## 2 The Space-time Radial Polynomial Basis Function Collocation Method
The derivative of Equation (4) with respect to \(x\) is taken to yield:  
\[\frac{\partial^{2}u}{\partial x^{2}}=\sum_{j=1}^{N_{S}}\Biggl{(}\sum_{k=1}^{N_{O}} \lambda_{jk}e^{-kt}\biggl{(}(k+2)r^{k}+k\Bigl{(}k+2)(x-x_{j}\Bigr{)}^{2}r_{j}^{k -2}\biggr{)}\Biggr{)} \tag{5}\]  
Taking the first derivative of Equation (3) with respect to \(t\), we may yield the following equation:  
\[\frac{\partial u}{\partial t}=\sum_{j=1}^{N_{S}}\Biggl{(}\sum_{k=1}^{N_{O}} \lambda_{jk}\Bigl{(}e^{-kt}\Bigl{(}k+2)(t-t_{j}\Bigr{)}r_{j}^{k}-ke^{-kt}r_{j}^ {k+2}\Bigr{)}\Biggr{)}. \tag{6}\]  
Substituting Equations (5) and (6) into the governing equation can yield the following equation:  
\[ \begin{array}{l}\alpha^{2}\sum_{j=1}^{N_{S}}\Biggl{(}\sum_{k=1}^{N_{O}} \lambda_{jk}e^{-kt}\biggl{(}(k+2)r^{k}+k\Bigl{(}k+2)(x-x_{j}\Bigr{)}^{2}r_{j}^ {k-2}\biggr{)}\Biggr{)}\\ -\sum_{j=1}^{N_{S}}\Biggl{(}\sum_{k=1}^{N_{O}}\lambda_{jk}\Bigl{(}e^{-kt} \Bigl{(}k+2\Bigr{)}(t-t_{j}\Bigr{)}r_{j}^{k}-ke^{-kt}r_{j}^{k+2}\Bigr{)}\Biggr{)} =0.\end{array}  \tag{7}\]  
To determine the coefficients in Equation (7), we impose the approximate solution to satisfy the given partial differential equation with the boundary conditions. We then achieve the following equations:  
\[[\mathbf{A}][\kappa]=[\mathbf{P}], \tag{8}\]  
where \(\kappa\) is a \(M\times 1\) vector of unknown coefficients, \(\mathbf{A}\) is a \(N\times M\) matrix for the inner and boundary collocation points, \(\mathbf{P}\) is a \(N\times 1\) vector of functions data at the inner and boundary points. The above equation can then be rewritten as follows:  
\[\left[ \begin{array}{c}\mathbf{A}_{I}\\ \mathbf{A}_{B}\end{array} \right][\kappa]=\left[ \begin{array}{c}\mathbf{P}_{I }\\ \mathbf{P}_{B}\end{array} \right], \tag{9}\]  
Figure 2: Illustration of the space–time collocation scheme: (**a**) The direct heat conduction problem (DHCP); (**b**) The backward heat conduction problem (BHCP). where \(\mathbf{A}_{I}\) is a \(N_{I}\times M\) matrix for the inner collocation points, \(\mathbf{A}_{B}\) is a \(N_{B}\times M\) matrix for the boundary collocation points, \(\mathbf{P}_{I}\) is a \(N_{I}\times 1\) vector of functions values at the inner collocation points, \(\mathbf{P}_{B}\) is a vector of boundary data with the size of \(N_{B}\times 1\), \(N_{I}\) is the number of inner collocation points, \(N_{B}\) is the number of boundary collocation points, and \(M=N_{S}\times N_{O}\). To solve the system of equations, the commercial program MATLAB backslash operator is adopted. To examine the stability of the proposed approach for solving the BHCP, the input data with noises are considered as:  
\[\hat{u}_{f}=u_{f}\times(1+e\times rand), \tag{10}\]  
\[\hat{u}_{b}=u_{b}\times(1+e\times rand), \tag{11}\]  
where \(rand\) denotes the random number, \(\hat{u}_{f}\) is the final time input data with noises, \(\hat{u}_{b}\) is the noise data on accessible boundary, \(e\) is the noise level, which is expressed as \(e=s/100\), and \(s\) is the percentage of noise. As expressed in Equations (10) and (11), the noisy data are considered in the simulation of the BHCP. To estimate the accuracy of the solutions, the root mean square error (RMSE) and maximum absolute error (MAE) are considered:  
\[\text{MAE}=\underset{1\leq i\leq N_{I}}{\text{maximum}}\;\big{|}u_{E}(x_{i},t_ {i})-u_{N}(x_{i},t_{i})\big{|}, \tag{12}\]  
\[\text{RMSE}=\sqrt{\frac{1}{N_{I}}\underset{i=1}{\overset{N_{I}}{\sum}}\;(u_{ E}(x_{i},t_{i})-u_{N}(x_{i},t_{i}))^{2}}, \tag{13}\]  
where \(u_{E}(x_{i},t_{i})\) and \(u_{N}(x_{i},t_{i})\) are the exact solution and numerical solution at the inner collocation points, respectively. For the RBFCM, the center, boundary, and inner points are required to be collocated. The center point usually shares the same position with the inner collocation point in the RBFCM. Besides, the numbers of the inner and center points must be the same for generating the determinate system of simultaneous equations.