This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form for all , where , , and . The Mann and Picard–Mann iteration schemes with -convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.

中文翻译：

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函数 zp−qz2+rz+sincw 的 Julia 和 Mandelbrot 集展示了 Mann 和 Picard-Mann 轨道以及 s 凸性

这篇研究论文介绍了一种通过采用迭代技术来可视化 Julia 和 Mandelbrot 集的新颖方法，这在创建分形中发挥着至关重要的作用。主要关注点是 for all 、 where 、 和 形式的复杂函数。整个研究中都使用了带有-凸性的 Mann 和 Picard-Mann 迭代方案。开发了创新的逃逸标准，以使用这些迭代方法生成 Julia 和 Mandelbrot 集。这些标准可作为确定迭代过程何时终止的指南，从而创建迷人的分形图案。该研究调查了迭代方案中参数变化对所得分形的形状、大小和颜色的影响。通过操纵这些参数，可以生成和可视化各种迷人的分形图案，涵盖各种美学可能性。此外，我们还讨论了与通过提议的迭代生成的 Julia 和 Mandelbrot 集相关的数值示例。我们还深入讨论了有关执行时间和平均迭代次数的讨论。