In recent papers, some fractional Newton-type methods have been proposed by using the Riemann–Liouville and Caputo fractional derivatives in their iterative schemes, with order $2\alpha$ or $1+\alpha$. In this manuscript, we introduce the Conformable fractional Newton-type method by using the so-called fractional derivative. The convergence analysis is made, proving its quadratic convergence, and the numerical results confirm the theory and improve the results obtained by classical Newton’s method. Unlike previous fractional Newton-type methods, this one involves a low computational cost, and the order of convergence is at least quadratic.

中文翻译：

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一种求解非线性方程的最优且低计算成本的分数牛顿型方法

在最近的论文中，通过在迭代方案中使用 Riemann-Liouville 和 Caputo 分数阶导数，提出了一些分数牛顿型方法，其阶数为 $2\alpha$ 或者 $1+\alpha$. 在这份手稿中，我们通过使用所谓的分数阶导数介绍了 Conformable 分数阶牛顿型方法。进行了收敛性分析，证明了其二次收敛性，数值结果证实了理论，改进了经典牛顿法得到的结果。与之前的分数牛顿型方法不同，这种方法计算成本低，收敛阶数至少是二次的。