## 4 Numerical Experiments and Performance Evaluation
The numerical results indicate that the proposed methods yield improved performance over existing techniques, particularly in terms of error reduction, memory efficiency, convergence rate, and the number of iterations required to reach the desired accuracy under chaos-optimized and fractional parameter settings. Moreover, Tables 14 and 15 show that increasing the fractional \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Metric** & **EF\({}^{[1]}\)** & **EF\({}^{[2]}\)** & **FS\({}^{[1]}\)** & **FS\({}^{[2]}\)** & **FS\({}^{[3]}\)** & **FS\({}^{[4]}\)** & **FS\({}^{[5]}\)** \\ \hline E-time & 7.6545 & 8.6546 & 2.0098 & 1.4045 & 2.9004 & 0.2015 & 0.0014 \\ M-usage & 87.545 & 89.085 & 65.983 & 45.345 & 33.765 & 32.545 & 19.435 \\ P-convergence & 25.65 & 29.23 & 89.07 & 90.07 & 93.05 & 94.09 & 97.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Dynamical analysis of fractional-order schemes \(\text{FS}^{[1]}\)–\(\text{FS}^{[5]}\) and \(\text{EF}^{[1]}\)–\(\text{EF}^{[2]}\) for solving Equation (52). Metrics include execution time, memory usage, and percentage convergence. order \(\gamma\) from 0.1 to 1.0 enhances the convergence speed, stability, and consistency relative to the EF\({}^{[1]}\)-EF\({}^{[2]}\) methods. Table 16 and Figure 11 summarize the overall performance of the proposed scheme under random perturbations and for \(\gamma\approx 1\), in comparison with existing methods. The results indicate that

* The number of iterations required for convergence is significantly reduced. * Higher relative accuracy is achieved, as reflected by a lower residual error in the fractional scheme. * A greater percentage of successful convergences is observed, along with reduced CPU and memory usage. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Metric** & **M-Usage** & **P-Con** & **M-Usage** & **P-Con** & **M-Usage** & **P-Con** & **M-Usage** & **P-Con** \\ \hline \(\gamma\) & 0.3 & & 0.5 & & 0.7 & & 0.9 & \\ EF\({}^{[1]}\) & 75.12 & 8.8 & 65.34 & 12.7 & 56.34 & 45.7 & 31.87 & 48.7 \\ EF\({}^{[2]}\) & 87.54 & 9.1 & 67.98 & 19.2 & 47.35 & 54.8 & 29.43 & 56.2 \\ FS\({}^{[1]}\) & 56.12 & 10.0 & 55.34 & 13.2 & 35.76 & 67.3 & 22.98 & 93.2 \\ FS\({}^{[2]}\) & 63.23 & 16.7 & 53.34 & 26.7 & 33.23 & 76.3 & 21.88 & 97.3 \\ FS\({}^{[3]}\) & 59.76 & 16.9 & 49.34 & 26.6 & 38.74 & 65.6 & 23.54 & 99.7 \\ FS\({}^{[4]}\) & 54.09 & 15.8 & 43.23 & 25.3 & 38.98 & 70.1 & 19.32 & 92.2 \\ FS\({}^{[5]}\) & 58.23 & 17.8 & 39.23 & 30.2 & 31.67 & 62.0 & 20.12 & 90.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Memory usage, percentage convergence, and stability analysis of fractional technique for solving (52). Figure 10: Basins of attraction of fractional-order schemes FS\({}^{[1]}-\)FS\({}^{[5]}\) and EF\({}^{[1]}-\)EF\({}^{[2]}\) for solving Equation (48). \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Metric** & **M-Err** & **C-Time** & **M-Err** & **C-Time** & **M-Err** & **C-Time** & **M-Err** & **C-Time** \\ \hline \(\gamma\) & 0.3 & & 0.5 & & 0.7 & & 0.9 & \\ EF\({}^{[1]}\) & \(1.1\times 10^{-3}\) & 7.1 & \(8.5\times 10^{-3}\) & 3.6 & \(8.5\times 10^{-6}\) & 1.5 & \(8.5\times 10^{-8}\) & 2.8 \\ EF\({}^{[2]}\) & \(0.8\times 10^{-3}\) & 6.4 & \(6.8\times 10^{-4}\) & 4.9 & \(6.8\times 10^{-7}\) & 1.7 & \(6.8\times 10^{-9}\) & 2.6 \\ FS\({}^{[1]}\) & \(0.9\times 10^{-2}\) & 6.6 & \(0.6\times 10^{-9}\) & 4.6 & \(0.6\times 10^{-41}\) & 0.1 & \(0.6\times 10^{-96}\) & 0.4 \\ FS\({}^{[2]}\) & \(6.7\times 10^{-5}\) & 4.6 & \(0.4\times 10^{-11}\) & 3.7 & \(0.4\times 10^{-31}\) & 0.7 & \(0.4\times 10^{-115}\) & 0.3 \\ FS\({}^{[3]}\) & \(3.5\times 10^{-4}\) & 4.9 & \(7.1\times 10^{-14}\) & 4.6 & \(3.3\times 10^{-39}\) & 0.8 & \(0.9\times 10^{-169}\) & 0.3 \\ FS\({}^{[4]}\) & \(3.4\times 10^{-6}\) & 5.7 & \(0.4\times 10^{-11}\) & 3.7 & \(6.4\times 10^{-31}\) & 0.7 & \(6.6\times 10^{-115}\) & 0.2 \\ FS\({}^{[5]}\) & \(1.2\times 10^{-5}\) & 4.9 & \(2.8\times 10^{-15}\) & 4.6 & \(5.5\times 10^{-43}\) & 0.8 & \(1.7\times 10^{-141}\) & 0.1 \\ \hline \hline \end{tabular}
\end{table}