## 5 Numerical Examples
### Example 2  
The governing equation is presented in Equation (1), where \(\alpha=0\), \(\beta=0\), \(\gamma=0\), and \(f(\mathbf{x})=-\cos x-\sin y\). The boundary conditions are  
\[\partial\Omega_{4}\ =\ \bigg{\{}\left(x,y\right)\left|x\ =\ 0.40\ +\ \rho_{4}( \theta)\cos\!\left(\theta\ +\ \frac{1}{2}\sin(8\theta)\right)\!,\ y\ =\ \rho_{4}( \theta)\sin\!\left(\theta\ +\ \frac{1}{2}\sin(8\theta)\right)\!\bigg{\}}, \tag{25}\] \[B_{1}u(\mathbf{x})\ =\ g(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1}, \tag{26}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Wave Number**} & \multicolumn{2}{c}{\(\omega\) = 50\%} & \multicolumn{2}{c}{\(\omega\) = 25\%} \\ \cline{2-5}  & **MAE** & **RMSE** & **MAE** & **RMSE** \\ \hline
100 & 4.30 \(\times\)\(10^{-11}\) & 3.42 \(\times\)\(10^{-13}\) & 1.01 \(\times\)\(10^{-10}\) & 8.04 \(\times\)\(10^{-13}\) \\ \hline
200 & 4.41 \(\times\)\(10^{-11}\) & 3  
\[B_{2}u(\mathbf{x})\ =\ h(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1}. \tag{27}\]  
Figure 12 depicts the domain for the analysis. The boundaries are defined as follows:  
\[\partial\Omega_{1}\ =\ \{(x,y)|x\ =\ \varrho_{1}(\theta)\cos(\theta),\ y\ =\ \varrho_{1}(\theta)\sin(\theta),\ 0\leq\theta\leq 2\pi\omega\},\ \omega\ =\ 50\%\text{ and }75\%, \tag{28}\]  
\[\partial\Omega_{2}\ =\ \{\ (x,y)|x\ =\ -0.40\ +\ \varrho_{2}(\theta)\cos(\theta),\ y \ =\ \varrho_{2}(\theta)\sin(\theta)\}, \tag{29}\]  
\[\partial\Omega_{3}\ =\ \{\ (x,y)|x\ =\ \varrho_{3}(\theta)\cos(\theta),\ y\ =\ \varrho_{3}(\theta)\sin(\theta)\}, \tag{30}\]  
\[\partial\Omega_{4}\ =\ \bigg{\{}\ (x,y)|x\ =\ 0.40\ +\ \varrho_{4}(\theta)\cos \bigg{(}\theta\ +\ \frac{1}{2}\sin(8\theta)\bigg{)},\ y\ =\ \varrho_{4}(\theta)\sin \bigg{(}\theta\ +\ \frac{1}{2}\sin(8\theta)\bigg{)}\bigg{\}}, \tag{31}\]  
in which  
\[\varrho_{1}(\theta)\ =\ \sqrt[1/3]{\cos(4\theta)\ +\ \sqrt{\frac{18}{5}-\sin^{2}(4 \theta)}}, \tag{32}\]  
\[\varrho_{2}(\theta)\ =\ \text{abs}[(\sec 3\ \theta)]^{\sin\theta\theta}, \tag{33}\]  
\[\varrho_{3}(\theta)\ =\ \Big{(}3+\cos\Big{(}\theta-\frac{\pi}{7}\Big{)}\Big{)} \sin(4\theta)/5+\sin(2\theta)\,, \tag{34}\]  
\[\varrho_{4}(\theta)\ =\ \sqrt[1/3]{\cos(3\theta)\ +\ \sqrt{2-\sin^{2}(3\theta)}}. \tag{35}\]  
The analytical solution is described as  
\[u(\mathbf{x})\ =\ \cos x\ +\ \sin y\ +\ 4. \tag{36}\]  
Two scenarios with different accessible boundaries are investigated. The locations of the collocation points for cases A and B are depicted in Figure 12a,b, respectively. The  
Figure 12: Location of the collocation points for Example 2: (**a**) case A and (**b**) case B.  
error of the proposed method with different terms of the PPs is investigated, as shown in Figure 13a. Highly accurate results are obtained while the terms are greater than 4. The RMSE is in the order of \(10^{-10}\) if the terms of the PPs range between 5 and 20. Consequently, the terms of the PPs are set as \(11\) in this example. Figure 13b illustrates the error versus the dilation parameter for case B. Accurate results are obtained for the dilation parameter from 2 to 5.  
The MAE and RMSE versus different noise levels for cases A and B are represented in Figure 14a,b, respectively. The RMSE values associated with the proposed approach for cases A and B are in the order of \(10^{-5}\) and \(10^{-5}\), respectively. Figure 15 displays the computed results with different noise levels. Results obtained indicate that our approach obtains highly accurate results, as shown in Figure 16.  
Figure 14: Error versus the noise level: (**a**) case A. (**b**) case B.  
Figure 13: Error versus the terms of the PPs and dilation parameter: (**a**) Error versus the terms of the PPs for case A. (**b**) Error versus the dilation parameter for case B.  
Figure 16: _Cont._  
Figure 15: Comparison of the results for Example 2.