## 4 Numerical Experiments and Performance Evaluation  
The performance of the proposed two-step fractional-order iterative method is evaluated on a set of nonlinear problems arising in engineering contexts. A comparative analysis is provided, including error estimates, computational time, and benchmarks against existing iterative methods. Key evaluation criteria include the stopping condition, convergence behavior, and computational efficiency. The proposed method is compared with conventional fractional iterative methods, such as EF\({}^{[1]}\) and EF\({}^{[2]}\), implemented in MATLAB 2016 using 64-digit floating-point arithmetic. The primary metrics used for assessing algorithmic performance are outlined below:  
1. **Stopping Criterion:** The algorithm terminates when the absolute difference between successive iterates satisfies the following condition: \[\epsilon=\left|x^{[j+1]}-x^{[j]}\right|<tol,\] (42) where \(\epsilon\) is the error tolerance, typically set to \(10^{-18}\). 2. **CPU Time:** The total computational time required for convergence is used as a measure of efficiency. Time is recorded using MATLAB's tic and toc functions. 3. **Error Analysis:** The following error metrics are considered: * **Percentage Error:** The relative error between two consecutive approximations is given by \[\text{Percentage Error}=\frac{\left|x^{[j+1]}-x^{[j]}\right|}{\left|x^{[j]} \right|}\times 100.\] (43)  
Figure 4: (**a–e**) Dynamical planes illustrating unstable convergence behavior of the fractional-order scheme. Convergence Rate: The ratio of errors between successive iterations is used to assess the rate at which convergence occurs. This is computed using Algorithms 1 and 2 and illustrated in the flowchart in Figure 5.
4. **Comparison with Existing Methods:** The proposed method is evaluated against existing classical techniques, such as the EF\({}^{[1]}\) and EF\({}^{[2]}\) methods. The comparison focuses on the following: * **Convergence Speed:** How quickly the method approaches the solution. * **Accuracy:** The degree to which the computed solution approximates the exact one. * **Computational Efficiency:** The computational resources required to achieve convergence. Figure 5: Flowchart of fractional-order scheme for solving nonlinear biomedical engineering models. ```
Input: Grid size \(N\gets 500\), fractional order \(\gamma\in(0,1)\) Output: Phase plot \(\angle A\) via pcolor(x, y, angle(A)) Step 1: Initialize the Complex Grid ;  Define \(x,y\leftarrow\texttt{linspace}(-2,2,N)\) ;  Construct meshgrid \([X,Y]\leftarrow\texttt{meshgrid}(x,y)\) ;  Form complex grid \(A\gets X+iY\) ; Step 2: Define Function and Fractional Derivative ;  Set \(\hbar(x)\gets 0.5x-\frac{0.9x}{1/\gamma}-0.3\ln(x+1)\) ;  Set \([D_{\tau n}^{\gamma}]\hbar(x)\leftarrow\frac{1}{\Gamma(\gamma+1)}\int_{0}^{x} \left[\frac{-0.0294-\frac{\phi}{12\gamma}}{(x-i)^{\gamma}}\right]dt\) ; Step 3: Iterate Using Fractional Newton-Type Method ; for\(k\gets 1\) to 20 do  Apply iteration formulas: \[y^{[j]} =x^{[j]}-\left(\Gamma(\gamma+1)\cdot\frac{\hbar(x^{[j]})}{[D_{ \tau n}^{\gamma}]\,\hbar(x^{[j]})}\right)^{1/\gamma},\] \[x^{[j+1]} =y^{[j]}-\left(\Gamma(\gamma+1)\cdot\frac{\phi(\zeta)}{1-2\cdot \frac{\hbar(y^{[j]})}{\hbar(x^{[j]})}}\right)^{1/\gamma},\]  where \(\alpha\in\mathbb{R}\).