## 4 Numerical Experiments and Performance Evaluation
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric** & **Iterations** & **C-Time** & **M-Usage** & **P-Con** & **Error** \\ \hline \multicolumn{6}{c}{\(\gamma=0.5\)} \\ Random & 67 & 6.030 & 87.56 & 09.73\% & \(8.5\times 10^{-2}\) \\ EF\({}^{[1]}\) & 57 & 5.324 & 65.34 & 13.63\% & \(7.5\times 10^{-2}\) \\ EF\({}^{[2]}\) & 50 & 5.025 & 64.43 & 10.43\% & \(0.5\times 10^{-2}\) \\ FS\({}^{[1]}\) & 55 & 5.543 & 55.54 & 13.43\% & \(0.6\times 10^{-2}\) \\ FS\({}^{[2]}\) & 51 & 4.543 & 36.65 & 15.23\% & \(0.4\times 10^{-3}\) \\ FS\({}^{[3]}\) & 45 & 5.546 & 54.65 & 17.34\% & \(0.1\times 10^{-2}\) \\ FS\({}^{[4]}\) & 48 & 5.436 & 53.76 & 18.74\% & \(8.0\times 10^{-2}\) \\ FS\({}^{[5]}\) & 47 & 4.543 & 61.78 & 15.03\% & \(0.3\times 10^{-1}\) \\ Random & 68 & 7.764 & 87.23 & 9.22\% & \(7.9\times 10^{-4}\) \\ EF\({}^{[1]}\) & 65 & 6.632 & 54.56 & 13.73\% & \(2.1\times 10^{-3}\) \\ EF\({}^{[2]}\) & 33 & 5654 & 45.13 & 43.23\% & \(1.5\times 10^{-4}\) \\ FS\({}^{[1]}\) & 37 & 3.234 & 52.04 & 30.03\% & \(2.4\times 10^{-5}\) \\ FS\({}^{[2]}\) & 33 & 4.124 & 43.08 & 37.73\% & \(3.4\times 10^{-5}\) \\ FS\({}^{[3]}\) & 31 & 3.653 & 57.11 & 35.23\% & \(3.4\times 10^{-6}\) \\ FS\({}^{[4]}\) & 33 & 3.554 & 58.23 & 53.73\% & \(5.7\times 10^{-5}\) \\ FS\({}^{[5]}\) & 29 & 3.543 & 54.65 & 41.03\% & \(0.3\times 10^{-5}\) \\  & & \(\gamma=0.9\) & & & \\ Random & 55 & 5.054 & 87.36 & 15.87\% & \(8.5\times 10^{-19}\) \\ EF\({}^{[1]}\) & 17 & 3.054 & 63.36 & 15.23\% & \(8.5\times 10^{-19}\) \\ EF\({}^{[2]}\) & 18 & 3.655 & 33.43 & 45.09\% & \(8.5\times 10^{-29}\) \\ FS\({}^{[1]}\) & 11 & 2.987 & 25.54 & 43.43\% & \(0.3\times 10^{-19}\) \\ FS\({}^{[2]}\) & 12 & 2.435 & 20.65 & 48.76\% & \(2.4\times 10^{-33}\) \\ FS\({}^{[3]}\) & 10 & 0.231 & 11.65 & 51.65\% & \(3.4\times 10^{-39}\) \\ FS\({}^{[4]}\) & 11 & 0.132 & 10.76 & 52.23\% & \(7.9\times 10^{-46}\) \\ FS\({}^{[5]}\) & 13 & 0.321 & 18.78 & 53.43\% & \(7.6\times 10^{-39}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Overall performance of fractional-order schemes FS\({}^{[1]}\)–FS\({}^{[5]}\) and EF\({}^{[1]}\)–EF\({}^{[2]}\), and a random perturbation strategy for solving Equation (48) for varying values of \(\gamma\). Results are compared in terms of iterations, CPU time, memory usage, percentage convergence, and error. \(C_{a}^{n}=5\) (drug concentration in \(\upmu M\)). * \(n=2\) (steepness factor). * \(R=60\) (observed response). By substituting them into (51), we obtain the nonlinear equation

\[f(x)=\frac{100x^{2}}{25+x^{2}}-60=0. \tag{52}\]

The Caputo-order fractional derivative of the function is given by

\[\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\hbar(x)=\big{[}D_{\gamma_{1}}^{\gamma} \big{]}\hbar(x)=\frac{1}{\Gamma(\gamma+1)}\int\limits_{0}^{x}\Bigg{[}\Bigg{(} \frac{200x}{25+x^{2}}-\frac{\left(200x\right)^{3}}{\left(25+x^{2}\right)^{2}} \Bigg{)}\frac{1}{\left(x-t\right)^{\gamma}}\Bigg{]}dt. \tag{53}\]

The exact root of this nonlinear model is \(\zeta=6.123724357\), and the corresponding initial guess is \(x^{[0]}=5.1\). Tables 12-15 present the numerical results obtained using fractional schemes with and without the chaos-optimized parameter for solving Equation (52). Tables 13-15 present cross-validation results for both training and testing phases, demonstrating the improved stability, consistency, memory efficiency, and accuracy of the proposed method compared with existing approaches. The dynamical behavior of the proposed scheme, compared with existing methods, is summarized in Table 13 and illustrated in Figure 10. As shown in Table 13, the proposed approach exhibits improved performance, characterized by reduced generation time for the basin of attraction (E-time) and a higher percentage of convergence. Table 14 presents the error and consistency analysis, while Table 15 reports the memory usage, percentage convergence, and stability assessment of the fractional schemes applied to the biomedical engineering model described in Section 4.3.