The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomial z³ − 1 on a complex plane.

本文的目的是直观地研究标准牛顿寻根方法中经典衍生物被分数Riemann-Liouville或Caputo衍生物代替的过程的动力学和稳定性。此外，代替标准的Picard迭代，使用了Mann，Khan，Ishikawa和S迭代。当将此过程应用于复杂平面上的多项式时，将生成图像，该图像显示出多项式零的吸引盆，或者表示获得任何多项式根所需的迭代次数的图像。这些图像称为多面体检查仪。在本文中，我们根据迭代次数使用了着色，从而揭示了由多项式扫描仪可视化的过程的收敛速度和动态特性。此外，为了研究方法的稳定性，我们使用吸引盆地。为了数值比较其中的修改后的求根方法，我们证明了它们对多项式的作用z ³ − 1在复平面上。