## 4 Numerical Examples
### Example 1  
The Fornberg-Whitham equation [9; 11] is solved in this subsection. This mathematical model is presented to describe the behavior of the breaking wave and has solitary or traveling wave solutions. The exact solution of the first numerical example is described as,  
\[{{u^{ext}}(x,t)=e^{(\frac{x}{2}-\frac{2t}{3})}}, \tag{23}\]  
and the initial condition and boundary conditions are represented as,  
\[{{U_{0}}(x)=e^{\frac{x}{2}},\ f_{1}(t)=e^{(\frac{x}{2}-\frac{2t}{3})},\ f_{2}(t )=e^{(\frac{b}{2}-\frac{2t}{3})}}. \tag{24}\]  
The following parameters are applied in this numerical example: \(a=-5\), \(b=5\), \(\text{dt}=0.5\), \(T=5\), and \(N_{D}=10\). In Figure 4, the numerical solutions are obtained by applying \(N_{T}=8421\) and \(n_{s}=23\). The numerical solutions at different time levels are compared with the exact solutions and demonstrated in Figure 4a. This figure shows that the numerical solutions agree well with the exact solutions. Figure 4b is a 3D surface plot of numerical solutions to observe the numerical behavior of the Fornberg-Whitham equation through time. Tables 2-4 are provided to verify the accuracy and parameter sensitivity of the proposed method. The MAEs and RMSEs are calculated using \(N_{T}\) and \(n_{s}\) at different time levels. In Table 2, the results for different time levels are displayed, showing the MAEs and RMSEs obtained for various \(N_{T}\). These error values demonstrate that increasing \(N_{T}\) leads to more accurate numerical solutions. Meanwhile, the convergence rate and CPU times are shown in Table 2. Since the proposed numerical scheme is a 3 rd-order  
Figure 4: (**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 1. [PERSON]'s method, according to the theoretical convergence rates of these two methods, the theoretical convergence rates of the proposed numerical scheme should be 2 nd-order convergence. From Table 2, the numerical performance is between 1.5-2. On the other hand, the CPU times are provided in Table 2 as well. Table 2 reveals that simulations are completed within 4 s when \(N_{T}\) is below 9000, and the errors table indicates that satisfactory numerical solutions can already be achieved at this \(N_{T}\). With an end time of 5 for the first example, the proposed numerical scheme can successfully complete the simulation within the problem's specified end time using approximately 9000 nodes. Despite the increase in CPU time as \(N_{T}\) increases, the CPU time remains below 10 s, demonstrating the efficiency of the proposed ST-GFDM in solving problems with long-time scales. In Table 3, MAEs and RMSEs are also presented using different \(n_{s}\). The errors indicate that the parameter \(n_{s}\) exhibits low sensitivity. Additionally, both tables demonstrate no accumulation of errors throughout the numerical simulation. This observation supports the effectiveness of the ST approach in mitigating the accumulation of numerical errors, a concern that hybrid numerical schemes often encounter when applied to time-dependent problems. In order to further validate the capability of the proposed numerical scheme, the numerical solutions located at specific positions (\(x\),\(t\)) are presented in Table 4 and compared with the exact solutions. In this table, the function values are taken to eight decimal places to show the precision accuracy of the proposed ST-GFDM. Additionally, the proposed ST-GFDM is compared with the HAM [10], ADM [7], RKHSM [7], MVIA-I [5], and the 2 nd-order finite difference method combined with the fourth-order Runge-Kutta method