## 4 Numerical Experimentation and Results
### Example 1  
Consider the following example that belongs to the class of problems with the so-called Runge phenomenon:  
\[y^{\prime}=144(1-2t)y^{2},\quad y(0)=\frac{1}{37}. \tag{31}\]  
The exact solution of Example (31) is \(y(t)=\frac{1}{36(2t-1)^{2}+1}\). In this example, the linear partitioning iterative method with  
\[\phi(t,y_{r})=144(1-2t)y_{r}^{2},\quad\psi(t,y_{r})=0\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(m\) & **Equidistant** & **Optimal Points** \\ \hline
3 & \(\begin{array}{c}1\;\;2\\ 3^{\prime}\;\;3\end{array}\) & \(\frac{1}{10}(5\pm\sqrt{5})\) \\ \hline
5 & \(\begin{array}{c}1\;\;2\;3\;\;4\\ 5^{\prime}\;\;5^{\prime}\;\;5^{\prime}\;\;5\end{array}\) & \(\frac{1}{2}\bigg{(}1\pm\sqrt{\frac{1}{21}\big{(}7\pm 2\sqrt{7}\big{)}} \bigg{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Intra-step points used in numerical experimentation. was used with the HBM schemes to solve (31) using \(h=0.01\) and up to 20 iterations, within the integration domain [0,1]. The convergence plot, computed as the maximum error per iteration in the last block \([t_{N-1},t_{N}]\), is shown in Figure 3. The convergence of the optimal overlapping (OOHBM) and the original optimal hybrid block technique (OHBM) with no overlapping grid for \(m=3\) and \(m=5\) are compared. The graph demonstrates that the overlapping HBM plateaus at a higher degree of accuracy than the non-overlapping HBM. This is one indicator of the overlapping HBM's higher precision. It is also apparent from Figure 3 that the linear partitioning HBM converges swiftly, with full convergence reached within 10 iterations for all HBM variants evaluated in this scenario. A comparison of the maximum absolute errors, computed from the exact solution, is displayed in Table 3. It can be seen from the table that the maximum error improves when the overlapping grid is applied. In this example, the improvement in the maximum error when comparing equally spaced and optimal hybrid block methods is not as significant as the difference between the original and proposed overlapping HBM. It can also be observed that the maximum error obtained using the overlapping HBM with equally spaced points is greater than the error of the original optimal HBM, for both \(m=3\) and \(m=5\). This appears to indicate that introducing grid overlap into HBM systems reduces the maximum error more than adjusting the type of intra-step points. The maximum error in each block \([t_{n},t_{n+1}]\) is defined as  
\[E_{n}=\max_{1\leq i\leq m}|y(x_{n+p_{i}})-y_{n+p_{i}}|. \tag{32}\]  
Plotting the profile of the maximum error in each block for \(n=0,1,\ldots,N-1\) reveals the distribution of the error as it traverses the complete domain of integration. Figure 4  
Figure 3: Comparison of the convergence plots for Example 1 when \(h=0.01\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{\(m=3\)} & \multicolumn{2}{c}{\(m=5\)} \\ \cline{2-4}  & **Original HBM** & **Proposed HBM** & **Original HBM** & **Proposed HBM** \\ \hline Equally spaced points & \(9.9970\times 10^{-7}\) & \(6.7542\times 10^{-8}\) & \(6.0825\times 10^{-10}\) & \(4.2735\times 10^{-11}\) \\ Optimal points & \(1.8106\times 10^{-7}\) & \(3.2112\times 10^{-8}\) & \(1.4701\times 10^{-10}\) & \(1.5855\times 10^{-11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum errors for Example 1 using different choices of intra-step points. depicts the maximum error profile \(E_{n}\) for 100 intervals (blocks) used during the deployment of the overlapping optimal HBM and original optimal HBM to Example 31. The graph illustrates the significant reduction in the maximum error when the overlapping grid method is implemented. The maximum error begins in the same location when \(n=0\). This is expected given that the initial interval is identical for both optimal overlapping (OOHBMs) and standard optimal non-overlapping HBMs (OHBMs). After \(n=0\), the maximum errors of the OOHBM are less than those of the OHBM when \(m=3\) and when \(m=5\).