## 2 The Space-time Radial Polynomial Basis Function Collocation Method  
The heat conduction problem is expressed as:  
\[\frac{\partial u}{\partial t}=\alpha^{2}\Delta u(\mathbf{x}),\mathbf{x}\in \Omega^{t} \tag{1}\]  
where \(\alpha^{2}\) is the thermal conductivity, \(\mathbf{x}\) is the space-time coordinate, \(t\) is time, \(u\) is the temperature, \(\Delta\) is the Laplacian operator, and \(\Omega^{t}\) is the space-time domain. The space-time boundary condition is as follows:  
\[u(\mathbf{x})=u_{b}(\mathbf{x})\text{ on }\partial\Omega^{t} \tag{2}\]  
where \(u_{b}\) is the space-time boundary data, and \(\partial\Omega^{t}\) is the space-time domain boundary. The heat conduction problems are categorized into initial value problems, which may be dealed with utilizing the conventional time-marching methods such as the implicit, explicit, Crank-Nicolson or Runge-Kutta schemes in Euclidean space domain. Figure 1 demonstrates the Euclidean space domain for describing the one-dimensional (1D) heat conduction problem. Different from the time-marching methods for dealing with the initial value problems, an innovative space-time collocation scheme is proposed in this study. The randomly unstructured source, boundary and inner points are collocated in the space-time domain, where both boundary as well as initial data may be regarded as space-time boundary conditions. The direct and backward heat conduction problems are converted into inverse boundary value problems such that the conventional time-marching scheme is not required. For example, Equation (1) is the governing equation of a direct heat conduction problem (DHCP) with assigned space-time boundary conditions, as depicted in Figure 2a, for modeling the transient heat conduction phenomena. Figure 2 denotes the two-dimensional (2D) space-time domain where three sets of points including the boundary, source, and inner points are collocated in the 2D space-time domain. The 1D Euclidean space domain, therefore, transform into the 2D space-time area. The boundary as well as initial data are given on the three space-time boundaries, which transform a 1D transient problem to a 2D inverse boundary value problem. Equation (1) is the governing equation of a backward heat conduction problem (BHCP) as well. Similarly, to solve the BHCP, the boundary conditions and the final time condition on the space-time boundary have to be provided, as depicted in Figure 2b. In this article, a new meshless method using the space-time radial polynomial is proposed. The temperature of the governing equation can be approximated utilizing the SRPBF as follows:  
\[u(x,\ t)=\sum_{j=1}^{N_{S}}\left(\sum_{k=1}^{N_{O}}\lambda_{jk}e^{-kt}r_{j}^{k +2}\right), \tag{3}\]  
where \(r_{j}=\left|\mathbf{x}-\mathbf{y}_{j}^{S}\right|\), \(\mathbf{x}\) is the collocation points, which may be the boundary or inner points, \(\mathbf{y}_{j}^{s}\) is the source points, \(N_{O}\) is the order of the SRPBF for approximating the solutions, and \(\lambda_{jk}\) is the coefficient to be determined. In the preceding equations, the SRPBF is a smooth and nonsingular function without the shape parameter. The SRPBF is applied on the space-time domain in which time is considered as another independent dimension. The derivative of Equation (3) with respect to \(x\) is taken to acquire:  
\[\frac{\partial u}{\partial x}=\sum_{j=1}^{N_{S}}\left(\sum_{k=1}^{N_{O}} \lambda_{jk}e^{-kt}\left(k+2\right)(x-x_{j})r_{j}^{k}\right). \tag{4}\]  
Figure 1: Configuration of the collocation points for the direct heat conduction problem.