## 3 The Proposed Meshless Numerical Scheme
### Space-Time Generalized Finite Difference Method
\right., \tag{15}\]where \(\left\{wx_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wt_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxx_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxt_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wtt_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxxx_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxxx_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxxx_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxxt_{i,j}\right\}_{j=0}^{n_{s}}\), \(\left\{wxt_{i,j}\right\}_{j=0}^{n_{s}}\), and \(\left\{wtt_{i,j}\right\}_{j=0}^{n_{s}}\) are the weighting coefficients, and obtained by solving Equation (14). Equation (15) is a significant part of the proposed ST-GFDM in that the partial derivatives are approximated by a linear combination of the weight coefficients and function values in a local support domain. Thus, the discretized governing equations and boundary conditions only relate to the central \(i\) node and its nearby \(n_{s}\) nodes in the local support domain. In addition, the numerical process of Equations (9)-(15) will apply for each node to obtain the weighting coefficients matrix \(\mathbf{E}_{i}\) of each node. Finally, the resultant matrix system, formed by the algebraic equations after discretization, is a sparse matrix. The following contexts demonstrate the numerical discretization for the governing equation and boundary conditions to form the nonlinear algebraic systems. Since the simulated start, \(N_{b1}\), \(N_{b2}\), \(N_{b3}\), and \(N_{b4}\) nodes have been set up along the \(\Gamma_{1}\), \(\Gamma_{2}\), \(\Gamma_{3}\) and \(\Gamma_{4}\), respectively. Meanwhile, \(N_{i}\) nodes are distributed inside the computational domain \(\Omega\). (see Figure 1a). The nodes along \(\Gamma_{1}\) are satisfied with the initial conditions \(U_{0}(x)\), and the nodes along \(\Gamma_{2}\) and \(\Gamma_{3}\) are satisfied with the boundary conditions \(f_{1}(a,t)\) and \(f_{2}(b,t)\), respectively. Therefore, the algebraic equations along the \(\Gamma_{1}\), \(\Gamma_{2}\) and \(\Gamma_{3}\) are described as,  
\[\left\{ \begin{array}{l}F_{i}=u_{i}-U_{0}(x_{i}),\,i=1,2,\ldots,N_{b1},\\ F_{i}=u_{i}-f_{2}(a,t_{i}),\,i=N_{b1}+1,N_{b1}+2,\ldots,N_{b1}+N_{b2},\\ F_{i}=u_{i}-f_{1}(b,t_{i}),\,i=N_{b1}+N_{b2}+1,N_{b1}+N_{b2}+2,\ldots,N_{b1}+N_ {b2}+N_{b3},\end{array} \right. \tag{16}\]  
The remaining domains, \(\Gamma_{4}\) and \(\Omega\), are both satisfied by the governing equation, Equation (1), and they can be discretized by Equation (15) as follows:  
\[F_{i}=\left\{ \begin{array}{l}\left(wt_{i,0}u_{i}+\sum\limits_{j=1}^{n_{s}} wt_{i,j}u_{i,j}\right)-\left(wxxt_{i,0}u_{i}+\sum\limits_{j=1}^{n_{s}}wxxt_{i,j}u_{ i,j}\right)+\ldots\\ \left(\alpha+\beta u_{i}^{\omega}\right)\left(wx_{i,0}u_{i}+\sum\limits_{j=1} ^{n_{s}}wx_{i,j}u_{i,j}\right)-\ldots\\ q\left(wx_{i,0}u_{i}+\sum\limits_{j=1}^{n_{s}}wx_{i,j}u_{i,j}\right)\left(wxx _{i,0}u_{i}+\sum\limits_{j=1}^{n_{s}}wxx_{i,j}u_{i,j}\right)-\ldots\\ u_{i}\left(wxxx_{i,0}u_{i}+\sum\limits_{j=1}^{n_{s}}wxxx_{i,j}u_{i,j}\right) \end{array} \right\}, \tag{17}\] \[i=N_{b1}+N_{b2}+N_{b3}+1,N_{b1}+N_{b2}+N_{b3}+2,\ldots,N_{B},N_{ B}+1,N_{B}+2,\ldots,N_{T},\]  
where \(N_{B}\) is a number of nodes set up on the whole boundary, and defined as \(N_{B}=N_{b1}+N_{b2}+N_{b3}+N_{b4}\); \(N_{T}\) is the number of total nodes in the computational domain, and defined as \(N_{T}=N_{B}+N_{i}\). In Equations (16) and (17), \(N_{T}\) algebraic equations are acquired and denoted as \(\mathbf{F}=\left[F_{1},F_{2},\ldots,F_{i}\right]^{\text{T}}\), \(i=1,2,\ldots,N_{T}\). Meanwhile, \(N_{T}\) unknown values are defined as \(\mathbf{U}=\left[u_{1},u_{2},\ldots,u_{i}\right]^{\text{T}}\), \(i=1,2,\ldots,N_{T}\). Then, the two-step [PERSON]'s method is applied to solve this nonlinear algebraic system, and the specific numerical process is described in the next subsection.