## 3 The Proposed Meshless Numerical Scheme
### Space-Time Generalized Finite Difference Method
By minimalizing \(B_{3}(u_{i})\) with respect to,  
\[\mathbf{D}_{u_{i}}=\left[\left.\frac{\partial u}{\partial x}\right|_{i}, \left.\frac{\partial u}{\partial x}\right|_{i},\left.\frac{\partial^{2}u}{ \partial x\partial t}\right|_{i},\left.\frac{\partial^{2}u}{\partial x\partial t }\right|_{i},\left.\frac{\partial^{2}u}{\partial x^{3}}\right|_{i},\left.\frac {\partial^{3}u}{\partial x^{2}ot}\right|_{i},\left.\frac{\partial^{3}u}{ \partial x\partial t^{2}}\right|_{i},\left.\frac{\partial^{3}u}{\partial x \partial t^{2}}\right|_{i},\left.\frac{\partial^{3}u}{\partial x\partial t^{2} }\right|_{i},\left.\frac{\partial^{3}u}{\partial x\partial t^{2}}\right|_{i},\left.\frac{\partial^{3}u}{\partial x\partial t^{2}}\right|_{i},\left.\frac{ \partial^{3}u}{\partial t^{3}}\right|_{i}\right]^{\mathsf{T}}, \tag{12}\]  
a linear system \(\mathbf{AD}_{u_{i}}=\mathbf{bU}_{i}\) is obtained. In this linear system, the matrix \(\mathbf{A}\) is denoted as \(\mathbf{A}=\mathbf{P}^{\mathsf{T}}\mathbf{W}^{2}\mathbf{P}\), where  
\[\mathbf{P}=\left[ \begin{array}{cccccccc}h_{i,1}&l_{i,1}&\frac{h_{i,1}^{2}}{ 2}&h_{i,1}l_{i,1}l_{i,1}&\frac{l_{i,1}^{2}}{2}&\frac{h_{i,1}^{3}}{2}&\frac{h_ {i,1}^{2}l_{i,1}}{2}&\frac{h_{i,1}l_{i,1}}{2}&\frac{h_{i,1}^{2}l_{i,1}}{2}&\frac {h_{i,1}^{2}}{2}&\frac{h_{i,1}^{2}}{2}\\ h_{i,2}&l_{i,2}&\frac{h_{i,2}^{2}}{2}&h_{i,2}l_{i,2}&\frac{h_{i,2}^{2}}{2}& \frac{h_{i,2}^{2}}{2}&\frac{h_{i,2}^{2}l_{i,2}}{2}&\frac{h_{i,2}^{2}l_{i,2}}{ 2}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ h_{i,j}&l_{i,j}&\frac{h_{i,j}^{2}}{2}&h_{i,j}l_{i,j}&\frac{l_{ij}^{2}}{2}& \frac{h_{ij}^{3}}{2}&\frac{h_{i,j}^{2}l_{i,j}}{2}&\frac{h_{i,j}^{2}l_{i,j}}{2} &\frac{h_{i,j}^{3}}{2}&\frac{h_{i,j}^{3}}{2}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ h_{i,n_{s}}&l_{i,n_{s}}&\frac{h_{i,n_{s}}^{2}}{2}&h_{i,n_{s}}l_{i,n_{s}}& \frac{l_{i,n_{s}}^{2}}{2}&\frac{h_{i,n_{s}}^{3}}{6}&\frac{l_{i,n_{s}}^{2}l_{i,n_ {s}}}{2}&\frac{h_{i,n_{s}}^{2}l_{i,n_{s}}}{2}&\frac{h_{i,n_{s}}^{3}}{6}\end{array}  \right., \tag{13}\]  
\(\mathbf{W}=diag(w_{i,1},w_{i,2},\cdots,w_{i,n_{s}})\); \(\mathbf{b}\) is a coefficient matrix whose size is \(9\times(n_{s}+1)\) and is written as \(\mathbf{b}=\left[\mathbf{P}^{\mathsf{T}}\mathbf{w}\quad\mathbf{P}^{\mathsf{ T}}\mathbf{W}^{2}\right]\); \(\mathbf{U}_{i}=\left[u_{i},u_{i,1},u_{i,2},\ldots,u_{i,n_{s}}\right]^{\mathsf{T}}\); and the superscript \(\mathsf{T}\) is a symbol of transpose. By solving this linear system, the approximate expression of \(\mathbf{D}_{u_{i}}\) can be written as,  
\[\mathbf{D}_{u_{i}}=\mathbf{A}^{-1}\mathbf{bU}_{i}=\mathbf{E}_{i}\mathbf{U}_{i}, \tag{14}\]  
where \(\mathbf{E}_{i}\) is the matrix of weighting coefficients within sized \(9\times(n_{s}+1)\). Then, the derivatives are denoted as follows:  
\[\left\{ \begin{array}{c}\left.\frac{\partial u}{\partial x}\right|_{i}=wx_{i, 0}u_{i}+\sum\limits_{j=1}^{n_{s}}wx_{i,ij}u_{i,j}\\ \left.\frac{\partial u}{\partial t}\right|_{i}=wt_{i,0}u_{i}+\sum\limits_{j=1}^{n _{s}}wt_{i,j}u_{i,j}\\ \left.\frac{\partial^{2}u}{\partial x^{2}}\right|_{i}=wxx_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wxx_{i,j}u_{i,j}\\ \left.\frac{\partial^{2}u}{\partial x\partial t}\right|_{i}=wxt_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wxt_{i,j}u_{i,j}\\ \left.\frac{\partial^{2}u}{\partial t^{2}}\right|_{i}=wtt_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wtt_{i,j}u_{i,j}\\ \left.\frac{\partial^{3}u}{\partial x^{3}}\right|_{i}=wxxx_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wxxx_{i,j}u_{i,j}\\ \left.\frac{\partial^{3}u}{\partial x^{2}ot}\right|_{i}=wxxt_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wxxt_{i,j}u_{i,j}\\ \left.\frac{\partial^{3}u}{\partial x\partial t^{2}}\right|_{i}=wxtt_{i,0}u_{i}+ \sum\limits_{j=1}^{n_{s}}wxtt_{i,j}u_{i,j}\\ \left.\frac{\partial^{2}u}{\partial t^{3}}\right|_{i}=wtt_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wtt_{i,j}u_{i,j}\\ \left.\frac{\partial^{2}u}{\partial t^{3}}\right|_{i}=wtt_{i,0}u_{i}+\sum \limits_{j=1}^{n_{s}}wtt_{i,j}u_{i,j}\end{array}