## 4 Numerical Experiments and Performance Evaluation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Metric** & **Iterations** & **C-Time** & **M-Usage** & **P-Con** & **M-Err** \\ \hline EF\({}^{[1]}\) & 22 & 5.432 & 67.424 & 19.342 & \(2.52\times 10^{-9}\) \\ EF\({}^{[2]}\) & 21 & 4.024 & 74.325 & 11.765 & \(0.53\times 10^{-7}\) \\ FS\({}^{[1]}\) & 23 & 3.142 & 65.932 & 17.439 & \(6.63\times 10^{-11}\) \\ FS\({}^{[2]}\) & 16 & 3.524 & 55.327 & 25.325 & \(0.43\times 10^{-21}\) \\ FS\({}^{[3]}\) & 17 & 4.442 & 31.403 & 45.897 & \(1.76\times 10^{-39}\) \\ FS\({}^{[4]}\) & 18 & 3.204 & 49.540 & 28.942 & \(5.47\times 10^{-25}\) \\ FS\({}^{[5]}\) & 15 & 3.642 & 41.542 & 22.342 & \(5.63\times 10^{-19}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of fractional schemes without chaos-optimized parameter tuning for solving Equation (45). The dynamical behavior of the proposed scheme, in comparison with existing methods, is presented in Table 3 and Figure 6. As shown in Table 3, the proposed approaches demonstrate improved performance, characterized by reduced generation time of the basins of attraction (E-time) and a higher percentage of convergence. In Table 4, C-time refers to the computational time in seconds, M-Err denotes the absolute residual error (as defined in (42)), and P-con represents the percentage convergence \begin{table}
\begin{tabular}{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\({}^{[1]}\) & \(8.9\times 10^{-3}\) & 5.4 & \(8.5\times 10^{-5}\) & 2.8 & \(8.5\times 10^{-15}\) & 1.5 & \(8.5\times 10^{-65}\) & 1.01 \\ EF\({}^{[2]}\) & \(6.8\times 10^{-2}\) & 4.0 & \(6.8\times 10^{-7}\) & 3.4 & \(8.5\times 10^{-15}\) & 1.4 & \(6.8\times 10^{-47}\) & 1.7 \\ FS\({}^{[1]}\) & \(1.7\times 10^{-6}\) & 3.1 & \(0.6\times 10^{-6}\) & 2.3 & \(0.6\times 10^{-46}\) & 0.0 & \(0.6\times 10^{-86}\) & 0.4 \\ FS\({}^{[2]}\) & \(3.0\times 10^{-5}\) & 3.5 & \(0.4\times 10^{-5}\) & 3.0 & \(0.4\times 10^{-45}\) & 0.5 & \(0.4\times 10^{-85}\) & 0.5 \\ FS\({}^{[3]}\) & \(3.1\times 10^{-9}\) & 4.4 & \(9.1\times 10^{-9}\) & 4.6 & \(1.1\times 10^{-59}\) & 0.4 & \(1.9\times 10^{-79}\) & 0.01 \\ FS\({}^{[4]}\) & \(1.7\times 10^{-1}\) & 3.0 & \(5.7\times 10^{-2}\) & 3.7 & \(5.7\times 10^{-65}\) & 0.4 & \(6.4\times 10^{-86}\) & 0.02 \\ FS\({}^{[5]}\) & \(0.5\times 10^{-7}\) & 3.6 & \(0.6\times 10^{-7}\) & 2.6 & \(5.9\times 10^{-59}\) & 0.3 & \(1.4\times 10^{-89}\) & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error and consistency analysis of fractional-order schemes EF\({}^{[1]}\)–EF\({}^{[2]}\) and FS\({}^{[1]}\)–FS\({}^{[5]}\) for solving Equation (45) across different values of fractional parameter \(\gamma\). Figure 6: Basins of attraction for fractional-order schemes FS\({}^{[1]}-\)FS\({}^{[5]}\) and EF\({}^{[1]}-\)EF\({}^{[2]}\) when solving Equation (45). \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Metric** & **EF\({}^{[1]}\)** & **EF\({}^{[2]}\)** & **FS\({}^{[1]}\)** & **FS\({}^{[2]}\)** & **FS\({}^{[3]}\)** & **FS\({}^{[4]}\)** & **FS\({}^{[5]}\)** \\ \hline E-time & 2.1348 & 2.3453 & 2.543 & 3.755 & 2.0054 & 4.7655 & 1.63254 \\ M-usage & 75.432 & 78.543 & 45.765 & 43.654 & 33.546 & 23.654 & 20.887 \\ P-convergence & 65.45 & 67.56 & 68.67 & 87.56 & 34.45 & 35.43 & 26.453 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dynamical performance analysis of fractional schemes FS\({}^{[1]}\)–FS\({}^{[5]}\) and EF\({}^{[1]}\)–EF\({}^{[2]}\) for solving Equation (45). ### Extended Glucose-Insulin Regulatory Model [31]

The glucose-insulin homeostasis model is fundamental for understanding the mechanisms that regulate blood glucose levels, particularly in the context of diabetes. It describes the dynamic relationship between glucose levels in the blood and insulin secretion by the pancreas. This regulation is governed by complex feedback loops in which insulin lowers glucose levels, while glucose concentration in turn influences insulin production.