## 4 Numerical Examples
### Example 1
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{5}{c}{\(t\) = 1 (n = 2)} & \multicolumn{2}{c}{\(t\) = 3 (n = 6)} & \multicolumn{2}{c}{\(t\) = 5 (n = 10)} \\ \hline  & \(u^{ext}\) & **ST-GFDM** & \(u^{ext}\) & **ST-GFDM** & \(u^{ext}\) & **ST-GFDM** \\ \hline \(x\) & \(-4\) & 0.06948345 & 0.06948238 & 0.01831564 & 0.01831622 & 0.00482795 & 0.00483149 \\  & \(-2\) & 0.18887560 & 0.18887409 & 0.04978707 & 0.04978139 & 0.01312373 & 0.01312198 \\  & \(0\) & 0.51341712 & 0.51342701 & 0.13533528 & 0.13532764 & 0.03567399 & 0.03566356 \\  & \(2\) & 1.39561243 & 1.39565330 & 0.36787944 & 0.36788606 & 0.09697197 & 0.09696040 \\  & \(4\) & 3.79366789 & 3.79371117 & 1.00000000 & 1.00001868 & 0.26359714 & 0.26359633 \\ \(L_{\infty,n}\) & & \(5.158\times 10^{-5}\) & & \(1.961\times 10^{-5}\) & & \(1.301\times 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparisons of the numerical results of Example 1 with exact solutions at various locations at different time levels (\(N_{T}\) = 8421 and \(n_{s}\) = 23). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{5}{c}{\(t\) = 1 (n = 2)} & \multicolumn{2}{c}{\(t\) = 3 (n = 6)} & \multicolumn{2}{c}{\(t\) = 5 (n = 10)} & \multicolumn{2}{c}{Convergence} & \multirow{2}{*}{**CPU-Time**} \\ \cline{2-2} \cline{6-7} \multicolumn{1}{c}{\(N_{T}\)} & \(L_{\infty,2}\) & **RMSE\({}_{2}\)** & \(L_{\infty,6}\) & **RMSE\({}_{6}\)** & \(L_{\infty,10}\) & **RMSE\({}_{10}\)** & **RMSE\({}_{10}\)** & **Are Used** \\ \hline
2211 & \(1.725\times 10^{-4}\) & \(2.424\times 10^{-5}\) & \(2.251\times 10^{-5}\) & \(4.081\times 10^{-6}\) & \(3.250\times 10^{-5}\) & \(5.891\times 10^{-6}\) & - & 0.7829 s \\
4215 & \(7.653\times 10^{-5}\) & \(8.978\times 10^{-6}\) & \(1.206\times 10^{-5}\) & \(1.780\times 10^{-6}\) & \(1.703\times 10^{-5}\) & \(2.693\times 10^{-6}\) & 1.918 & 1.5629 s \\
8421 & \(3.267\times 10^{-5}\) & \(3.071\times 10^{-6}\) & \(6.677\times 10^{-6}\) & \(8.885\times 10^{-7}\) & \(9.153\times 10^{-6}\) & \(1.214\times 10^{-6}\) & 1.742 & 3.5874 s \\
12,025 & \(2.147\times 10^{-5}\) & \(1.773\times 10^{-6}\) & \(5.152\times 10^{-6}\) & \(6.536\times 10^{-7}\) & \(6.891\times 10^{-6}\) & \(8.334\times 10^{-7}\) & 1.543 & 5.9342 s \\
18,631 & \(1.313\times 10^{-5}\) & \(9.162\times 10^{-7}\) & \(3.856\times 10^{-6}\) & \(4.747\times 10^{-7}\) & \(4.927\times 10^{-6}\) & \(5.949\times 10^{-7}\) & 1.528 & 10.6508 s \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical errors of Example 1 with \(n_{s}\) = 20 using different \(N_{T}\) at different \(t\), convergence rates, and CPU-time. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{5}{c}{\(t\) = 1 (n = 2)} & \multicolumn{2}{c}{\(t\) = 3 (n = 6)} & \multicolumn{2}{c}{\(t\) = 5 (n = 10)} \\ \hline \(n_{s}\) & \(L_{\infty,2}\) & **RMSE\({}_{2}\)** & \(L_{\infty,6}\) & **RMSE\({}_{6}\)** & \(L_{\infty,10}\) & **RMSE\({}_{10}\)** \\ \hline
20 & \(3.267\times 10^{-5}\) & \(3.071\times 10^{-6}\) & \(6.677\times 10^{-6}\) & \(8.855\times 10^{-7}\) & \(9.153\times 10^{-6}\) & \(1.214\times 10^{-6}\) \\
23 & \(5.158\times 10^{-5}\) & \(5.737\times 10^{-6}\) & \(1.961\times 10^{-5}\) & \(2.011\times 10^{-6}\) & \(1.301\times 10^{-5}\) & \(1.563\times 10^{-6}\) \\
26 & \(5.093\times 10^{-5}\) & \(5.713\times 10^{-6}\) & \(3.067\times 10^{-5}\) & \(3.064\times 10^{-6}\) & \(1.264\times 10^{-5}\) & \(FDM-RK4). Those comparisons are shown in Table 5. In these comparative data, it can be seen that the proposed ST-GFDM performs better than the HAM and ADM. Although the accuracy is weaker than the RKHSM and MVIA-I, solutions with numerical errors below \(10^{-6}\) or even \(10^{-7}\) can be obtained, which is already an acceptable numerical result for engineering applications. To compare with the original numerical scheme for time-dependent problems, the FDM-RK4 is applied to obtain the numerical results within \(\Delta x\)= 0.5 and \(\Delta t\) = 0.0001. The numerical solutions of the FDM-RK4 show that the results have a maximum absolute error of the order of \(10^{-1}\).