## 3 Validation of the Proposed Method  
### Accuracy and Convergence Analysis  
To investigate the accuracy of the source collocation scheme, a DHCP is carried out. The governing equation of the DHCP is expressed as Equation (1). The initial data are given by:  
\[u(x,\ t=0)=\sin(2x). \tag{15}\]  
The boundary conditions on the right and left boundaries are described as the following equations:  
\[u(x=L,\ t)=0, \tag{16}\]  
\[u(x=0,\ t)=0. \tag{17}\]  
We consider the following exact solution:  
\[u(x,\ t)=e^{-4\alpha^{2}t}\sin(2x). \tag{18}\]  
In this validation example, the space domain is 1 in length, the final elapsed time is 1, and the thermal conductivity is 1. The numbers of the inner collocation and source points are 400 and 100, respectively. The terms of the SRPBF and numbers of the boundary collocation points are listed in Table 1. To investigate the correctness of the source collocation scheme, the source points placed in and outside the 2D space-time domain are considered. For the inner source collocation scheme, the source points are placed in the 2D space-time domain, as depicted in Figure 2a. As for the outer source collocation scheme, we place the source points outside the 2D space-time domain, as depicted in Figure 3.  
To illustrate the accuracy of the proposed approach, the MAE of the computed results is evaluated. The relationship between the terms of the SRPBF and the MAE with the consideration of the inner and outer source collocation schemes is demonstrated in Figure 4.  
Figure 3: The outer source collocation scheme.  
For the inner source collocation scheme, it seems that the MAE of the order of \(10^{-7}\) to \(10^{-9}\) may be yielded when the terms of the SRPBF is in the range of 9 to 16. For the outer source collocation scheme, it is clear that the MAE of the order of \(10^{-10}\) to \(10^{-12}\) may be yielded when the terms of the SRPBF is in the range of 9 to 16. In addition, it seems that the terms of the SRPBF in the outer source collocation scheme are not very sensitive while the number of terms is greater than 8. From Figure 4, it appears that the computed results with high accuracy utilizing our approach with the consideration of the outer source collocation scheme can be acquired, which means that the source points collocated outside the 2D space-time domain may obtain more accurate results.  
Since the position of the source point may affect the accuracy of the proposed approach, we further conduct an example to investigate the dilation parameter to the accuracy. In this example, the source points collocated at different locations outside the 2D space-time domain are utilized. The numbers of boundary collocation points, inner collocation points, and source point are considered to be 400, 400, and 100, respectively. The terms of the SRPBF are determined to be 8.  
Figure 5 demonstrates the dilation parameter versus the MAE. It is found that highly accurate solutions in the order of \(10^{-8}\) to \(10^{-11}\) may be acquired when the dilation parameter ranges from 2 to 7, which indicates the location of the source points may be not sensitive to the accuracy of the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(N_{B}\) & \(N_{I}\) & \(N_{S}\) & \(N_{O}\) & \multicolumn{2}{c}{MAE} \\ \cline{3-6}
**The Size of the** & & & & \multicolumn{2}{c}{**The Inner Source**} & \multicolumn{2}{c}{**The Outer Source**} \\
**A Matrix** & & & & **Collocation Scheme** & **Collocation Scheme** \\ \hline \(500\times 500\) & 100 & & 5 & \(1.45\times 10^{-6}\) & \(2.19\times 10^{-7}\) \\ \(600\times 600\) & 200 & & 6 & \(9.99\times 10^{-8}\) & \(6.48\times 10^{-9}\) \\ \(700\times 700\) &  
outer source collocation scheme. Accordingly, we adopt the outer source collocation scheme for the following numerical examples.