## 2 Fractional-Order Scheme Development, Construction, and Analysis
The update rule is given by  
\[x^{[j+1]}=x^{[j]}-\left(\left(\Gamma(\gamma+1)\frac{\hbar\left(x^{[j]}\right)}{ \left[D_{\gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})}\right)\left[\frac{1}{1- \gamma\frac{\hbar\left(x^{[j]}\right)}{1+\hbar\left(x^{[j]}\right)}}\right] \right)^{1/\gamma}. \tag{7}\]  
This method achieves a convergence order of \(\gamma\in(0,1]\), where \(\gamma\) is the fractional parameter governing the behavior of the iteration. [PERSON] et al. [20] later proposed a modified fractional iterative scheme, denoted by EF\({}^{[1]}\), which achieves a higher convergence order. The iterative update is defined as  
\[\left. \begin{array}{l}y^{[j]}=x^{[j]}-\Gamma(\gamma+1)\frac{\hbar\left(x^{[j] }\right)}{\left[D_{\gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})-\hbar\left(x^{[j ]}\right)},\\ x^{[j+1]}=y^{[j]}-\Gamma(\gamma+1)\frac{\hbar\left(y^{[j]}\right)}{\left[D_{ \gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})-\hbar\left(y^{[j]}\right)},\end{array} \right\} \tag{8}\]  
for \(\alpha\in\mathbb{R}\). In [21], a fractional Caputo-type iterative method, denoted by EF\({}^{[2]}\), was introduced. This method achieves a convergence order of \(2\gamma+1\). The update rule is given by  
\[\left. \begin{array}{l}y^{[j]}=x^{[j]}-\left(\Gamma(\gamma+1)\frac{\hbar\left( x^{[j]}\right)}{\left[D_{\gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})}\right)^{1/ \gamma},\\ x^{[j+1]}=y^{[j]}-\left(\Gamma(\gamma+1)\frac{\hbar\left(y^{[j]}\right)}{ \left[D_{\gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})}\right)^{1/\gamma}.\end{array} \right\} \tag{9}\]  
Here, we extend this approach by introducing a fractional-order variant that employs the Caputo-type fractional derivative. The update rule of the proposed scheme is given by  
\[\left. \begin{array}{l}y^{[j]}=x^{[j]}-\left(\Gamma(\gamma+1)\frac{\hbar\left( x^{[j]}\right)}{\left[D_{\gamma_{1}}^{\gamma}\right]\hbar(x^{[j]})}\right)^{1/ \gamma}.\\ x^{[j+1]}=y^{[j]}-\left(\Gamma(\gamma+1)\frac{\phi(\varsigma)}{\hbar\left(y ^{[j]}\right)}{1-2\frac{\hbar\left(y^{[j]}\right)}{\hbar\left(x^{[j]}\right) }}\right)^{1/\gamma},\end{array} \right\} \tag{10}\]  
where \(\varsigma=\frac{\hbar\left(y^{[j]}\right)}{\left[D_{\gamma_{1}}^{\gamma}\right] \hbar\left(x^{[j]}\right)}\). This formulation preserves the structure of the conventional scheme while enhancing its stability through fractional-order dynamics. For ease of reference in the following sections, we denote this fractional generalization by the MS\({}^{[\gamma]}\) scheme. In what follows, we analyze the convergence behavior of the method under the assumptions that \(\gamma\in(0,1]\) and \(\hbar(\cdot)\) is sufficiently smooth. Theorem 1.1: _Let_  
\[\hbar:\Gamma\subseteq\mathbb{R}\rightarrow\mathbb{R} \tag{11}\]  
_be a continuous function, where \(\left[D_{\gamma_{1}}^{m_{\gamma}}\right]\hbar(x)\) is defined and of order \(m\) for any \(\gamma\geq 0\) and \(\gamma\in(0,1]\). Let \(\xi\) denote the exact root of \(\hbar(x)\), i.e., \(\hbar(\xi)=0\)._Assume that the initial guess \(x^{[0]}\) is sufficiently close to \(\xi\) and that the function \(\phi\) satisfies \(\phi(0)=0\), \(\phi^{\prime}(0)=1\), and \(\phi^{\prime\prime}(0)<\infty\). Then, the iterative scheme  
\[x^{[j+1]}=y^{[j]}-\left(\Gamma(\gamma+1)\frac{\phi(\zeta)}{1-2\frac{\hbar(y^{[ j]})}{\hbar(x^{[j]})}}\right)^{1/\gamma}, \tag{12}\]  
converges with order at least \(2\gamma+1\). The corresponding error expression is given by  
\[e_{x}^{[j]}=\left( \begin{array}{c}3\frac{(2^{\gamma})^{2}\Gamma(\gamma+1/2)c _{2}c_{2}}{\Gamma(\gamma)\gamma\sqrt{\pi}}-\\ \frac{(2^{\gamma})^{4}c_{2}}{\gamma^{2}\pi\left(\Gamma(\gamma)\right)^{2}} \Big{(}\Gamma\Big{(}\gamma+\frac{1}{2}\Big{)}\Big{)}^{2}-2\,c_{2}\end{array}  \right)\Big{[}\varepsilon^{[j]}\Big{]}^{2\gamma+1}+O\bigg{(}\Big{[} \varepsilon^{[j]}\Big{]}^{3\gamma+1}\bigg{)}, \tag{13}\]  
where \(c_{\kappa}=\frac{\Gamma(\gamma+1)}{\Gamma(\kappa\gamma+1)}\frac{\left[D_{ \xi}^{\gamma}\right]\hbar(\xi)}{\left[D_{\xi}^{\gamma}\right]\hbar(\xi)}, \kappa\geq 2\). Proof.