## 4 Numerical Experiments and Performance Evaluation

Table 14: Error and consistency analysis of fractional schemes for solving Equation (52). Figure 11: Convergence iterations, CPU time, memory usage, percentage convergence, and residual error of fractional-order approaches for solving Equation (52) with different \(\gamma\) values. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric** & **Iterations** & **C-Time** & **M-Usage** & **P-Con** & **Error** \\ \hline \multicolumn{5}{c}{\(\gamma=0.5\)} \\ Random & 67 & 7.876 & 87.56 & 7.73\% & \(8.5\times 10^{-2}\) \\ EF\({}^{[1]}\) & 57 & 6.324 & 77.34 & 9.63\% & \(7.5\times 10^{-1}\) \\ EF\({}^{[2]}\) & 50 & 6.087 & 75.23 & 9.43\% & \(0.5\times 10^{-1}\) \\ 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\% & \(0.3\times 10^{-1}\) \\ \multicolumn{5}{c}{\(\gamma=0.7\)} \\ Random & 65 & 7.632 & 78.87 & 7.73\% & \(5.7\times 10^{-3}\) \\ EF\({}^{[1]}\) & 45 & 6.632 & 54.43 & 9.73\% & \(9.0\times 10^{-3}\) \\ EF\({}^{[2]}\) & 46 & 5654 & 45.13 & 8.55\% & \(0.5\times 10^{-3}\) \\ 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\% & \(0.3\times 10^{-5}\) \\ \multicolumn{5}{c}{\(\gamma=0.9\)} \\ Random & 55 & 7.054 & 87.36 & 9.54\% & \(1.0\times 10^{-3}\) \\ EF\({}^{[1]}\) & 33 & 4.054 & 73.36 & 12.03\% & \(2.1\times 10^{-5}\) \\ EF\({}^{[2]}\) & 37 & 6.655 & 67.43 & 11.32\% & \(8.5\times 10^{-7}\) \\ 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 & 51.6

### Lung Volume and Pressure (Static Compliance Curve) [33]

The lung volume-pressure relationship, commonly described by the static compliance curve, is fundamental in characterizing the mechanical properties of the lungs, particularly their capacity to expand in response to applied pressure. This relationship is governed by the elastic properties of lung tissue and airway resistance and is typically expressed using a nonlinear function such as the arctangent model:

\[f(x)=A\arctan(B(x-C))-D, \tag{54}\]

where \(x\) denotes the applied pressure, and \(A,B,C,D\in\mathbb{R}\). The curve generally displays saturation behavior at higher pressures, where additional increases in pressure result in minimal changes in lung volume. This nonlinearity is essential to accurately modeling both healthy and pathological lung conditions, including chronic obstructive pulmonary disease (COPD) and pulmonary fibrosis. It also plays a crucial role in the design and optimization of mechanical ventilation systems. For steady-state analysis, the model can be approximated by the following nonlinear function by setting the parameters \(A=0.1\), \(B=0.3\), \(C=0.8\), and \(D=0.9\):

\[\hbar(x)=0.1\arctan(0.3(x-0.8))-0.9. \tag{55}\]

The Caputo-order fractional derivative of this function is expressed as

\[\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\hbar(x)=\big{[}D_{\gamma_{1}}^{\gamma} \big{]}\hbar(x)=\frac{1}{\Gamma(\gamma+1)}\int\limits_{0}^{x}\bigg{[}\frac{0.0 3}{1+0.09(t-0.8)^{2}}\frac{1}{(x-t)^{\gamma}}\bigg{]}dt. \tag{56}\]

The exact root of this nonlinear model is \(\xi=0.7077\), with an initial starting value of \(x^{[0]}=-1.3\). Tables 17-19 present the numerical results obtained from fractional schemes, both with and without the chaos-optimized parameter, used to solve Equation (55).