## 3 Validation of the Proposed Method
### Comparison with the Radial Basis Function Collocation Method (RBFCM)  
To compare the computed results with those obtained from the RBFCM based on the multiquadric (MQ) basis function using the time-marching scheme, we consider a 1D benchmark DHCP. The initial as well as boundary data are expressed as Equations (15) to (17). The exact solution as shown in Equation (18) is considered.  
In this comparison example, the space domain is \(\pi\) m in length, the final elapsed time is 2 h, and the thermal conductivity is 1 Wm\({}^{-1}\)K\({}^{-1}\). To apply the conventional RBFCM based on the MQ function utilizing the time-marching method, the spatial and temporal discretizations have to be separately considered. For the spatial discretization, the length of the space is divided into segments, which have the equal length of \(\Delta x=0.039\) m. For the temporal discretization, the implicit difference scheme is adopted where the time interval (\(\Delta t\)) is 0.033 h. Figure 3 displays the collocation points of the 1D DHCP. We consider the number of boundary points in space domain and in time domain to be 80 and 120, respectively. Therefore, we have \(\Delta x=0.039\) m and \(\Delta t=0.033\) h, which are exactly the same with those in the conventional RBFCM using the time-marching scheme. As displayed in Figure 3, the numbers of the inner, source, and boundary points are 900, 100, and 200, respectively. The terms of the SRPBF and the dilation parameter are determined to be 11 and 4, respectively.  
To validate the proposed approach, the computed results using our approach and the RBFCM utilizing the time-marching scheme are compared with the exact solution. Figure 6 presents the results computed using the conventional RBFCM, the proposed approach and the exact solution. It is clear that good agreement obtained by the both proposed approach and conventional RBFCM based on the MQ function using the time-marching scheme may be found.  
Figure 7 demonstrates the accuracy of our approach and the RBFCM based on the MQ function utilizing the time-marching method. The numerical solutions of the RBFCM based on the MQ function are computed utilizing the optimal shape parameter, where the shape parameter is considered to be 3. As for the proposed method, the terms of the SRPBF are 11. From Figure 7, it is found that the MAE of the proposed method is in the order of \(10^{-7}\) to \(10^{-11}\). As for the RBFCM based on the MQ function with the time-marching scheme, the MAE is in the order of \(10^{-4}\) to \(10^{-6}\). It is obvious that our approach may acquire better solutions than the RBFCM based on the MQ function with the time-marching scheme.  
Figure 5: The maximum absolute error versus the dilation parameter.  
Figure 8 shows the relationship between the absolute error and simulation time. It is found that the absolute error may significantly increase with the simulation time in the time-marching method. The absolute error of the proposed approach, however, always remains in the order of \(10^{-9}\) to \(10^{-10}\) with the simulation time, which demonstrates that our approach may avoid the error propagation. It may be especially advantageous for DHCP requiring long simulation time.  
To compare the computational efficiency of the SRPBF with that of RBPCM, an efficiency analysis is carried out. From the results of the efficiency analysis, it is found that the central processing unit (CPU) time of the SRPBF and RBPCM is 121 (s) and 140 (s), respectively. Results obtained demonstrate that the proposed method may be more effective than the RBPCM.  
Figure 6: Comparison of the computed results with the exact solution.  
Figure 7: Comparison of the error of the radial basis function collocation method (RBPCM) with the time–marching scheme and our approach at different time: (**a**) t = 0.5 h; (**b**) t = 1.0 h; (**c**) t = 1.5 h; (**d**) t = 2.0 h.