# Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications
## 7 Numerical Results
With the scheme based on Legendre polynomials and Legendre wavelets, the authors of [43] solved integral Equation (60) and obtained the result with the maximum absolute error \(E_{m}=10^{-4}\) at \(N=11\), \(M=8\), and \(k=3\). Our method gave the result with the maximum absolute error from \(E_{m}=8.14\times 10^{-9}\) to \(E_{m}=8.22\times 10^{-15}\) at \(m=\overline{4,8}\). \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 & 4.02\(E-13\) & 9.56\(E-17\) & 6.04\(E-24\) \\ \hline
0.1 & 0.995 & 0.995 & 0.995 & 0.995 & 3.63\(E-13\) & 8.65\(E-17\) & 5.46\(E-24\) \\ \hline
0.2 & 0.980 & 0.980 & 0.980 & 0.980 & 3.29\(E-13\) & 7.83\(E-17\) & 4.94\(E-24\) \\ \hline
0.3 & 0.955 & 0.955 & 0.955 & 0.955 & 2.98\(E-13\) & 7.08\(E-17\) & 4.47\(E-24\) \\ \hline
0.4 & 0.921 & 0.921 & 0.921 & 0.921 & 2.69\(E-13\) & 6.41\(E-17\) & 4.05\(E-24\) \\ \hline
0.5 & 0.878 & 0.878 & 0.878 & 0.878 & 2.44\(E-13\) & 5.80\(E-17\) & 3.66\(E-24\) \\ \hline
0.6 & 0.825 & 0.825 & 0.825 & 0.825 & 2.20\(E-13\) & 5.25\(E-17\) & 3.31\(E-24\) \\ \hline
0.7 & 0.765 & 0.765 & 0.765 & 0.765 & 1.99\(E-13\) & 4.75\(E-17\) & 3.00\(E-24\) \\ \hline
0.8 & 0.697 & 0.697 & 0.697 & 0.697 & 1.80\(E-13\) & 4.30\(E-17\) & 2.71\(E-24\) \\ \hline
0.9 & 0.622 & 0.622 & 0.622 & 0.622 & 1.63\(E-13\) & 3.89\(E-17\) & 2.45\(E-24\) \\ \hline
1.0 & 0.540 & 0.540 & 0.540 & 0.540 & 1.48\(E-13\) & 3.52\(E-17\) & 2.22\(E-24\) \\ \hline  & & & **MAE** & 4.02\(E-13\) & 9.56\(E-17\) & 6.04\(E-24\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of solving the integral equation in Example 6. **Example 8**. : _In (50) \(K(x,s)=x\times s,f(x)=e^{x}-x,\lambda=1,a=0,b=1\). Then integral Equation (50) will take the following form:_  
\[y(x)-\int_{0}^{1}x\times s\times y(s)ds=e^{x}-x,x\in[0,1] \tag{61}\]  
_The exact solution of integral Equation (61) is as follows:_  
\[y_{ex}(x)=e^{x}.\]  
The results for this example are shown in Table 8. \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 & \(-0.186\) & \(-0.186\) & \(-0.186\) & \(-0.186\) & \(6.00E-32\) & \(2.30E-31\) & \(2.70E-31\) \\ \hline
0.1 & \(-0.205\) & \(-0.205\) & \(-0.205\) & \(-0.205\) & \(0.00E+00\) & \(6.00E-32\) & \(1.00E-31\) \\ \hline
0.2 & \(-0.227\) & \(-0.227\) & \(-0.227\) & \(-0.227\) & \(1.00E-32\) & \(2.90E-31\) & \(3.00E-31\) \\ \hline
0.3 & \(-0.250\) & \(-0.250\) & \(-0.250\) & \(-0.250\) & \(2.20E-31\) & \(2.60E-31\) & \(1.80E-31\) \\ \hline
0.4 & \(-0.277\) & \(-0.277\) & \(-0.277\) & \(-0.277\) & \(4.90E-31\) & \(6.50E-31\) & \(6.60E-31\) \\ \hline
0.5 & \(-0.306\) & \(-0.306\) & \(-0.306\) & \(-0.306\) & \(3.40E-31\) & \(5.30E-31\) & \(6.10E-31\) \\ \hline
0.6 & \(-0.338\) & \(-0.338\) & \(-0.338\) & \(-0.338\) & \(3.30E-31\) & \(5.20E-31\) & \(6.60E-31\) \\ \hline
0.7 & \(-0.374\) & \(-0.374\) & \(-0.374\) & \(-0.374\) & \(1.20E-31\) & \(5.00E-31\) & \(4.90E-31\) \\ \hline
0.8 & \(-0.413\) & \(-0.413\) & \(-0.413\) & \(-0.413\) & \(1.20E-31\) & \(6.00E-32\) & \(1.70E-31\) \\ \hline
0.9 & \(-0.456\) & \(-0.456\) & \(-0.456\) & \(-0.456\) & \(5.00E-32\) & \(5.00E-32\) & \(1.30E-31\) \\ \hline
1.0 & \(-0.504\) & \(-0.504\) & \(-0.504\) & \(-0.504\) & \(6.00E-32\) & \(8.00E-32\) & \(6.00E-32\) \\ \hline  & & & & **MAE** & \(4.90E-31\) & \(6.50E-31\) & \(6.60E-31\) \\ \hline \end{tabular}
\end{table} 
Table 8: Results of solving the integral equation in Example 8.