)} & \multicolumn{2}{c}{\(2.09E-18\)} \\ \hline \end{tabular}
\end{table} 
Table 4: Results of solving the integral equation in Example 4. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{x}_{i}\)} & \multirow{2}{*}{**ES**} & \multicolumn{4}{c}{**QQF**} & \multicolumn{4}{c}{**Absolute Error**} \\ \cline{3-8}  & & \(\mathbf{m=2}\) & \(\mathbf{m=4}\) & \(\mathbf{m=8}\) & \(\mathbf{m=2}\) & \(\mathbf{m=4}\) & \(\mathbf{m=8}\) \\ \hline
0.0 & 0.000 & 0.000 & 0.000 & 0.000 & \(4.30E-33\) & \(2.47E-32\) & \(2.47E-32\) \\ \hline
0.1 & \(-0.072\) & \(-0.072\) & \(-0.072\) & \(-0.072\) & \(1.10E-32\) & \(5.00E-33\) & \(5.00E-33\) \\ \hline
0.2 & \(-0.096\) & \(-0.096\) & \(-0.096\) & \(-0.096\) & \(7. Our method gave the result with the value of maximum absolute error \(E_{m}=2.09\times 10^{-18}\) at \(m=8\). **Example 5**. : _In (50), \(K(x,s)=e^{x+s},f(x)=e^{-x},\lambda=-1,a=0,b=1\).