## 5 Numerical Examples
### Example 3  
The governing equation is presented in Equation (1), where \(\alpha=y^{3}\sin x\), \(\beta=-y^{2}\cos x\), \(\gamma=x^{2}y\) and \(f(\mathbf{x})=y\cos(x)+x\cos(y)\). The following analytical solution is adopted for validation  
\[u(x,y)\ =\ y\cos(x)\ +\ x\cos(y). \tag{37}\]  
The boundary data are expressed as  
\[B_{1}u(\mathbf{x})\ =\ g(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1},\ \text{and} \tag{38}\]  
\[B_{2}u(\mathbf{x})\ =\ h(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1}. \tag{39}\]  
Figure 17 depicts a multiply connected domain for this example. The boundaries are defined as follows:  
\[\partial\Omega_{1}\ =\ \{\ (x,y)|x\ =\ \rho_{1}(\theta)\cos(\theta),\ y=\rho_{1}( \theta)\sin(\theta),\ 0\ \leq\ \theta\ \leq\ 2\pi\alpha\},\omega\ =\ 75\%\ \text{and}\ 50\%, \tag{40}\]  
\[\partial\Omega_{2}\ =\ \{\ (x,y)|x\ =\ -0.22+\rho_{2}(\theta)\cos(\theta),\ y=\ \rho_{2}(\theta)\sin(\theta)\}, \tag{41}\]  
\[\partial\Omega_{3}\ =\ \{\ (x,y)|x\ =\ \rho_{3}(\theta)\cos(\theta),\ y=\ 0.25\ +\ \rho_{3}(\theta)\sin(\theta)\}, \tag{42}\]  
\[\partial\Omega_{4}\ =\ \{\ (x,y)|x=0.25+\rho_{4}(\theta)\cos(\theta),\ y=\ \rho_{4}(\theta)\sin(\theta)\},\ \text{and} \tag{43}\]  
\[\partial\Omega_{5}\ =\ \{\ (x,y)|x=\rho_{5}(\theta)\cos(\theta),\ y=-0.25+\rho_{5}( \theta)\sin(\theta)\}, \tag{44}\]  
in which  
\[\rho_{1}(\theta)\ =\ \sqrt[1/3]{\cos(4\theta)\ +\ \sqrt{\frac{18}{5}-\sin^{2}(4 \theta)}}, \tag{45}\]  
\[\rho_{2}(\theta)\ =\ \text{abs}(\sec(3\theta))^{\sin(\theta\theta)}, \tag{46}\]  
\[\rho_{3}(\theta)\ =\ \left(3\ +\ \cos\left(\theta-\frac{\pi}{7}\right)\right) \sin(4\theta)/5\ +\ \sin(2\theta)\,, \tag{47}\]  
\[\rho_{4}(\theta)\ =\ 1\ +\ \frac{1}{10}\tan h(10\sin(2\theta)),\ \text{and} \tag{48}\]  
\[\rho_{5}(\theta)\ =\ \sqrt[1/3]{\cos(3\theta)\ +\ \sqrt{2-\sin^{2}(3\theta)}}. \tag{49}\]  
In this example, the dilation parameter and the PPs terms are 2 and 11, respectively. Two scenarios with different accessible boundaries are investigated. The configurations ofthe collocation points for cases C and D are illustrated in Figure 17. The percentages of the accessible boundary for cases C and D are 75% and 50%, respectively.  
The error of the proposed approach with different terms of the PPs is investigated, as shown in Figure 18a. Highly accurate results are acquired while the terms are greater than 4. The RMSE is in the order of \(10^{-9}\), with the terms ranging from 5 to 20. Figure 18b illustrates the error versus the terms for case D. Accurate results may be achieved while the dilation parameter is from 2 to 5. Furthermore, the dilation parameter may be less sensitive to accuracy.  
The MAE and RMSE versus different noise levels for cases C and D are shown in Table 3. Figure 19 displays the computed results with the different noise levels. Our method yields highly accurate results shown in Figure 20. The MAE of our method can reach \(10^{-3}\) with \(\delta~{}=~{}0.03\), as depicted in Figure 21.  
Figure 17: Location of the collocation points for Example 3: (**a**) case C and (**b**) case D.  
Figure 18: Error versus the terms of the PPs and dilation parameter: (**a**) Error versus the terms of the PPs and (**b**) Error versus the dilation parameter.  
Figure 19: Comparison of the results for Example 3.