## 5 Conclusions  
This study uses the proposed ST-GFDM to numerically solve nonlinear shallow water wave equations, specifically the modified Camassa-Holm equation, the modified Degasperis-Procesi equation, the Fornberg-Whitham equation, and its modified form. The ST-GFDM discretizes the governing equations in the ST domain, eliminating the need for mesh generation and simplifying the numerical procedures. The resulting nonlinear algebraic system is solved using the two-step Newton's method. Adopting ST-GFDM makes the computational process more straightforward and avoids the complexities associated with mesh generation. One significant innovation of the ST-GFDM is treating the temporal derivative as a spatial derivative. The ST-GFDM enables the discretization of all partial derivatives and the efficient handling of mixed derivatives. Due to the Taylor series expansion and the moving least-squares method, the proposed meshless numerical scheme forms a nonlinear algebraic system, and this system has a sparse Jacobian matrix. Then, the two-step Newton-Raphson method is applied to efficiently solve this nonlinear algebraic system. To deal with long-time-scale numerical problems, the time-marching method is applied for moving the ST domain and saving computational resource, thereby addressing the shortcomings of the ST approach.  
Four numerical examples are presented to verify the proposed ST-GFDM scheme. In the first numerical case, by varying different parameters, it is observed that increasing the \(N_{T}\) leads to more accurate numerical solutions. Additionally, adjusting the parameter \(n_{s}\) in a supporting domain demonstrates that \(n_{s}\) has low sensitivity. The numerical solutions have good agreements with the exact solutions and the solutions of other approximation methods in four examples. Furthermore, the accumulation of numerical errors during the time progression is insignificant. The proposed ST-GFDM scheme offers several advantages, including accurate treatment of transient problems, computational efficiency,  
Figure 8: (**a**) The comparisons of the \(u^{ext}\) and \(u^{num}\) at different \(t\) and (**b**) the surface plot of \(u^{num}\) in the entire \(x\)-\(t\) domain of Example 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(x\) & \(t\) & \(u^{ext}\) & **ST-GFDM** & **VIM [2]** & **MAEs** \\ \hline
6 & 0.05 & \(-\)0.020951831 & \(-\)0.020948113 & \(-\)0.018566923 & 3.717910 \(\times\) 10\({}^{-6}\) \\
8 & 0.05 & \(-\)0.002851900 & \(-\)0.002848801 & \(-\)0.002682964 & 3.099397 \(\times\) 10\({}^{-6}\) \\
6 & 0.1 & \(-\)0.023714184 & \(-\)0.023719627 & \(-\)0.018635025 & 5.443036 \(\times\) 10\({}^{-6}\) \\
8 & 0.1 & \(-\)0.003229560 & \(-\)0.003227788 & \(-\)0.002516809 & 1.771923 \(\times\) 10\({}^{-6}\) \\ \hline \end{tabular}
\end{table}
Table 8: Comparisons of the numerical results of Example 4 with the exact solution and the other numerical methods at various locations at \(t\) = 0.05 and \(t\) = 0.1.  
avoidance of mesh generation, and flexibility in handling complex PDEs. This observation highlights the scheme's capability to effectively solve transient problems while maintaining numerical accuracy. Future research can explore the application of the ST-GFDM scheme in computational fluid dynamics and tackle more challenging problem domains.  
Conceptualization, P.-W.L. and S.H.; methodology, P.-W.L.; software, S.H. and M.Z.; validation, S.H. and M.Z.; writing--original draft preparation, P.-W.L.; writing--review and editing, P.-W.L.; funding acquisition, P.-W.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation of Shandong Province of China, grant number ZR2021 QA012.  
Not applicable.  
The data are available from the corresponding author upon reasonable request.  
The authors sincerely thank the Natural Science Foundation of Shandong Province, China, for providing financial support.  
The authors declare no conflict of interest.