Search

Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q (z) = z p + log c t for all z ∈ ℂ , where p ∈ N ∖ { 1 } , t ∈ [ 1 , ∞) , c ∈ ℂ ∖ { 0 } , using a four-step iteration scheme extended with s -convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments. [ABSTRACT FROM AUTHOR]

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 F (z) = z p − q z 2 + r z + sin c w for all z ∈ ℂ , where p ∈ N ∖ { 1 } , q ∈ ℂ , r , c ∈ ℂ ∖ { 0 } and w ∈ [ 1 , ∞). The Mann and Picard–Mann iteration schemes with s -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. [ABSTRACT FROM AUTHOR]