## 5 Numerical Examples  
### Example 1  
An inverse problem of the stationary convection-diffusion equation with nine cavities enclosed by a multiply connected domain was considered. Figure 7 shows a hollow circular domain with eight circular holes with a radius of 0.125. The radii of the outer and inner circles of the hollow circular domain are 1.0 and 0.5, respectively.  
The governing equation is presented in Equation (1), where \(\alpha=0\), \(\beta=0\), \(\gamma=-900\), and \(f(\mathbf{x})=(1+\gamma)(e^{x}+e^{y})\). The boundary conditions are  
\[B_{1}u(\mathbf{x}) = g(\mathbf{x}),\,\mathbf{x}\in\Omega_{1}, \tag{22}\] \[B_{2}u(\mathbf{x}) = h(\mathbf{x}),\,\mathbf{x}\in\Omega_{1}. \tag{23}\]  
The boundary data are given using the following exact solution:  
\[u(\mathbf{x}) = e^{x}\ +\ e^{y}. \tag{24}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Noise Level**} & \multicolumn{2}{c}{\(\omega\) = 75\%} & \multicolumn{2}{c}{\(\omega\) = 50\%} \\ \cline{2-5}  & **MAE** & **RMSE** & **MAE** & **RMSE** \\ \hline \(10^{-6}\) & 2.57 \(\times\)\(10^{-6}\) & 5.00 \(\times\)\(10^{-8}\) & 8.43 \(\times\)\(10^{-6}\) & 8.67 \(\times\)\(10^{-8}\) \\ \hline \(10^{-5}\) & 2.58 \(\times\)\(10^{-6}\) & 5.54 \(\times\)\(10^{-7}\) & 3.62 \(\times\)\(10^{-5}\) & 7.77 \(\times\)\(10^{-7}\) \\ \hline \(10^{-4}\) & 4.69 \(\times\)\(10^{-5}\) & 1.01 \(\times\)\(10^{-6}\) & 7.74 \(\times\)\(10^{-5}\) & 1.66 \(\times\)\(10^{-6}\) \\ \hline \(10^{-3}\) & 8.10 \(\times\)\(10^{-5}\) & 1.74 \(\times\)\(10^{-6}\) & 7.67 \(\times\)\(10^{-5}\) & 1.65 \(\times\)\(10^{-6}\) \\ \hline \(0.01\) & 2.40 \(\times\)\(10^{-3}\) & 5.21 \(\times\)\(10^{-5}\) & 3.20 \(\times\)\(10^{-3}\) & 6.91 \(\times\)\(10^{-5}\) \\ \hline \(0.02\) & 2.90 \(\times\)\(10^{-3}\) & 6.28 \(\times\)\(10^{-5}\) & 5.20 \(\times\)\(10^{-3}\) & 1.12 \(\times\)\(10^{-4}\) \\ \hline \(0.03\) & 6.90 \(\times\)\(10^{-3}\) & 1.49 \(\times\)\(10^{-4}\) & 7.20 \(\times\)\(10^{-5}\) & 1.55 \(\times\)\(10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 1: Errors of the proposed method with various noise levels.  
Figure 6: Comparison of results: (**a**) analytical solution, (**b**) numerical solution, and (**c**) MAE.  
Figure 8 depicts the MAE and RMSE versus the terms of the PPs. The MAE and RMSE rapidly decrease with an increase in terms, and accurate results may be obtained while the terms of the PPs are greater than 5. Figure 9 presents the error versus the dilation parameter, indicating that the value of the dilation parameter is not sensitive to the computed results and the MAE and RMSE from the proposed method are in \(10^{-11}\) and \(10^{-13}\), respectively.  
Figure 8: Error versus the terms the PPRBF.  
Figure 7: Location of the collocation points for Example 1.  
Additionally, we examine the influence of the location of the inner points. The layouts of the uniformly and randomly distributed inner points are depicted in Figure 9(a),b, respectively. Furthermore, we consider the specified data to be polluted by random noise when the noise level is \(\delta~{}=~{}0.03\). Figure 11 depicts the comparison of solutions along the hole boundary \(\delta~{}=~{}0.03\). It was found that the proposed method generates results consistent with those of the analytical solution. The MAE values of the uniformly and randomly distributed inner points are in the order of \(10^{-4}\).  
Figure 10: Layouts of the (**a**) uniformly distributed inner points and (**b**) randomly distributed inner points.  
Figure 9: Error versus the dilation parameter.  
Table 2 lists the errors of the proposed method with various wave numbers, and the percentages of the accessible boundary are set as \(\omega=50\%\) and \(\omega=25\%\). As displayed in Table 2, the MAEs utilizing the proposed method with \(\omega=50\%\) and \(\omega=25\%\) for \(\gamma=1000\) are in the order of \(10^{-14}\) and \(10^{-13}\), respectively. The results demonstrate that the boundary data on the inaccessible boundary can be recovered even considering a large wave number.