## 4 Numerical Examples
### The Benchmark BHCP  
In the last numerical example, we consider a 1D benchmark BHCP [27]. The governing equation is expressed as Equation (1). The boundary conditions on the right and left boundaries are written as Equations (19) and (20). We consider the exact solution to be:  
\[u(x,\ t)=e^{-\pi^{2}t}\sin(\pi x). \tag{24}\]  
We consider the final elapsed time to be 0.25, length of the space domain to be 1, and thermal conductivity to be 1. Since the conventional RBFCM may provide only spatial approximation of the numerical solutions, it limits its application in inverse problems. [PERSON] [PERSON] was firstly proposed the RBFCM based on the global space-time MQ function [27]. The accuracy of our approach is then compared with that of the space-time MQ method developed by [PERSON] and [PERSON] [27].  
We collocate with the inner and boundary collocation points by using Equations (3) and (7). The boundary points are uniformly provided on the lateral and top sides of the 2D space-time domain. The numbers of boundary collocation points, inner collocation points, and source point are considered to be 121, 690, and 80, respectively. The terms of the SRPBF are determined to be 8.  
The space-time MQ method has also been applied to solve this BHCP [27]. We then compare the accuracy of our approach with those of the space-time MQ method. Based on their results, the RMSE using the space-time MQ method can reach to the order of \(10^{-7}\)[27]. However, the RMSE using our approach may reach to the order of \(10^{-10}\). It appears that highly accurate initial temperature may be recovered utilizing our approach.  
The input boundary conditions polluted by random noise are further considered. We consider the noise level to be 0, 0.01, 0.1, and 0.4, which are the same of those adopted by [PERSON] and [PERSON] [27]. Figure 12 displays the estimated initial temperature with different noise level. The MAE for noise level 0, 0.01, 0.1, and 0.4 is \(1.58\times 10^{-7}\), \(4.41\times 10^{-5}\), \(7.88\times 10^{-4}\), and \(1.42\times 10^{-3}\), as depicted in Figure 13. It seems that high accurate initial temperature may be recovered utilizing our approach even when the measured data polluted by random noise are considered. Figure 14 demonstrates the RMSE versus different noise level. From Figure 14, the RMSE associated with the space-time MQ method is of the order of \(10^{-2}\) to \(10^{-3}\). However, the RMSE associated with our approach is of the order of \(10^{-3}\) to \(10^{-7}\). It appears that the RMSE associated with our method may provide better solutions than the space-time MQ method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Final Elapsed Time** & **Maximum Absolute Error (MAE)** & **Root Mean Square Error (RMSE)** \\ \hline
0.2 & \(5.46\times 10^{-9}\) & \(2.61\times 10^{-11}\) \\
0.4 & \(1.06\times 10^{-9}\) & \(3.90\times 10^{-12}\) \\
0.6 & \(2.88\times 10^{-8}\) & \(9.76\times 10^{-11}\) \\
0.8 & \(1.24\times 10^{-7}\) & \(5.28\times 10^{-10}\) \\
1.0 & \(9.00\times 10^{-6}\) & \(3.88\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: MAE and RMSE of the proposed approach for solving the BHCP.  
Figure 12: The estimated initial temperature with noise level: (**a**) the space-time multiquadric (MQ) method; (**b**) this study.  
Figure 13: Absolute error of the computed temperature: (**a**) \(\mathbf{e}=0\), (**b**) \(\mathbf{e}=0.01\), (**c**) \(\mathbf{e}=0.1\), (**d**) \(\mathbf{e}=0.4\).