## 2 Fractional-Order Scheme Development, Construction, and Analysis  
Fractional calculus--particularly the Caputo derivative--provides a solid foundation for modeling systems with memory and hereditary characteristics, which are common in fields such as biology, control theory, and materials science. In contrast to classical derivatives, the Caputo definition allows for the use of initial conditions in the same form as those for integer-order models, thereby increasing its practical relevance and interpretability in engineering and biomedical contexts. This feature often makes it preferable to alternative definitions, such as the Riemann-Liouville derivative, especially when solving fractional differential equations. In this section, we briefly review essential concepts from fractional calculus and examine existing fractional-order iterative schemes for solving Equation (1). **Definition 1** (Caputo Fractional Derivative). : _Let_  
\[\hbar:\mathbb{R}\to\mathbb{R},\quad\hbar\in\mathbb{C}^{+\infty}([\gamma,x]), \quad-\infty<\gamma<x<+\infty,\quad\gamma\geq 0, \tag{2}\]  
_where \(m=\lfloor\gamma\rfloor+1\)._  
_To ensure sufficient smoothness of the function, the Caputo derivative of order \(\gamma\in(0,1]\) involves a convolution with a power-law kernel applied to the classical derivative._  
_The Caputo fractional derivative of order \(\gamma\) is defined as follows [15]:_  
\[ \begin{split}&\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\hbar(x)= \begin{cases}\frac{1}{\Gamma(m-\gamma)}\int\limits_{0}^{\tau}\!\!\left[\frac{ d^{m}}{dt^{m}}\hbar(t)\cdot\frac{1}{(x-t)^{\gamma-m+1}}\right]dt,&\gamma\
otin \mathbb{N},\\ &\frac{d^{m-1}}{dt^{m-1}}\hbar(x),\gamma=m-1\in\mathbb{N}\cup\{0\}. \end{cases}\end{split}  \tag{3}\]  
_Here, \(\Gamma(x)\) denotes the Gamma function [16], which is defined as_  
\[\Gamma(x)=\int\limits_{0}^{+\infty}u^{x-1}e^{-u}\,du,\quad x>0. \tag{4}\]  
The Taylor series theorem provides a local approximation of a function near its root and serves as the theoretical basis for many iterative methods used to solve nonlinear equations. By expanding a function into its Taylor series and incorporating higher-order terms, one can derive more accurate and efficient iterative schemes. In this study, a generalized Taylor expansion based on fractional derivatives is employed to construct high-order iterative methods with enhanced convergence properties. Suppose that \(\left[D_{\gamma_{1}}^{\theta\gamma}\right]\hbar(x)\in C([\gamma_{1},\gamma_{2}])\) for \(\theta=1,\ldots,n+1\), where \(\gamma\in(0,1]\). Then, the generalized Taylor formula [17] is given by  
\[\hbar(x)=\left[\sum\limits_{i=0}^{n}\!\left[D_{\gamma_{1}}^{i\gamma}\right] \hbar(\gamma_{1})\frac{(x-\gamma_{1})^{i\gamma}}{\Gamma(i\gamma+1)}+\left[D_{ \gamma_{1}}^{(n+1)\gamma}\right]\hbar(\xi)\frac{(x-\gamma_{1})^{(n+1)\gamma}} {\Gamma((n+1)\gamma+1)}\right], \tag{5}\]  
where  
\[\gamma_{1}\leq\xi\leq x,\quad\forall x\in(\gamma_{1},\gamma_{2}],\]  
\[\big{[}D_{\gamma_{1}}^{\eta\gamma}\big{]}=\big{[}\big{[}D_{\gamma_{1}}^{ \gamma}\big{]}\cdot\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\cdot\big{[}D_{\gamma _{1}}^{\gamma}\big{]}\ (n\text{-times})\big{]},\]  
and \(\left[\Gamma(\cdot)\right]^{n}=\Gamma^{n}(\cdot)\). [PERSON] et al. [18] proposed a Caputo-type modification of [PERSON]'s method:  
\[x^{[j+1]}=x^{[j]}-\Bigg{(}\Gamma(\gamma+1)\frac{\hbar(x^{[j]})}{\big{[}D_{ \gamma_{1}}^{\gamma}\big{]}\hbar(x^{[j]})}\Bigg{)}^{1/\gamma}, \tag{6}\]  
where \(\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\hbar(x_{i})\approx\left[D_{\xi}^{\gamma }\right]\hbar(\xi)\). For any \(\gamma\in\mathbb{R}\), the method exhibits a convergence order of \(\gamma+1\). [PERSON] et al. [19] introduced a one-step fractional iterative method that incorporates the Caputo derivative to account for memory effects in fractional-order systems.