## 2 Fractional-Order Scheme Development, Construction, and Analysis
\begin{array}{c}-\frac{c_{2}c_{3}(3^{7})^{3}\sqrt{3}}{( \Gamma(\gamma))^{7}\gamma^{3}\pi^{2}}\Gamma\Big{(}\ \gamma+\frac{1}{3}\Big{)}\ \Gamma\left(\ \gamma+\frac{2}{3}\right)\\ +\frac{c_{2}(3^{7})^{3}\sqrt{3}}{2(\Gamma(\gamma)\gamma^{2}\gamma^{2}\sqrt{ \pi}(2\gamma)^{2}}\frac{\Gamma\left(\ \gamma+\frac{1}{3}\right)}{\Gamma\left(\ \gamma+\frac{2}{3}\right)}\\ +\frac{c_{3}^{3}}{\Gamma(\gamma)\gamma}-2\ \frac{c_{3}^{3}(2\gamma)^{2} \Gamma(\gamma+1/2)}{(\Gamma(\gamma))^{2}\gamma^{2}\sqrt{\pi}}-\\ \frac{c_{3}^{3}(2\gamma)^{4}}{(\Gamma(\gamma))^{3}\gamma\pi}\Big{(}\ \Gamma\left(\ \gamma+\frac{1}{3}\right)\Big{)}\ 2+\\ 2\ \frac{c_{3}^{3}(2\gamma)^{2}}{(\Gamma(\gamma)\gamma)^{4}\pi^{3/2}}+\frac{c_{ 2}(2\gamma)^{2}c_{3}}{(\Gamma(\gamma))^{2}\gamma^{2}\sqrt{\pi}}\Gamma\Big{(} \ \gamma+\frac{1}{2}\Big{)}\end{array} \right).\]  
The function \(\phi(\xi)\) can be expanded in a Taylor series as  
\[\phi(\xi)=\phi(0)+\phi^{\prime}(0)\xi+\frac{\phi^{\prime\prime}(0)}{2!}\xi^{2 }+\cdots. \tag{22}\]  
Substituting the expansion of \(\xi\) into (22) yields  
\[\phi(\xi)=\left( \begin{array}{c}\phi(0)+\left(-\frac{\phi^{\prime}(0)c_{2}}{ \Gamma(\gamma)\gamma}+\frac{\phi^{\prime}(0)(2\gamma^{2})^{2}c_{2}}{(\Gamma( \gamma))^{7}2^{7}\sqrt{\pi}}\Gamma\Big{(}\gamma+\frac{1}{2}\Big{)}\right) \Big{[}\varepsilon^{[i]}\Big{]}^{\gamma+1}\\ +\Lambda_{17}\Big{[}\varepsilon^{[i]}\Big{]}^{2\gamma+1}+O\Big{(}\Big{[} \varepsilon^{[i]}_{y}\Big{]}^{3\gamma+1}\Big{)}\end{array} \right), \tag{23}\]  
where  
\[\Lambda_{17}=\left( \begin{array}{c}\frac{\phi^{\prime}(0)c_{2}(3\gamma)^{3} \sqrt{3}}{2\left(\Gamma(\gamma)\right)^{7}2^{7}\sqrt{\pi}(2\gamma)^{2}}\frac {\Gamma\left(\gamma+\frac{1}{3}\right)\Gamma\left(\gamma+\frac{1}{3}\right) }{\Gamma\left(\gamma+\frac{1}{3}\right)}\\ -\frac{\phi^{\prime}(0)c_{3}}{\Gamma(\gamma)\gamma}+2\frac{\phi^{\prime}(0)(2 \gamma^{2})^{4}\Gamma(\gamma+1/2)c_{2}}{(\Gamma(\gamma))^{7}2^{7}\sqrt{\pi}} -2\frac{\phi^{\prime}(0)(2\gamma^{2})^{4}\Gamma(\gamma+1/2))^{2}c_{2}^{2}}{( \Gamma(\gamma))^{3}\gamma^{3}\pi}\end{array} \right).\]  
By multiplying (23) by (20), we obtain  
\[\frac{\phi(\xi)}{1-2\frac{\hbar(y^{[i]})}{\hbar(x^{[i]})}}=\left( \begin{array} []{c}\phi(0)+\left(2\ \frac{(2^{7})^{2}\Gamma(\gamma+1/2)c_{2}\hbar(0)}{\Gamma(\gamma)\gamma\sqrt{ \pi}}-2\ c_{2}\hbar(0)\right)\Big{[}\varepsilon^{[i]}\Big{]}^{\gamma+1}\\ +\Lambda_{14}\Big{[}\varepsilon^{[i]}\Big{]}^{2\gamma+1}+O\Big{(}\Big{[} \varepsilon^{[j]}_{y}\Big{]}^{3\gamma+1}\Big{)}\end{array} \right), \tag{24}\]  
where  
\[\Lambda_{14}=\left( \begin{array}{c}2\ \frac{(2^{7})^{4}(\Gamma(\gamma+1/2))^{2}c_{2} ^{2}\phi(0)}{\gamma^{2}\pi\ (\Gamma(\gamma))^{2}}-8\ \frac{(2^{7})^{2}\Gamma(\gamma+1/2)c_{2} ^{2}\phi(0)}{\Gamma(\gamma)\gamma\sqrt{\pi}}\\ +\frac{c_{3}(3^{7})^{3}\sqrt{3}\phi(0)}{\Gamma(\gamma)\gamma\sqrt{\pi}(2\gamma )^{2}}\frac{\Gamma\left(\ \gamma+\frac{1}{3}\right)\Gamma\left(\ \gamma+\frac{2}{3}\right)}{\Gamma\left(\ \gamma+\frac{1}{2}\right)}+6\ c_{2}^{2}\phi(0)\\ -2\ c_{3}\phi(0)-\frac{\phi^{\prime}(0)c_{2}}{\Gamma(\gamma)\gamma}+\frac{\phi^{ \prime}(0)(2\gamma^{2})c_{2}\Gamma\left(\ \gamma+\frac{1}{2}\right)}{(\Gamma(\gamma))^{2}\gamma^{2}\sqrt{\pi}}\end{array}  \right).\]  
By substituting (24) into the second step of (10), we find  
\[x^{[j+1]}-\xi=y^{[j]}-\xi-\left(\Gamma(\gamma+1)\frac{\phi(\xi)}{1-2\frac{\hbar (y^{[i]})}{\hbar(x^{[i]})}}\right)^{\frac{1}{\gamma}}, \tag{25}\]which leads to the following expression for the error:  
\[e_{x}^{[i]}=\left( \begin{array}{c}-\Gamma(\gamma)\gamma\phi(0)+\left(-2\,\frac{(2 ^{7})^{2}\Gamma(\gamma+1/2)c_{2}\phi(0)}{\sqrt{\pi}}+2\,\Gamma(\gamma)\gamma\,c _{2}\phi(0)\right)\left[e^{[i]}\right]^{\gamma}\\ \hskip 28.452756 pt+\Lambda_{15}\left[e^{[i]}\right]^{\gamma+1}+O\left(\left[e _{y}^{[j]}\right]^{2\gamma+1}\right)\end{array} \right), \tag{26}\]  
where  
\[\Lambda_{15}=\left(