## 4 Validation of the Proposed RBF  
A benchmark problem enclosed by a simply connected domain was adopted for the validation. We adopted the simplified form of Equation (1), which is the Laplace equation for the inverse problem for the validation as  
\[\
abla^{2}u(\mathbf{x})\ =\ 0. \tag{16}\]The boundary shape is  
\[ \begin{split}\partial\Omega\ =\ \{(x,y)|x\ =\ \rho(\theta)\cos(\theta),\,y =\ \rho(\theta)\sin(\theta),\,0\ \leq\ \theta\ \leq 2\pi\omega\},\\ \rho(\theta)\ =\ \sqrt[1/3]{\cos\left(3\theta\right)\ +\ \sqrt{2-\sin^{2} \left(3\theta\right)}}.\end{split}  \tag{17}\]  
The boundary conditions are expressed as follows:  
\[B_{1}u(\mathbf{x})\ =\ g(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1}, \tag{18}\]  
\[B_{2}u(\mathbf{x})\ =\ h(\mathbf{x}),\ \mathbf{x}\in\partial\Omega_{1}. \tag{19}\]  
The overspecified boundary conditions are assigned using the analytical solution:  
\[u(\mathbf{x})\ =\ e^{y}\sin x\ +\ e^{x}\sin y. \tag{20}\]  
The sources are placed outside the boundary using the equation. \[\mathbf{x}_{j}^{s}\ =\ \eta\rho_{j}^{s}\Big{(}\cos\theta_{j}^{s},\sin\theta_{j}^{ s}\Big{)},\ j\ =\ 1,\dots,M_{\text{O}}, \tag{21}\]  
where \(\eta\) is the dilation parameter, \(\rho_{j}^{s}\) is the radius of the sources at the \(j\)th source point, and \(\theta_{j}^{s}\) is the angle of the sources at the \(j\)th source point. For obtaining better accuracy, the centers are often regarded as fictitious sources that are randomly scattered within the domain. In this study, we attempted to increase the accuracy by locating the sources in different ways. For investigating the accuracy, three different scenarios for locating the source points are considered, as illustrated in Figure 2. The first scenario is that the sources are randomly located inside the domain shown in Figure 2a. The second scenario is that the sources are simultaneously located within and outside the computational domain depicted in Figure 2b. The third scenario is that the sources are collocated outside the domain demonstrated in Figure 2c. The number of the source points for three different scenarios, as shown in Figure 2, is actually equal to investigating the accuracy of the proposed three collocation scenarios. Figure 2: _Cont_. In this study, the convergence analysis for the number of source points is conducted to find the optimal number of source points. The results obtained demonstrate that promising numerical solutions can be achieved when the number of source points is greater than 300. Accordingly, the number of source points can then be determined. After determining the optimal number of source points based on the convergence analysis, another convergence analysis for the terms of the basis functions is then carried out to find the optimal terms of the basis functions, as shown in Figure 3. Figure 3 illustrates the RMSE of the three scenarios for locating the sources. As depicted in Figure 3, the third scenario exhibits higher accuracy in the order of \(10^{-11}\) than those of the first and the second scenarios. Accordingly, we may conclude that the sources collocated outside the domain are suggested for locating the sources. Figure 4 depicts the error versus the dilation parameter values for the third scenario. The MAE fluctuates from \(10^{-7}\) to \(10^{-11}\), while the dilation parameter ranges from 1.1 to 5. Figure 5 illustrates the error versus the proportion of the accessible boundary. From Figure 5, the value of the accessible boundary to be 1 means that the entire set of boundary values is prescribed. Additionally, the accessible boundary to be 1 represents that the problem becomes a forward problem. For the forward problem, we found the solution within the domain similar to the boundary value problem in which the entire boundary values are given. On the other hand, the accessible boundary value less than 1 represents that the problem becomes an inverse problem. The inverse problems of the convection-diffusion equation involve the findings of the solution within the domain as well as missing boundary values.