# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 7 Numerical Results
\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 & \(-0.186\) & \(-0.186\) & \(-0.186\) & \(6.00E-32\) & \(2.30E-31\) & \(2.70E-31\) \\ \hline
0.1 & \(-0.205\) & \(-0.205\) & \(-0.205\) & \(-0.205\) & \(0.00E+00\) & \(6. The author [37] solved integral Equation (61) using the Pell-Lucas series method and obtained the result with the maximum absolute error \(E_{m}=7.5\times 10^{-8}\) at \(N=11\). Our method gave the result with the maximum absolute error \(E_{m}=3.57\times 10^{-11}-2.25\times 10^{-18}\) at \(m=\overline{4,8}\). **Example 9**. : _In (50) \(K(x,s)=e^{\frac{x-s}{2}},f(x)=x,\lambda=\frac{1}{2},a=0,b=1\). Then the integral Equation (50) will take the following form:_  
\[y(x)-\frac{1}{2}\int_{0}^{1}e^{\frac{x-s}{2}}y(s)ds=x,x\in[0,1]. \tag{62}\]  
_The exact solution of integral Equation (62) is as follows:_  
\[y_{ex}(x)=x+(4\times\sqrt{e}-6)\times e^{\frac{x-1}{2}}.\]  
The results for this example are shown in Table 9.  
The authors of [38] solved integral Equation (62) using the Taylor series expansion method and obtained the result with the maximum absolute error \(E_{m}=8.88\times 10^{-16}\) in \(N=11\) and \(p=12\). Our method gave the result with the maximum absolute error \(E_{m}=3.84\times 10^{-17}-2.43\times 10^{-21}\) at \(m=\overline{6,8}\). **Example 10**. : _In (50) \(K(x,s)=x+s,f(x)=e^{x}+(1-e)\times x-1,\lambda=1,a=0,b=1\). Then integral Equation (50) will take the following form:_  
\[y(x)-\int_{0}^{1}(x+s)\times y(s)ds=e^{x}+(1-e)\times x-1,x\in[0,1]. \tag{63}\]  
_The exact solution of integral Equation (63) is as follows:_  
\[y_{ex}(x)=e^{x}.\]  
The results for this example are shown in Table 10. \begin{table}
\begin{tabular}{c 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 & 1.000 & 1.000 & 1.000 & 1.000 & 0.00\(E+00\) & 0.00\(E+00\) & 0.00\(E+00\) \\ \hline
0.1 & 1.105 & 1.105 & 1.105 & 1.105 & 1.50\(E-08\) & 3.57\(E-12\) & 2.25\(E-19\) \\ \hline
0.2 & 1.221 & 1.221 & 1.221 & 1.221 & 3.00\(E-08\) & 7.14\(E-12\) & 4.51\(E-19\) \\ \hline
0.3 & 1.350 & 1.350 & 1.350 & 1.350 & 4.50\(E-08\) & 1.07\(E-11\) & 6.76\(E-19\) \\ \hline
0.4 & 1.492 & 1.492 & 1.492 & 1.492 & 6.00\(E-08\) & 1.43\(E-11\) & 9.01\(E-19\) \\ \hline
0.5 & 1.649 & 1.649 & 1.649 & 1.649 & 7.50\(E-08\) & 1.78\(E-11\) & 1.13\(E-18\) \\ \hline
0.6 & 1.822 & 1.822 & 1.822 & 1.822 & 8.99\(E-08\) & 2.14\(E-11\) & 1.35\(E-18\) \\ \hline
0.7 & 2.014 & 2.014 & 2.014 & 2.014 & 1.05\(E-07\) & 2.50\(E-11\) & 1.58\(E-18\) \\ \hline
0.8 & 2.226 & 2.226 & 2.226 & 2.226 & 1.20\(E-07\) & 2.86\(E-11\) & 1.80\(E-18\) \\ \hline
0.9 & 2.460 & 2.460 & 2.460 & 2.460 & 1.35\(E-07\) & 3.21\(E-11\) & 2.03\(E-18\) \\ \hline
1.0 & 2.718 & 2.718 & 2.718 & 2.718 & 1.50\(E-07\) & 3.57\(E-11\) & 2.25\(E-18\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of solving the integral equation in Example 9. The authors of [38] solved integral Equation (63) and obtained the result with the maximum absolute error \(E_{m}=5.6\times 10^{-9}\) in \(N=11\) and \(p=8\). The authors of [39], in addition to integral Equation (60), solved integral Equation (63) and obtained the result with the maximum absolute error \(E_{m}=2.84\times 10^{-6}\) at \(N=80\). The authors of [46] solved integral Equation (63) using general Legendre wavelets and obtained the result with the highest absolute error \(E_{m}=0.7\times 10^{-13}\) in \(M=11\) and \(N=4\). Our method gave the result with the maximum absolute error \(E_{m}=8.82\times 10^{-18}\) at \(m=8\).