## 4 Numerical Experiments and Performance Evaluation
Furthermore, the data in Tables 9 and 10 show that increasing the value of \(\gamma\) from 0.1 to 1.0 leads to notable improvements in convergence rate, stability, and consistency compared with the EF\({}^{[1]}\) and EF\({}^{[2]}\) methods. Table 11 and Figure 9 present the overall performance evaluation. The results indicate that when \(\gamma\approx 1\), the proposed method demonstrates improved behavior under random perturbations compared with existing approaches. Specifically,

* The number of iterations required by the proposed scheme decreases significantly. * Higher relative accuracy is achieved, as evidenced by a lower residual error of the fractional scheme. * The method attains a higher percentage of convergence while requiring less CPU time and memory. \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 & 80.7 \\ EF\({}^{[2]}\) & 87.54 & 9.1 & 67.98 & 19.2 & 47.35 & 54.8 & 29.43 & 86.2 \\ 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 10: Memory usage, percentage convergence, and stability analysis of fractional schemes for solving Equation (48) with varying values of \(\gamma\). Figure 9: Convergence iterations, CPU time, memory usage, percentage convergence, and residual error analysis of fractional-order schemes FS\({}^{[1]}-\)FS\({}^{[5]}\) and EF\({}^{[1]}-\)EF\({}^{[2]}\), and a random initialization strategy for solving Equation (48) with different values of \(\gamma\). ### Drug Dose-Response Model [32]

A drug dose-response model describes how a biological system responds to varying concentrations of a drug. Key parameters in this model include the maximum effect (\(E_{max}\)), the half-maximal effective concentration (\(C_{a}^{n}\)), and the exponent (\(n\)), which reflects the steepness of the dose-response curve. The model captures how the pharmacological effect increases with dosage until a saturation point is reached. Solving the corresponding nonlinear equation allows for the determination of the drug concentration required to achieve a target biological response. The dose-response relationship can be modeled using the following nonlinear equation:

\[R(x)=\frac{E_{max}x^{n}}{C_{a}^{n}+x^{n}}, \tag{50}\]

where \(R(x)\) is the observed response, and the objective is to determine the drug concentration \(x\) that produces this response. Rearranging the equation yields the following nonlinear form:

\[f(x)=\frac{E_{max}x^{n}}{C_{a}^{n}+x^{n}}-R. \tag{51}\]

We consider the following parameter values:

* \(E_{max}=100\) (maximum effect, e.g., % inhibition).